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a/_weave/lecture02/optimizing/index.html +++ b/_weave/lecture02/optimizing/index.html @@ -10,7 +10,7 @@

Optimizing Serial Code

Chris Rackauckas
Septe end @btime inner_rows!(C,A,B) 
-7.654 μs (0 allocations: 0 bytes)
+7.587 μs (0 allocations: 0 bytes)
 
 function inner_cols!(C,A,B)
   for j in 1:100, i in 1:100
@@ -19,7 +19,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime inner_cols!(C,A,B)
 
-2.414 μs (0 allocations: 0 bytes)
+2.386 μs (0 allocations: 0 bytes)

Lower Level View: The Stack and the Heap

Locally, the stack is composed of a stack and a heap. The stack requires a static allocation: it is ordered. Because it's ordered, it is very clear where things are in the stack, and therefore accesses are very quick (think instantaneous). However, because this is static, it requires that the size of the variables is known at compile time (to determine all of the variable locations). Since that is not possible with all variables, there exists the heap. The heap is essentially a stack of pointers to objects in memory. When heap variables are needed, their values are pulled up the cache chain and accessed.

Heap Allocations and Speed

Heap allocations are costly because they involve this pointer indirection, so stack allocation should be done when sensible (it's not helpful for really large arrays, but for small values like scalars it's essential!)

 function inner_alloc!(C,A,B)
   for j in 1:100, i in 1:100
@@ -29,7 +29,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime inner_alloc!(C,A,B)
 
-234.700 μs (10000 allocations: 625.00 KiB)
+234.498 μs (10000 allocations: 625.00 KiB)
 
 function inner_noalloc!(C,A,B)
   for j in 1:100, i in 1:100
@@ -39,7 +39,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime inner_noalloc!(C,A,B)
 
-2.413 μs (0 allocations: 0 bytes)
+2.386 μs (0 allocations: 0 bytes)

Why does the array here get heap-allocated? It isn't able to prove/guarantee at compile-time that the array's size will always be a given value, and thus it allocates it to the heap. @btime tells us this allocation occurred and shows us the total heap memory that was taken. Meanwhile, the size of a Float64 number is known at compile-time (64-bits), and so this is stored onto the stack and given a specific location that the compiler will be able to directly address.

Note that one can use the StaticArrays.jl library to get statically-sized arrays and thus arrays which are stack-allocated:

 using StaticArrays
 function static_inner_alloc!(C,A,B)
@@ -50,7 +50,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime static_inner_alloc!(C,A,B)
 
-2.411 μs (0 allocations: 0 bytes)
+2.388 μs (0 allocations: 0 bytes)

Mutation to Avoid Heap Allocations

Many times you do need to write into an array, so how can you write into an array without performing a heap allocation? The answer is mutation. Mutation is changing the values of an already existing array. In that case, no free memory has to be found to put the array (and no memory has to be freed by the garbage collector).

In Julia, functions which mutate the first value are conventionally noted by a !. See the difference between these two equivalent functions:

 function inner_noalloc!(C,A,B)
   for j in 1:100, i in 1:100
@@ -60,7 +60,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime inner_noalloc!(C,A,B)
 
-2.413 μs (0 allocations: 0 bytes)
+2.385 μs (0 allocations: 0 bytes)
 
 function inner_alloc(A,B)
   C = similar(A)
@@ -71,7 +71,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime inner_alloc(A,B)
 
-6.807 μs (2 allocations: 78.17 KiB)
+6.803 μs (2 allocations: 78.17 KiB)

To use this algorithm effectively, the ! algorithm assumes that the caller already has allocated the output array to put as the output argument. If that is not true, then one would need to manually allocate. The goal of that interface is to give the caller control over the allocations to allow them to manually reduce the total number of heap allocations and thus increase the speed.

Julia's Broadcasting Mechanism

Wouldn't it be nice to not have to write the loop there? In many high level languages this is simply called vectorization. In Julia, we will call it array vectorization to distinguish it from the SIMD vectorization which is common in lower level languages like C, Fortran, and Julia.

In Julia, if you use . on an operator it will transform it to the broadcasted form. Broadcast is lazy: it will build up an entire .'d expression and then call broadcast! on composed expression. This is customizable and documented in detail. However, to a first approximation we can think of the broadcast mechanism as a mechanism for building fused expressions. For example, the Julia code:

 A .+ B .+ C;

under the hood lowers to something like:

@@ -86,29 +86,29 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime unfused(A,B,C);
 
-6.622 μs (4 allocations: 156.34 KiB)
+9.294 μs (4 allocations: 156.34 KiB)
 
 fused(A,B,C) = A .+ B .+ C
 @btime fused(A,B,C);
 
-4.630 μs (2 allocations: 78.17 KiB)
+4.433 μs (2 allocations: 78.17 KiB)

Note that we can also fuse the output by using .=. This is essentially the vectorized version of a ! function:

 D = similar(A)
 fused!(D,A,B,C) = (D .= A .+ B .+ C)
 @btime fused!(D,A,B,C);
 
-3.436 μs (0 allocations: 0 bytes)
+3.441 μs (0 allocations: 0 bytes)

Note on Broadcasting Function Calls

Julia allows for broadcasting the call () operator as well. .() will call the function element-wise on all arguments, so sin.(A) will be the elementwise sine function. This will fuse Julia like the other operators.

Note on Vectorization and Speed

In articles on MATLAB, Python, R, etc., this is where you will be told to vectorize your code. Notice from above that this isn't a performance difference between writing loops and using vectorized broadcasts. This is not abnormal! The reason why you are told to vectorize code in these other languages is because they have a high per-operation overhead (which will be discussed further down). This means that every call, like +, is costly in these languages. To get around this issue and make the language usable, someone wrote and compiled the loop for the C/Fortran function that does the broadcasted form (see numpy's Github repo). Thus A .+ B's MATLAB/Python/R equivalents are calling a single C function to generally avoid the cost of function calls and thus are faster.

But this is not an intrinsic property of vectorization. Vectorization isn't "fast" in these languages, it's just close to the correct speed. The reason vectorization is recommended is because looping is slow in these languages. Because looping isn't slow in Julia (or C, C++, Fortran, etc.), loops and vectorization generally have the same speed. So use the one that works best for your code without a care about performance.

(As a small side effect, these high level languages tend to allocate a lot of temporary variables since the individual C kernels are written for specific numbers of inputs and thus don't naturally fuse. Julia's broadcast mechanism is just generating and JIT compiling Julia functions on the fly, and thus it can accommodate the combinatorial explosion in the amount of choices just by only compiling the combinations that are necessary for a specific code)

Heap Allocations from Slicing

It's important to note that slices in Julia produce copies instead of views. Thus for example:

 A[50,50]
 
-0.17778218519466293
+0.4378374168708473

allocates a new output. This is for safety, since if it pointed to the same array then writing to it would change the original array. We can demonstrate this by asking for a view instead of a copy.

 @show A[1]
 E = @view A[1:5,1:5]
 E[1] = 2.0
 @show A[1]
 
-A[1] = 0.5530951950733606
+A[1] = 0.37233258859606955
 A[1] = 2.0
 2.0

However, this means that @view A[1:5,1:5] did not allocate an array (it does allocate a pointer if the escape analysis is unable to prove that it can be elided. This means that in small loops there will be no allocation, while if the view is returned from a function for example it will allocate the pointer, ~80 bytes, but not the memory of the array. This means that it is O(1) in cost but with a relatively small constant).

Asymptotic Cost of Heap Allocations

Heap allocations have to locate and prepare a space in RAM that is proportional to the amount of memory that is calculated, which means that the cost of a heap allocation for an array is O(n), with a large constant. As RAM begins to fill up, this cost dramatically increases. If you run out of RAM, your computer may begin to use swap, which is essentially RAM simulated on your hard drive. Generally when you hit swap your performance is so dead that you may think that your computation froze, but if you check your resource use you will notice that it's actually just filled the RAM and starting to use the swap.

But think of it as O(n) with a large constant factor. This means that for operations which only touch the data once, heap allocations can dominate the computational cost:

@@ -130,7 +130,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 plot(ns,alloc,label="=",xscale=:log10,yscale=:log10,legend=:bottomright,
      title="Micro-optimizations matter for BLAS1")
 plot!(ns,noalloc,label=".=")
-

However, when the computation takes O(n^3), like in matrix multiplications, the high constant factor only comes into play when the matrices are sufficiently small:

+

However, when the computation takes O(n^3), like in matrix multiplications, the high constant factor only comes into play when the matrices are sufficiently small:

 using LinearAlgebra, BenchmarkTools
 function alloc_timer(n)
     A = rand(n,n)
@@ -149,7 +149,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 plot(ns,alloc,label="*",xscale=:log10,yscale=:log10,legend=:bottomright,
      title="Micro-optimizations only matter for small matmuls")
 plot!(ns,noalloc,label="mul!")
-

Though using a mutating form is never bad and always is a little bit better.

Optimizing Memory Use Summary

  • Avoid cache misses by reusing values

  • Iterate along columns

  • Avoid heap allocations in inner loops

  • Heap allocations occur when the size of things is not proven at compile-time

  • Use fused broadcasts (with mutated outputs) to avoid heap allocations

  • Array vectorization confers no special benefit in Julia because Julia loops are as fast as C or Fortran

  • Use views instead of slices when applicable

  • Avoiding heap allocations is most necessary for O(n) algorithms or algorithms with small arrays

  • Use StaticArrays.jl to avoid heap allocations of small arrays in inner loops

Julia's Type Inference and the Compiler

Many people think Julia is fast because it is JIT compiled. That is simply not true (we've already shown examples where Julia code isn't fast, but it's always JIT compiled!). Instead, the reason why Julia is fast is because the combination of two ideas:

  • Type inference

  • Type specialization in functions

These two features naturally give rise to Julia's core design feature: multiple dispatch. Let's break down these pieces.

Type Inference

At the core level of the computer, everything has a type. Some languages are more explicit about said types, while others try to hide the types from the user. A type tells the compiler how to to store and interpret the memory of a value. For example, if the compiled code knows that the value in the register is supposed to be interpreted as a 64-bit floating point number, then it understands that slab of memory like:

Importantly, it will know what to do for function calls. If the code tells it to add two floating point numbers, it will send them as inputs to the Floating Point Unit (FPU) which will give the output.

If the types are not known, then... ? So one cannot actually compute until the types are known, since otherwise it's impossible to interpret the memory. In languages like C, the programmer has to declare the types of variables in the program:

void add(double *a, double *b, double *c, size_t n){
+

Though using a mutating form is never bad and always is a little bit better.

Optimizing Memory Use Summary

  • Avoid cache misses by reusing values

  • Iterate along columns

  • Avoid heap allocations in inner loops

  • Heap allocations occur when the size of things is not proven at compile-time

  • Use fused broadcasts (with mutated outputs) to avoid heap allocations

  • Array vectorization confers no special benefit in Julia because Julia loops are as fast as C or Fortran

  • Use views instead of slices when applicable

  • Avoiding heap allocations is most necessary for O(n) algorithms or algorithms with small arrays

  • Use StaticArrays.jl to avoid heap allocations of small arrays in inner loops

Julia's Type Inference and the Compiler

Many people think Julia is fast because it is JIT compiled. That is simply not true (we've already shown examples where Julia code isn't fast, but it's always JIT compiled!). Instead, the reason why Julia is fast is because the combination of two ideas:

  • Type inference

  • Type specialization in functions

These two features naturally give rise to Julia's core design feature: multiple dispatch. Let's break down these pieces.

Type Inference

At the core level of the computer, everything has a type. Some languages are more explicit about said types, while others try to hide the types from the user. A type tells the compiler how to to store and interpret the memory of a value. For example, if the compiled code knows that the value in the register is supposed to be interpreted as a 64-bit floating point number, then it understands that slab of memory like:

Importantly, it will know what to do for function calls. If the code tells it to add two floating point numbers, it will send them as inputs to the Floating Point Unit (FPU) which will give the output.

If the types are not known, then... ? So one cannot actually compute until the types are known, since otherwise it's impossible to interpret the memory. In languages like C, the programmer has to declare the types of variables in the program:

void add(double *a, double *b, double *c, size_t n){
   size_t i;
   for(i = 0; i < n; ++i) {
     c[i] = a[i] + b[i];
@@ -172,7 +172,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `f`
-define i64 @julia_f_3082(i64 signext %0, i64 signext %1) #0 {
+define i64 @julia_f_3070(i64 signext %0, i64 signext %1) #0 {
 top:
 ; ┌ @ int.jl:87 within `+`
    %2 = add i64 %1, %0
@@ -184,7 +184,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `f`
-define double @julia_f_3084(double %0, double %1) #0 {
+define double @julia_f_3072(double %0, double %1) #0 {
 top:
 ; ┌ @ float.jl:409 within `+`
    %2 = fadd double %0, %1
@@ -204,7 +204,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define i64 @julia_g_3086(i64 signext %0, i64 signext %1) #0 {
+define i64 @julia_g_3074(i64 signext %0, i64 signext %1) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 6 within `g`
@@ -249,7 +249,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `f`
-define double @julia_f_3545(double %0, i64 signext %1) #0 {
+define double @julia_f_3533(double %0, i64 signext %1) #0 {
 top:
 ; ┌ @ promotion.jl:422 within `+`
 ; │┌ @ promotion.jl:393 within `promote`
@@ -290,7 +290,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define double @julia_g_3548(double %0, i64 signext %1) #0 {
+define double @julia_g_3536(double %0, i64 signext %1) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 5 within `g`
@@ -381,7 +381,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `ff`
-define double @julia_ff_3667(double %0, i64 signext %1) #0 {
+define double @julia_ff_3655(double %0, i64 signext %1) #0 {
 top:
 ; ┌ @ promotion.jl:422 within `+`
 ; │┌ @ promotion.jl:393 within `promote`
@@ -537,7 +537,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define void @julia_g_3806([2 x double]* noalias nocapture noundef nonnull s
+define void @julia_g_3794([2 x double]* noalias nocapture noundef nonnull s
 ret([2 x double]) align 8 dereferenceable(16) %0, [2 x double]* nocapture n
 oundef nonnull readonly align 8 dereferenceable(16) %1, [2 x double]* nocap
 ture noundef nonnull readonly align 8 dereferenceable(16) %2) #0 {
@@ -657,7 +657,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define [2 x float] @julia_g_3818([2 x float]* nocapture noundef nonnull rea
+define [2 x float] @julia_g_3806([2 x float]* nocapture noundef nonnull rea
 donly align 4 dereferenceable(8) %0, [2 x float]* nocapture noundef nonnull
  readonly align 4 dereferenceable(8) %1) #0 {
 top:
@@ -763,7 +763,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define void @julia_g_3837([2 x {}*]* noalias nocapture noundef nonnull sret
+define void @julia_g_3825([2 x {}*]* noalias nocapture noundef nonnull sret
 ([2 x {}*]) align 8 dereferenceable(16) %0, [2 x {}*]* nocapture noundef no
 nnull readonly align 8 dereferenceable(16) %1, [2 x {}*]* nocapture noundef
  nonnull readonly align 8 dereferenceable(16) %2) #0 {
@@ -796,25 +796,25 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
    store {}** %14, {}*** %13, align 8
    %15 = bitcast {}*** %tls_pgcstack to {}***
    store {}** %gcframe2.sub, {}*** %15, align 8
-   call void @"j_+_3839"([2 x {}*]* noalias nocapture noundef nonnull sret(
-[2 x {}*]) %9, [2 x {}*]* nocapture nonnull readonly %1, i64 signext 4)
+   call void @"j_+_3827"([2 x {}*]* noalias nocapture noundef nonnull sret(
+[2 x {}*]) %7, [2 x {}*]* nocapture nonnull readonly %1, i64 signext 4)
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 6 within `g`
 ; ┌ @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd
 :2 within `f`
-   call void @"j_+_3840"([2 x {}*]* noalias nocapture noundef nonnull sret(
-[2 x {}*]) %5, i64 signext 2, [2 x {}*]* nocapture readonly %9)
+   call void @"j_+_3828"([2 x {}*]* noalias nocapture noundef nonnull sret(
+[2 x {}*]) %5, i64 signext 2, [2 x {}*]* nocapture readonly %7)
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 7 within `g`
 ; ┌ @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd
 :2 within `f`
-   call void @"j_+_3841"([2 x {}*]* noalias nocapture noundef nonnull sret(
-[2 x {}*]) %7, [2 x {}*]* nocapture readonly %5, [2 x {}*]* nocapture nonnu
+   call void @"j_+_3829"([2 x {}*]* noalias nocapture noundef nonnull sret(
+[2 x {}*]) %9, [2 x {}*]* nocapture readonly %5, [2 x {}*]* nocapture nonnu
 ll readonly %2)
    %16 = bitcast [2 x {}*]* %0 to i8*
-   %17 = bitcast {}** %6 to i8*
+   %17 = bitcast {}** %8 to i8*
    call void @llvm.memcpy.p0i8.p0i8.i64(i8* noundef nonnull align 8 derefer
 enceable(16) %16, i8* noundef nonnull align 16 dereferenceable(16) %17, i64
  16, i1 false)
@@ -844,28 +844,28 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 b = MyComplex(2.0,1.0)
 @btime g(a,b)
 
-20.439 ns (1 allocation: 32 bytes)
+20.388 ns (1 allocation: 32 bytes)
 MyComplex(9.0, 2.0)
 
 a = MyParameterizedComplex(1.0,1.0)
 b = MyParameterizedComplex(2.0,1.0)
 @btime g(a,b)
 
-21.304 ns (1 allocation: 32 bytes)
+21.142 ns (1 allocation: 32 bytes)
 MyParameterizedComplex{Float64}(9.0, 2.0)
 
 a = MySlowComplex(1.0,1.0)
 b = MySlowComplex(2.0,1.0)
 @btime g(a,b)
 
-121.043 ns (5 allocations: 96 bytes)
+113.240 ns (5 allocations: 96 bytes)
 MySlowComplex(9.0, 2.0)
 
 a = MySlowComplex2(1.0,1.0)
 b = MySlowComplex2(2.0,1.0)
 @btime g(a,b)
 
-853.314 ns (14 allocations: 288 bytes)
+519.620 ns (14 allocations: 288 bytes)
 MySlowComplex2(9.0, 2.0)

Note on Julia

Note that, because of these type specialization, value types, etc. properties, the number types, even ones such as Int, Float64, and Complex, are all themselves implemented in pure Julia! Thus even basic pieces can be implemented in Julia with full performance, given one uses the features correctly.

Note on isbits

Note that a type which is mutable struct will not be isbits. This means that mutable structs will be a pointer to a heap allocated object, unless it's shortlived and the compiler can erase its construction. Also, note that isbits compiles down to bit operations from pure Julia, which means that these types can directly compile to GPU kernels through CUDAnative without modification.

Function Barriers

Since functions automatically specialize on their input types in Julia, we can use this to our advantage in order to make an inner loop fully inferred. For example, take the code from above but with a loop:

 function r(x)
@@ -880,7 +880,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime r(x)
 
-5.756 μs (300 allocations: 4.69 KiB)
+5.878 μs (300 allocations: 4.69 KiB)
 604.0

In here, the loop variables are not inferred and thus this is really slow. However, we can force a function call in the middle to end up with specialization and in the inner loop be stable:

 s(x) = _s(x[1],x[2])
@@ -896,7 +896,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime s(x)
 
-299.899 ns (1 allocation: 16 bytes)
+296.107 ns (1 allocation: 16 bytes)
 604.0

Notice that this algorithm still doesn't infer:

 @code_warntype s(x)
@@ -928,7 +928,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `fff`
-define i64 @julia_fff_3931(i64 signext %0) #0 {
+define i64 @julia_fff_3919(i64 signext %0) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 8 within `fff`
@@ -942,7 +942,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `fff`
-define double @julia_fff_3933(double %0) #0 {
+define double @julia_fff_3921(double %0) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 8 within `fff`
@@ -957,7 +957,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
   C[i,j] = A[i,j] + B[i,j]
 end
 
-769.551 μs (30000 allocations: 468.75 KiB)
+792.921 μs (30000 allocations: 468.75 KiB)

This is very slow because the types of A, B, and C cannot be inferred. Why can't they be inferred? Well, at any time in the dynamic REPL scope I can do something like C = "haha now a string!", and thus it cannot specialize on the types currently existing in the REPL (since asynchronous changes could also occur), and therefore it defaults back to doing a type check at every single function which slows it down. Moral of the story, Julia functions are fast but its global scope is too dynamic to be optimized.

Summary

  • Julia is not fast because of its JIT, it's fast because of function specialization and type inference

  • Type stable functions allow inference to fully occur

  • Multiple dispatch works within the function specialization mechanism to create overhead-free compile time controls

  • Julia will specialize the generic functions

  • Making sure values are concretely typed in inner loops is essential for performance

Overheads of Individual Operations

Now let's dig even a little deeper. Everything the processor does has a cost. A great chart to keep in mind is this classic one. A few things should immediately jump out to you:

  • Simple arithmetic, like floating point additions, are super cheap. ~1 clock cycle, or a few nanoseconds.

  • Processors do branch prediction on if statements. If the code goes down the predicted route, the if statement costs ~1-2 clock cycles. If it goes down the wrong route, then it will take ~10-20 clock cycles. This means that predictable branches, like ones with clear patterns or usually the same output, are much cheaper (almost free) than unpredictable branches.

  • Function calls are expensive: 15-60 clock cycles!

  • RAM reads are very expensive, with lower caches less expensive.

Bounds Checking

Let's check the LLVM IR on one of our earlier loops:

 function inner_noalloc!(C,A,B)
   for j in 1:100, i in 1:100
@@ -969,7 +969,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `inner_noalloc!`
-define nonnull {}* @"japi1_inner_noalloc!_3942"({}* %function, {}** noalias
+define nonnull {}* @"japi1_inner_noalloc!_3930"({}* %function, {}** noalias
  nocapture noundef readonly %args, i32 %nargs) #0 {
 top:
   %stackargs = alloca {}**, align 8
@@ -1997,7 +1997,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 6 within `inner_noalloc!`
-  ret {}* inttoptr (i64 139788390752264 to {}*)
+  ret {}* inttoptr (i64 139763090710536 to {}*)
 
 oob:                                              ; preds = %L5.us.us.us.po
 stloop, %L2.split.us.split, %L2.split.us.split.us.split, %L5.us.us.us.postl
@@ -2098,17 +2098,17 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 end
 @btime inner_noalloc!(C,A,B)
 
-2.411 μs (0 allocations: 0 bytes)
+2.386 μs (0 allocations: 0 bytes)
 
 @btime inner_noalloc_ib!(C,A,B)
 
-2.347 μs (0 allocations: 0 bytes)
+2.342 μs (0 allocations: 0 bytes)

SIMD

Now let's inspect the LLVM IR again:

 @code_llvm inner_noalloc_ib!(C,A,B)
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `inner_noalloc_ib!`
-define nonnull {}* @"japi1_inner_noalloc_ib!_3978"({}* %function, {}** noal
+define nonnull {}* @"japi1_inner_noalloc_ib!_3966"({}* %function, {}** noal
 ias nocapture noundef readonly %args, i32 %nargs) #0 {
 top:
   %stackargs = alloca {}**, align 8
@@ -4013,7 +4013,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
   br i1 %.not.not24, label %L36, label %L2
 
 L36:                                              ; preds = %L25
-  ret {}* inttoptr (i64 139788390752264 to {}*)
+  ret {}* inttoptr (i64 139763090710536 to {}*)
 }

If you look closely, you will see things like:

%wide.load24 = load <4 x double>, <4 x double> addrspac(13)* %46, align 8
 ; â””
@@ -4022,7 +4022,7 @@ 
Optimizing Serial Code
 
Chris Rackauckas
 
Septe
 @code_llvm fma(2.0,5.0,3.0)
 
 ;  @ floatfuncs.jl:439 within `fma`
-define double @julia_fma_3979(double %0, double %1, double %2) #0 {
+define double @julia_fma_3967(double %0, double %1, double %2) #0 {
 common.ret:
 ; ┌ @ floatfuncs.jl:434 within `fma_llvm`
    %3 = call double @llvm.fma.f64(double %0, double %1, double %2)
@@ -4070,7 +4070,7 @@ 
Inlining

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 4 within `qinline`
-define double @julia_qinline_3982(double %0, double %1) #0 {
+define double @julia_qinline_3970(double %0, double %1) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 7 within `qinline`
@@ -4107,17 +4107,17 @@ 
Inlining

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 11 within `qnoinline`
-define double @julia_qnoinline_3984(double %0, double %1) #0 {
+define double @julia_qnoinline_3972(double %0, double %1) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 14 within `qnoinline`
-  %2 = call double @j_fnoinline_3986(double %0, i64 signext 4)
+  %2 = call double @j_fnoinline_3974(double %0, i64 signext 4)
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 15 within `qnoinline`
-  %3 = call double @j_fnoinline_3987(i64 signext 2, double %2)
+  %3 = call double @j_fnoinline_3975(i64 signext 2, double %2)
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 16 within `qnoinline`
-  %4 = call double @j_fnoinline_3988(double %3, double %1)
+  %4 = call double @j_fnoinline_3976(double %3, double %1)
   ret double %4
 }
 

@@ -4136,7 +4136,7 @@ 
Inlining

 
 
 
-21.605 ns (1 allocation: 16 bytes)
+20.982 ns (1 allocation: 16 bytes)
 9.0
 

 
@@ -4148,7 +4148,7 @@ 
Inlining

 
 
 
-25.319 ns (1 allocation: 16 bytes)
+25.017 ns (1 allocation: 16 bytes)
 9.0
 

 
@@ -4193,7 +4193,7 @@ 
Note on Benchmarking

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `cheat`
-define double @julia_cheat_4016() #0 {
+define double @julia_cheat_4004() #0 {
 top:
   ret double 9.000000e+00
 }
diff --git a/_weave/lecture03/jl_V4oHyq/sciml_35_1.png b/_weave/lecture03/jl_V4oHyq/sciml_35_1.png
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diff --git a/_weave/lecture03/sciml/index.html b/_weave/lecture03/sciml/index.html
index defb266e..0dc0722a 100644
--- a/_weave/lecture03/sciml/index.html
+++ b/_weave/lecture03/sciml/index.html
@@ -19,11 +19,11 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 simpleNN(rand(10))
 
 
 5-element Vector{Float64}:
-  6.191491803887638
- 10.961350935082876
- -0.086794697902561
-  4.329029452777256
- -0.4819728343992912
+ -1.1205287975184102
+ -6.95758884176028
+  4.900880949623628
+  3.7834125354726122
+  4.191471397681439
 
 
This is our direct definition of a neural network. Notice that we choose to use tanh as our activation function between the layers.
 
Defining Neural Networks with Flux.jl
 
One of the main deep learning libraries in Julia is Flux.jl. Flux is an interesting library for scientific machine learning because it is built on top of language-wide automatic differentiation libraries, giving rise to a programming paradigm known as differentiable programming, which means that one can write a program in a manner that it has easily accessible fast derivatives. However, due to being built on a differentiable programming base, the underlying functionality is simply standard Julia code,
 
To learn how to use the library, consult the documentation. A Google search will bring up the Flux.jl Github repository. From there, the blue link on the README brings you to the package documentation. This is common through Julia so it's a good habit to learn!
 
In the documentation you will find that the way a neural network is defined is through a Chain of layers. A Dense layer is the kind we defined above, which is given by an input size, an output size, and an activation function. For example, the following recreates the neural network that we had above:
 
 using Flux
 NN2 = Chain(Dense(10 => 32,tanh),
@@ -32,11 +32,11 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 NN2(rand(10))
 
 
 5-element Vector{Float32}:
- -0.40341762
- -0.32073504
- -0.60663885
-  0.6273459
-  0.3418348
+ -0.41757214
+  0.3268494
+  0.6551256
+  0.3942203
+  0.5987754
 
 
Notice that Flux.jl as a library is written in pure Julia, which means that every piece of this syntax is just sugar over some Julia code that we can specialize ourselves (this is the advantage of having a language fast enough for the implementation of the library and the use of the library!)
 
For example, the activation function is just a scalar Julia function. If we wanted to replace it by something like the quadratic function, we can just use an anonymous function to define the scalar function we would like to use:
 
 NN3 = Chain(Dense(10 => 32,x->x^2),
             Dense(32 => 32,x->max(0,x)),
@@ -44,11 +44,11 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 NN3(rand(10))
 
 
 5-element Vector{Float32}:
- -0.18383282
- -0.030633058
- -0.021082453
- -0.08232622
- -0.22971418
+ -0.02532567
+ -0.14441662
+ -0.051778786
+ -0.38974416
+ -0.10805663
 
 
The second activation function there is what's known as a relu. A relu can be good to use because it's an exceptionally fast operation and satisfies a form of the universal approximation theorem (UAT). However, a downside is that its derivative is not continuous, which could impact the numerical properties of some algorithms, and thus it's widely used throughout standard machine learning but we'll see reasons why it may be disadvantageous in some cases in scientific machine learning.
 
Digging into the Construction of a Neural Network Library
 
Again, as mentioned before, this neural network NN2 is simply a function:
 
 simpleNN(x) = W[3]*tanh.(W[2]*tanh.(W[1]*x + b[1]) + b[2]) + b[3]
 
 
@@ -96,26 +96,26 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 denselayer_f(rand(32))
 
 
 32-element Vector{Float32}:
-  0.015360817
- -0.46573928
-  0.84118056
-  0.6889729
- -0.04318807
- -0.486555
-  0.7456628
- -0.1116474
- -0.8562411
-  0.14605747
+ -0.5618108
+ -0.6442508
+  0.4827489
+ -0.2672533
+ -0.87355
+ -0.030600643
+  0.333129
+ -0.16809873
+ -0.03701947
+  0.38146716
   â‹®
-  0.07789154
-  0.35291886
-  0.27881867
- -0.68542886
- -0.3935992
-  0.657173
- -0.18403019
-  0.063687466
- -0.15492688
+ -0.39834422
+  0.8219398
+ -0.55398756
+ -0.031825528
+  0.17183039
+  0.72180593
+ -0.7143836
+  0.49238148
+ -0.7401128
 
 
So okay, Dense objects are just functions that have weight and bias matrices inside of them. Now what does Chain do?
 
 @which Chain(1,2,3)
 
 Chain(xs...) in Flux at /home/runner/.julia/packages/Flux/Wz6D4/src/layers/basic.jl:39 
Again, for our explanations here we will look at the slightly simpler code From and earlier version of the Flux package:
 
@@ -169,52 +169,52 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 loss() = sum(abs2,sum(abs2,NN(rand(10)).-1) for i in 1:100)
 loss()
 
 
-4269.761f0
+2630.6902f0
 
 
This loss function takes 100 random points in $[0,1]^{10}$ and then computes the output of the neural network minus 1 on each of the values, and sums up the squared values (abs2). Why the squared values? This means that every computed loss value is positive, and so we know that by decreasing the loss this means that, on average our neural network outputs are closer to 1. What are the weights? Since we're using the Flux callable struct style from above, the weights are those inside of the NN chain object, which we can inspect:
 
 NN[1].weight # The W matrix of the first layer
 
 
 32×10 Matrix{Float32}:
-  0.210141   -0.0318616   0.0161381  …   0.320312   -0.331868    0.187002
- -0.102589    0.208266   -0.18678        0.259299   -0.0850212   0.325602
-  0.0954246   0.29503     0.155045      -0.13743    -0.226485   -0.352321
-  0.094165    0.147697   -0.0402125      0.376543   -0.0652325  -0.0910839
- -0.0960281  -0.239555    0.0736242      0.179233    0.0747602  -0.231401
-  0.346408   -0.229251    0.247542   …  -0.227708   -0.342039   -0.3638
- -0.0874939  -0.146548    0.162742      -0.194633   -0.280423   -0.374489
-  0.340297    0.264646    0.097446       0.312401   -0.212878    0.307074
-  0.365767   -0.366916    0.261565      -0.0138854  -0.105345   -0.265025
-  0.228151    0.247341   -0.135668       0.247047    0.140386   -0.19907
-  ⋮                                  ⋱                          
-  0.186276    0.115302   -0.303067      -0.0957572  -0.202029    0.199585
- -0.0958866   0.139531    0.182455      -0.0669233  -0.190238    0.280298
- -0.108114    0.258266    0.052903   …   0.191436   -0.12086    -0.186751
-  0.2797     -0.0260807   0.256827       0.363836   -0.308047    0.104556
- -0.132173   -0.372458    0.122993       0.102172   -0.186959    0.195341
- -0.0277494  -0.15691     0.231842      -0.35223    -0.130247   -0.256816
- -0.309665   -0.313657   -0.345432       0.338976    0.302093    0.373044
- -0.368304    0.298401   -0.0528614  …  -0.196604    0.361721    0.234842
-  0.147451    0.324855   -0.175287       0.0836575   0.331776   -0.0320324
+ -0.0912051   0.171404   -0.225056   …   0.26575    -0.2033       0.19557
+ -0.023353   -0.193015    0.33744        0.135803    0.305452     0.330386
+ -0.21381     0.222574    0.146357       0.302459   -0.204567     0.177094
+ -0.236288   -0.0437021   0.149399      -0.322455    0.0413544    0.376834
+  0.0779307  -0.363982   -0.256043       0.150865    0.310071     0.141492
+  0.310811   -0.0969612  -0.1377     …   0.377674   -0.372038     0.237749
+  0.155738   -0.272288    0.148163      -0.175247    0.0568673   -0.302354
+ -0.338095    0.375844   -0.0522764     -0.0950616   0.241941     0.177569
+ -0.313516    0.0510053  -0.283776      -0.262828    0.127621     0.0381291
+ -0.238924   -0.019392   -0.345777       0.123743   -0.171073     0.286924
+  ⋮                                  ⋱                           
+  0.361331   -0.300791    0.340877       0.051346    0.250142     0.236606
+  0.256587   -0.144472   -0.250453       0.0694984  -0.00493134   0.242588
+  0.321232    0.133965    0.080042   …   0.0232834  -0.357839     0.175813
+  0.29538     0.0922859   0.246678      -0.102874   -0.237856    -0.340453
+  0.344922    0.306977    0.155636      -0.132782    0.127812    -0.0488943
+ -0.0161293  -0.25159    -0.220966       0.190733    0.0313222   -0.100159
+  0.0903141   0.251727    0.341801       0.127054   -0.243489    -0.0498583
+  0.0220798  -0.314075   -0.0345888  …  -0.0257494   0.109001    -0.0628161
+ -0.321938    0.144522   -0.0387606      0.17195     0.200624    -0.272704
 
 
Now let's grab all of the parameters together:
 
 p = Flux.params(NN)
 
 
-Params([Float32[0.21014117 -0.031861562 … -0.33186755 0.18700212; -0.102589
-46 0.20826595 … -0.0850212 0.32560205; … ; -0.36830446 0.29840133 … 0.36172
-083 0.23484248; 0.14745119 0.3248552 … 0.33177575 -0.03203242], Float32[0.0
-, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 0.0, 
-0.0, 0.0, 0.0, 0.0, 0.0], Float32[-0.18695994 -0.04698782 … -0.055100985 -0
-.1341884; 0.0074982448 0.11674296 … -0.22039233 -0.05518888; … ; -0.0007238
-363 0.2559794 … -0.29302862 -0.30272353; 0.087237954 0.087170936 … 0.295242
-6 -0.19100738], Float32[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …
-  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Float32[0.029854774 -0
-.06820957 … 0.09176909 -0.11000199; 0.2896224 0.2912958 … 0.034562368 -0.17
-458537; … ; 0.07649842 0.26569244 … -0.07407499 0.37595725; -0.28656915 0.0
-42809594 … -0.24949594 -0.33653733], Float32[0.0, 0.0, 0.0, 0.0, 0.0]])
+Params([Float32[-0.091205075 0.17140365 … -0.20329987 0.19557032; -0.023352
+979 -0.19301474 … 0.3054521 0.33038554; … ; 0.022079762 -0.31407467 … 0.109
+00141 -0.062816136; -0.3219381 0.14452209 … 0.20062439 -0.27270395], Float3
+2[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 
+0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Float32[0.019839598 -0.2168316 … -0.2577663 
+0.30027536; -0.19628644 0.07071809 … -0.117501654 0.20883556; … ; 0.0368188
+55 -0.012596854 … -0.24249776 -0.19441248; 0.2561112 -0.19482055 … -0.30460
+06 -0.06886764], Float32[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  
+…  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Float32[-0.12173528 -
+0.3812579 … -0.3371982 -0.27681643; 0.33429277 0.32999685 … -0.3093447 -0.2
+4638811; … ; 0.38407874 -0.25517553 … -0.011652403 0.027645448; -0.18082629
+ 0.22756484 … 0.22212335 0.053659596], Float32[0.0, 0.0, 0.0, 0.0, 0.0]])
 
 
That's a helper function on Chain which recursively gathers all of the defining parameters. Let's now find the optimal values p which cause the neural network to be the constant 1 function:
 
 Flux.train!(loss, p, Iterators.repeated((), 10000), ADAM(0.1))
 
 
Now let's check the loss:
 
 loss()
 
 
-3.377496f-5
+3.805218f-5
 
 
This means that NN(x) is now a very good function approximator to f(x) = ones(5)!
 
So Why Machine Learning? Why Neural Networks?
 
All we did was find parameters that made NN(x) act like a function f(x). How does that relate to machine learning? Well, in any case where one is acting on data (x,y), the idea is to assume that there exists some underlying mathematical model f(x) = y. If we had perfect knowledge of what f is, then from only the information of x we can then predict what y would be. The inference problem is to then figure out what function f should be. Therefore, machine learning on data is simply this problem of finding an approximator to some unknown function!
 
So why neural networks? Neural networks satisfy two properties. The first of which is known as the Universal Approximation Theorem (UAT), which in simple non-mathematical language means that, for any ϵ of accuracy, if your neural network is large enough (has enough layers, the weight matrices are large enough), then it can approximate any (nice) function f within that ϵ. Therefore, we can reduce the problem of finding missing functions, the problem of machine learning, to a problem of finding the weights of neural networks, which is a well-defined mathematical optimization problem.
 
Why neural networks specifically? That's a fairly good question, since there are many other functions with this property. For example, you will have learned from analysis that $a_0 + a_1 x + a_2 x^2 + \ldots$ arbitrary polynomials can be used to approximate any analytic function (this is the Taylor series). Similarly, a Fourier series
 
\[ f(x) = a_0 + \sum_k b_k \cos(kx) + c_k \sin(kx) \]
 
can approximate any continuous function f (and discontinuous functions also can have convergence, etc. these are the details of a harmonic analysis course).
 
That's all for one dimension. How about two dimensional functions? It turns out it's not difficult to prove that tensor products of universal approximators will give higher dimensional universal approximators. So for example, tensoring together two polynomials:
 
\[ a_0 + a_1 x + a_2 y + a_3 x y + a_4 x^2 y + a_5 x y^2 + a_6 x^2 y^2 + \ldots \]
 
will give a two-dimensional function approximator. But notice how we have to resolve every combination of terms. This means that if we used n coefficients in each dimension d, the total number of coefficients to build a d-dimensional universal approximator from one-dimensional objects would need $n^d$ coefficients. This exponential growth is known as the curse of dimensionality.
 
The second property of neural networks that makes them applicable to machine learning is that they overcome the curse of dimensionality. The proofs in this area can be a little difficult to parse, but what they boil down to is proving in many cases that the growth of neural networks to sufficiently approximate a d-dimensional function grows as a polynomial of d, rather than exponential. This means that there's some dimensional cutoff where for $d>cutoff$ it is more efficient to use a neural network. This can be problem-specific, but generally it tends to be the case at least by 8 or 10 dimensions.
 
Neural networks have a few other properties to consider as well:
 
     
  1. The assumptions of the neural network can be encoded into the neural architectures. A neural network where the last layer has an activation function x->x^2 is a neural network where all outputs are positive. This means that if you want to find a positive function, you can make the optimization easier by enforcing this constraint. A lot of other constraints can be enforced, like tanh activation functions can make the neural network be a smooth (all derivatives finite) function, or other activations can cause finite numbers of learnable discontinuities.
     
  2. Generating higher dimensional forms from one dimensional forms does not have good symmetry. For example, the two-dimensional tensor Fourier basis does not have a good way to represent $sin(xy)$. This property of the approximator is called (non)isotropy and more detail can be found in this wonderful talk about function approximation for multidimensional integration (cubature). Neural networks are naturally not aligned to a basis.
     
  3. Neural networks are "easy" to compute. There's good software for them, GPU-acceleration, and all other kinds of tooling that make them particularly simple to use.
     
  4. There are proofs that in many scenarios for neural networks the local minima are the global minima, meaning that local optimization is sufficient for training a neural network. Global optimization (which we will cover later in the course) is much more expensive than local methods like gradient descent, and thus this can be a good property to abuse for faster computation.
     
 
From Machine Learning to Scientific Machine Learning: Structure and Science
 
This understanding of a neural network and their libraries directly bridges to the understanding of scientific machine learning and the computation done in the field. In scientific machine learning, neural networks and machine learning are used as the basis to solve problems in scientific computing. Scientific computing, as a discipline also known as Computational Science, is a field of study which focuses on scientific simulation, using tools such as differential equations to investigate physical, biological, and other phenomena.
 
What we wish to do in scientific machine learning is use these properties of neural networks to improve the way that we investigate our scientific models.
 
Aside: Why Differential Equations?
 
Why do differential equations come up so often in as the model in the scientific context? This is a deep question with quite a simple answer. Essentially, all scientific experiments always have to test how things change. For example, you take a system now, you change it, and your measurement is how the changes you made caused changes in the system. This boils down to gather information about how, for some arbitrary system $y = f(x)$, how $\Delta x$ is related to $\Delta y$. Thus what you learn from scientific experiments, what is codified as scientific laws, is not "the answer", but the answer to how things change. This process of writing down equations by describing how they change precisely gives differential equations.
 
Solving ODEs with Neural Networks: The Physics-Informed Neural Network
 
Now let's get to our first true SciML application: solving ordinary differential equations with neural networks. The process of solving a differential equation with a neural network, or using a differential equation as a regularizer in the loss function, is known as a physics-informed neural network, since this allows for physical equations to guide the training of the neural network in circumstances where data might be lacking.
 
Background: A Method for Solving Ordinary Differential Equations with Neural Networks
 
This is a result first due to Lagaris et. al from 1998. The idea is to solve differential equations using neural networks by representing the solution by a neural network and training the resulting network to satisfy the conditions required by the differential equation.
 
Let's say we want to solve a system of ordinary differential equations
 
\[ u' = f(u,t) \]
 
with $t \in [0,1]$ and a known initial condition $u(0)=u_0$. To solve this, we approximate the solution by a neural network:
 
\[ NN(t) \approx u(t) \]
 
If $NN(t)$ was the true solution, then it would hold that $NN'(t) = f(NN(t),t)$ for all $t$. Thus we turn this condition into our loss function. This motivates the loss function:
 
\[ L(p) = \sum_i \left(\frac{dNN(t_i)}{dt} - f(NN(t_i),t_i) \right)^2 \]
 
The choice of $t_i$ could be done in many ways: it can be random, it can be a grid, etc. Anyways, when this loss function is minimized (gradients computed with standard reverse-mode automatic differentiation), then we have that $\frac{dNN(t_i)}{dt} \approx f(NN(t_i),t_i)$ and thus $NN(t)$ approximately solves the differential equation.
 
Note that we still have to handle the initial condition. One simple way to do this is to add an initial condition term to the cost function. This would look like:
 
\[ L(p) = (NN(0) - u_0)^2 + \sum_i \left(\frac{dNN(t_i)}{dt} - f(NN(t_i),t_i) \right)^2 \]
 
While that would work, it can be more efficient to encode the initial condition into the function itself so that it's trivially satisfied for any possible set of parameters. For example, instead of directly using a neural network, we can use:
 
\[ g(t) = u_0 + tNN(t) \]
 
as our solution. Notice that $g(t)$ is thus a universal approximator for all continuous functions such that $g(0)=u_0$ (this is a property one should prove!). Since $g(t)$ will always satisfy the initial condition, we can train $g(t)$ to satisfy the derivative function then it will automatically be a solution to the derivative function. In this sense, we can use the loss function:
 
\[ L(p) = \sum_i \left(\frac{dg(t_i)}{dt} - f(g(t_i),t_i) \right)^2 \]
 
where $p$ are the parameters that define $g$, which in turn are the parameters which define the neural network $NN$ that define $g$. Thus this reduces down, once again, to simply finding weights which minimize a loss function!
 
Coding Up the Method
 
Now let's implement this method with Flux. Let's define a neural network to be the NN(t) above. To make the problem easier, let's look at the ODE:
 
\[ u' = \cos 2\pi t \]
 
and approximate it with the neural network from a scalar to a scalar:
 
 using Flux
 NNODE = Chain(x -> [x], # Take in a scalar and transform it into an array
@@ -223,7 +223,7 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
            first) # Take first value, i.e. return a scalar
 NNODE(1.0)
 
 
-0.10085204f0
+-0.23149511f0
 
 
Instead of directly approximating the neural network, we will use the transformed equation that is forced to satisfy the boundary conditions. Using u0=1.0, we have the function:
 
 g(t) = t*NNODE(t) + 1f0
 
 
@@ -247,23 +247,23 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 display(loss())
 Flux.train!(loss, Flux.params(NNODE), data, opt; cb=cb)
 
 
-0.5089291687042273
-0.4660090972283474
-0.37798210626977025
-0.18461971004026578
-0.04371968207719241
-0.019536930111555596
-0.014619263249474197
-0.012650337532635803
-0.011729013939159596
-0.011283318492406378
-0.010946978192004615
+0.5708075887278983
+0.4843225444558577
+0.431599836626712
+0.26459017110776495
+0.05310078897817342
+0.011030989241838655
+0.007069401923025258
+0.00634987145212996
+0.00601922323173909
+0.0057577987424845736
+0.005500448268481607
 
 
How well did this do? Well if we take the integral of both sides of our differential equation, we see it's fairly trivial:
 
\[ \int g' = g = \int \cos 2\pi t = C + \frac{\sin 2\pi t}{2\pi} \]
 
where we defined $C = 1$. Let's take a bunch of (input,output) pairs from the neural network and plot it against the analytical solution to the differential equation:
 
 using Plots
 t = 0:0.001:1.0
 plot(t,g.(t),label="NN")
 plot!(t,1.0 .+ sin.(2Ï€.*t)/2Ï€, label = "True Solution")
-
  
We see that it matches very well, and we can keep improving this fit by increasing the size of the neural network, using more training points, and training for more iterations.
 
Example: Harmonic Oscillator Informed Training
 
Using this idea, differential equations encoding physical laws can be utilized inside of loss functions for terms which we have some basis to believe should approximately follow some physical system. Let's investigate this last step by looking at how to inform the training of a neural network using the harmonic oscillator.
 
Let's assume that we are taking measurements of (position,force) in some real one-dimensional spring pushing and pulling against a wall.
 
 
But instead of the simple spring, let's assume we had a more complex spring, for example, let's say $F(x) = -kx + 0.1sin(x)$ where this extra term is due to some deformities in the metal (assume mass=1). Then by Newton's law of motion we have a second order ordinary differential equation:
 
\[ x'' = -kx + 0.1 \sin(x) \]
 
We can use the DifferentialEquations.jl package to solve this differential equation and see what this system looks like:
 
+
  
We see that it matches very well, and we can keep improving this fit by increasing the size of the neural network, using more training points, and training for more iterations.
 
Example: Harmonic Oscillator Informed Training
 
Using this idea, differential equations encoding physical laws can be utilized inside of loss functions for terms which we have some basis to believe should approximately follow some physical system. Let's investigate this last step by looking at how to inform the training of a neural network using the harmonic oscillator.
 
Let's assume that we are taking measurements of (position,force) in some real one-dimensional spring pushing and pulling against a wall.
 
 
But instead of the simple spring, let's assume we had a more complex spring, for example, let's say $F(x) = -kx + 0.1sin(x)$ where this extra term is due to some deformities in the metal (assume mass=1). Then by Newton's law of motion we have a second order ordinary differential equation:
 
\[ x'' = -kx + 0.1 \sin(x) \]
 
We can use the DifferentialEquations.jl package to solve this differential equation and see what this system looks like:
 
 using DifferentialEquations
 k = 1.0
 force(dx,x,k,t) = -k*x + 0.1sin(x)
@@ -300,7 +300,7 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 loss() = sum(abs2,NNForce(position_data[i]) - force_data[i] for i in 1:length(position_data))
 loss()
 
 
-0.0021075816164434196
+0.001639101475417397
 
 
Our random parameters do not do so well, so let's train!
 
 opt = Flux.Descent(0.01)
 data = Iterators.repeated((), 5000)
@@ -314,24 +314,24 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 display(loss())
 Flux.train!(loss, Flux.params(NNForce), data, opt; cb=cb)
 
 
-0.0021075816164434196
-0.0016626665734058707
-0.0014129151050661442
-0.0012004879578227366
-0.001019683666213778
-0.00086574214000748
-0.0007346664655364099
-0.0006230813332454321
-0.0005281206353467606
-0.0004473482765489638
-0.00037868342158744984
+0.001639101475417397
+0.001281078565346881
+0.001078228101263855
+0.000907144491383387
+0.000762834587654969
+0.0006411230663627027
+0.0005385050286698408
+0.00045202585782455135
+0.0003791885175632164
+0.00031788120953395697
+0.0002663162119004075
 
 
The neural network almost exactly matched the dataset, but how well did it actually learn the real force function? Let's plot it to see:
 
 learned_force_plot = NNForce.(positions_plot)
 
 plot(plot_t,force_plot,xlabel="t",label="True Force")
 plot!(plot_t,learned_force_plot,label="Predicted Force")
 scatter!(t,force_data,label="Force Measurements")
-
  
Ouch. The problem is that a neural network can approximate any function, so it approximated a function that fits the data, but not the correct function. We somehow need to have more data... but where can we get more data?
 
Well, even a first year undergrad in physics will know Hooke's law, which is that the idealized spring should satisfy $F(x) = -kx$. This is a decent assumption for the evolution of the system:
 
+
  
Ouch. The problem is that a neural network can approximate any function, so it approximated a function that fits the data, but not the correct function. We somehow need to have more data... but where can we get more data?
 
Well, even a first year undergrad in physics will know Hooke's law, which is that the idealized spring should satisfy $F(x) = -kx$. This is a decent assumption for the evolution of the system:
 
 force2(dx,x,k,t) = -k*x
 prob_simplified = SecondOrderODEProblem(force2,1.0,0.0,(0.0,10.0),k)
 sol_simplified = solve(prob_simplified)
@@ -342,7 +342,7 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 loss_ode() = sum(abs2,NNForce(x) - (-k*x) for x in random_positions)
 loss_ode()
 
 
-10.993295820275433
+8.460564225345507
 
 
If this term is zero, then $F(x) = -kx$, which is approximately true. So now let's put these together:
 
 λ = 0.1
 composed_loss() = loss() + λ*loss_ode()
@@ -367,15 +367,15 @@ 
Introduction to Scientific Machine Learning through Physics-Inf
 plot!(plot_t,learned_force_plot,label="Predicted Force")
 scatter!(t,force_data,label="Force Measurements")
 
 
-1.0997082654491308
-0.0005850661060708156
-0.0005522938565740613
-0.0005232864079531569
-0.0004973263251085195
-0.00047389858231739466
-0.000452611856433718
-0.0004331665468480585
-0.00041531905615030537
-0.00039886843159506777
-0.000383654997635387
-
  
And there we go: we have used knowledge of physics to help inform our neural network training process!
 
Conclusion
 
In this lecture we motivated machine learning not as a process of predicting from data but as a process for learning arbitrary nonlinear functions. Neural networks were just one choice of possible function. We then demonstrated how differential equations could be solved using this function approximation technique and then put together these two domains, solving differential equations and approximating data, into a single process to allow for physical knowledge to be embedded into the training process of a neural network, thus arriving at a physics-informed neural network. This is just one method in scientific machine learning which we will be exploring in more detail, demonstrating how we can utilize scientific knowledge to improve fits and allow for data-efficient machine learning.
 
 
 Published from sciml.jmd using Weave.jl v0.10.12 on 2024-07-26. 
 

\ No newline at end of file
+0.8463227387464513
+0.000210408027752647
+0.00020519506792570903
+0.00020026862808211442
+0.00019559928791649267
+0.00019116211020453466
+0.00018693271539265902
+0.00018289622008194647
+0.00017903277909909004
+0.00017533236251209575
+0.0001717819891828354
+
  
And there we go: we have used knowledge of physics to help inform our neural network training process!
 
Conclusion
 
In this lecture we motivated machine learning not as a process of predicting from data but as a process for learning arbitrary nonlinear functions. Neural networks were just one choice of possible function. We then demonstrated how differential equations could be solved using this function approximation technique and then put together these two domains, solving differential equations and approximating data, into a single process to allow for physical knowledge to be embedded into the training process of a neural network, thus arriving at a physics-informed neural network. This is just one method in scientific machine learning which we will be exploring in more detail, demonstrating how we can utilize scientific knowledge to improve fits and allow for data-efficient machine learning.
 
 
 Published from sciml.jmd using Weave.jl v0.10.12 on 2024-07-26. 
 

\ No newline at end of file
diff --git a/_weave/lecture04/dynamical_systems/index.html b/_weave/lecture04/dynamical_systems/index.html
index 71ca50c7..e5fe7753 100644
--- a/_weave/lecture04/dynamical_systems/index.html
+++ b/_weave/lecture04/dynamical_systems/index.html
@@ -113,7 +113,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 end
 @time solve_system_save(lorenz,[1.0,0.0,0.0],p,1000)
 
-0.000050 seconds (1.00 k allocations: 86.062 KiB)
+0.000054 seconds (1.00 k allocations: 86.062 KiB)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -138,7 +138,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 
 @time solve_system_save_push(lorenz,[1.0,0.0,0.0],p,1000)
 
-0.012133 seconds (4.50 k allocations: 336.016 KiB, 99.40% compilation tim
+0.011324 seconds (4.50 k allocations: 336.016 KiB, 99.43% compilation tim
 e)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
@@ -164,7 +164,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

The first time Julia compiles the function, and the second is a straight call.

 @time solve_system_save_push(lorenz,[1.0,0.0,0.0],p,1000)
 
-0.000057 seconds (1.01 k allocations: 99.984 KiB)
+0.000056 seconds (1.01 k allocations: 99.984 KiB)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -190,7 +190,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 using BenchmarkTools
 @btime solve_system_save(lorenz,[1.0,0.0,0.0],p,1000)
 
-38.372 μs (1001 allocations: 86.06 KiB)
+27.531 μs (1001 allocations: 86.06 KiB)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -215,7 +215,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 
 @btime solve_system_save_push(lorenz,[1.0,0.0,0.0],p,1000)
 
-44.223 μs (1006 allocations: 99.98 KiB)
+33.333 μs (1006 allocations: 99.98 KiB)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -248,7 +248,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 end
 @btime solve_system_save_matrix(lorenz,[1.0,0.0,0.0],p,1000)
 
-69.190 μs (2001 allocations: 179.66 KiB)
+66.845 μs (2001 allocations: 179.66 KiB)
 3×1000 Matrix{Float64}:
  1.0  0.8   0.752    0.80096   0.920338   …   1.98201    1.67886    1.4744
  0.0  0.56  0.9968   1.39785   1.81805        0.466287   0.656559   0.85300
@@ -265,7 +265,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 end
 @btime solve_system_save_matrix_view(lorenz,[1.0,0.0,0.0],p,1000)
 
-34.535 μs (1002 allocations: 101.61 KiB)
+34.956 μs (1002 allocations: 101.61 KiB)
 3×1000 Matrix{Float64}:
  1.0  0.8   0.752    0.80096   0.920338   …   1.98201    1.67886    1.4744
  0.0  0.56  0.9968   1.39785   1.81805        0.466287   0.656559   0.85300
@@ -282,7 +282,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 end
 @btime solve_system_save_matrix_resize(lorenz,[1.0,0.0,0.0],p,1000)
 
-1.198 ms (2318 allocations: 11.65 MiB)
+1.043 ms (2318 allocations: 11.65 MiB)
 3×1000 Matrix{Float64}:
  1.0  0.8   0.752    0.80096   0.920338   …   1.98201    1.67886    1.4744
  0.0  0.56  0.9968   1.39785   1.81805        0.466287   0.656559   0.85300
@@ -388,7 +388,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

which would compute f and then take the values of du and update u with them, but that's 3 extra operations than required, whereas u,du = du,u will change u to be a pointer to the updated memory and now du is an "empty" cache array that we can refill (this decreases the computational cost by ~33%). Let's see what the cost is with this newest version:

 @btime solve_system(lorenz,[1.0,0.0,0.0],p,1000)
 
-37.210 μs (1000 allocations: 78.12 KiB)
+25.919 μs (1000 allocations: 78.12 KiB)
 3-element Vector{Float64}:
   1.4744010677851374
   0.8530017039412324
@@ -396,7 +396,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 
 @btime solve_system_mutate(lorenz,[1.0,0.0,0.0],p,1000)
 
-7.301 μs (3 allocations: 240 bytes)
+7.291 μs (3 allocations: 240 bytes)
 3-element Vector{Float64}:
   1.4744010677851374
   0.8530017039412324
@@ -445,7 +445,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 
 @btime solve_system_save(lorenz,@SVector[1.0,0.0,0.0],p,1000)
 
-8.766 μs (2 allocations: 23.48 KiB)
+8.850 μs (2 allocations: 23.48 KiB)
 1000-element Vector{SVector{3, Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -511,7 +511,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 
 @btime solve_system_save(lorenz,@SVector[1.0,0.0,0.0],p,1000)
 
-6.723 μs (2 allocations: 23.48 KiB)
+6.721 μs (2 allocations: 23.48 KiB)
 1000-element Vector{SVector{3, Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -536,7 +536,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

And we can get down to non-allocating for the loop:

 @btime solve_system(lorenz,@SVector([1.0,0.0,0.0]),p,1000)
 
-6.502 μs (1 allocation: 32 bytes)
+6.496 μs (1 allocation: 32 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
   1.4744010677851374
   0.8530017039412324
@@ -552,7 +552,7 @@ 
How Iteration Works, An Introduction to Discrete Dynamics

 u = Vector{typeof(@SVector([1.0,0.0,0.0]))}(undef,1000)
 @btime solve_system_save!(u,lorenz,@SVector([1.0,0.0,0.0]),p,1000)
 
-6.498 μs (0 allocations: 0 bytes)
+6.500 μs (0 allocations: 0 bytes)
 1000-element Vector{SVector{3, Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
diff --git a/_weave/lecture04/jl_lshY2p/dynamical_systems_11_1.png b/_weave/lecture04/jl_fP27tE/dynamical_systems_11_1.png
similarity index 100%
rename from _weave/lecture04/jl_lshY2p/dynamical_systems_11_1.png
rename to _weave/lecture04/jl_fP27tE/dynamical_systems_11_1.png
diff --git a/_weave/lecture05/parallelism_overview/index.html b/_weave/lecture05/parallelism_overview/index.html
index 2e2ef5b4..38bf7652 100644
--- a/_weave/lecture05/parallelism_overview/index.html
+++ b/_weave/lecture05/parallelism_overview/index.html
@@ -127,7 +127,7 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 u = [Vector{Float64}(undef,3) for i in 1:1000]
 @btime solve_system_save_iip!(u,lorenz_mt!,[1.0,0.0,0.0],p,1000);
 
-1.202 ms (5995 allocations: 608.84 KiB)
+1.203 ms (5995 allocations: 608.84 KiB)

Parallelism doesn't always make things faster. There are two costs associated with this code. For one, we had to go to the slower heap+mutation version, so its implementation starting point is slower. But secondly, and more importantly, the cost of spinning a new thread is non-negligible. In fact, here we can see that it even needs to make a small allocation for the new context. The total cost is on the order of It's on the order of 50ns: not huge, but something to take note of. So what we've done is taken almost free calculations and made them ~50ns by making each in a different thread, instead of just having it be one thread with one call stack.

The moral of the story is that you need to make sure that there's enough work per thread in order to effectively accelerate a program with parallelism.

Data-Parallel Problems

So not every setup is amenable to parallelism. Dynamical systems are notorious for being quite difficult to parallelize because the dependency of the future time step on the previous time step is clear, meaning that one cannot easily "parallelize through time" (though it is possible, which we will study later).

However, one common way that these systems are generally parallelized is in their inputs. The following questions allow for independent simulations:

  • What steady state does an input u0 go to for some list/region of initial conditions?

  • How does the solution very when I use a different p?

The problem has a few descriptions. For one, it's called an embarrassingly parallel problem since the problem can remain largely intact to solve the parallelism problem. To solve this, we can use the exact same solve_system_save_iip!, and just change how we are calling it. Secondly, this is called a data parallel problem, since it parallelized by splitting up the input data (here, the possible u0 or ps) and acting on them independently.

Multithreaded Parameter Searches

Now let's multithread our parameter search. Let's say we wanted to compute the mean of the values in the trajectory. For a single input pair, we can compute that like:

 using Statistics
 function compute_trajectory_mean(u0,p)
@@ -137,7 +137,7 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 end
 @btime compute_trajectory_mean(@SVector([1.0,0.0,0.0]),p)
 
-7.647 μs (3 allocations: 23.52 KiB)
+7.612 μs (3 allocations: 23.52 KiB)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -151,7 +151,7 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 end
 @btime compute_trajectory_mean2(@SVector([1.0,0.0,0.0]),p)
 
-7.464 μs (3 allocations: 112 bytes)
+7.469 μs (3 allocations: 112 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -165,7 +165,7 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 end
 @btime compute_trajectory_mean3(@SVector([1.0,0.0,0.0]),p)
 
-7.424 μs (1 allocation: 32 bytes)
+7.421 μs (1 allocation: 32 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -178,7 +178,7 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 compute_trajectory_mean4(u0,p) = _compute_trajectory_mean4(_u_cache,u0,p)
 @btime compute_trajectory_mean4(@SVector([1.0,0.0,0.0]),p)
 
-7.429 μs (1 allocation: 32 bytes)
+7.426 μs (1 allocation: 32 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -187,50 +187,50 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 ps = [(0.02,10.0,28.0,8/3) .* (1.0,rand(3)...) for i in 1:1000]
 
 1000-element Vector{NTuple{4, Float64}}:
- (0.02, 9.106049390932355, 3.303812946184277, 2.0289376601917164)
- (0.02, 4.735606575537758, 24.169178529667747, 1.3480009969264561)
- (0.02, 1.8782542650718392, 22.557130006439053, 2.0243816234869705)
- (0.02, 5.302261087000523, 27.07006533567516, 2.655762043438801)
- (0.02, 5.670338250773302, 4.441306677748524, 0.9560932489779355)
- (0.02, 1.4007298019106085, 8.924613440852292, 1.7845149466502728)
- (0.02, 4.210072135010345, 12.885641724584875, 1.4106354338111666)
- (0.02, 6.178913314158388, 14.077198942044106, 2.6422469838602893)
- (0.02, 9.022764003735679, 5.959401451034244, 0.2915953150851287)
- (0.02, 7.637994682331543, 2.280140073955992, 1.3174704247968336)
+ (0.02, 2.0494785010995353, 12.87474091643628, 2.221379168745697)
+ (0.02, 3.0141933992052494, 5.011093405297812, 0.9795778033401599)
+ (0.02, 6.6087512736768526, 22.554785148961784, 0.3707891781853476)
+ (0.02, 0.5160432877281274, 2.1442560677379037, 1.7411393467364724)
+ (0.02, 8.558522987629267, 18.545203958656234, 1.9346962153448815)
+ (0.02, 9.54542624443799, 14.894432021569349, 2.2001967050663778)
+ (0.02, 7.164909712680697, 14.058087153752206, 1.9721359183413034)
+ (0.02, 4.503455496371159, 15.638037179709096, 0.5478848102231814)
+ (0.02, 5.0067292534598415, 23.498092029722358, 1.8321033765018702)
+ (0.02, 2.043018481497864, 24.8343511074286, 0.40227226208229894)
  â‹®
- (0.02, 1.4295983611840468, 22.723025540961917, 0.5603759621647108)
- (0.02, 3.5691263501247104, 11.614453902298626, 0.9865390285286525)
- (0.02, 6.347641844244725, 11.287551194973615, 2.170255707975067)
- (0.02, 4.3184493713301455, 24.303906061742534, 0.09612994921103102)
- (0.02, 6.080688519020651, 7.21028271370196, 0.8896348073525427)
- (0.02, 6.3902301771886725, 13.890419308445491, 0.5118246742927562)
- (0.02, 3.0250459886141687, 16.614712639906436, 2.5380729556522454)
- (0.02, 1.4197562136945086, 12.915115904241887, 2.1974841648464785)
- (0.02, 2.760108862922477, 8.843016809441602, 0.6191363970184215)
+ (0.02, 9.459925043278687, 14.913762836267876, 1.946411348397799)
+ (0.02, 3.7058223528064884, 9.956749354099607, 2.5430008163316744)
+ (0.02, 5.6913112833876225, 12.407156482636111, 1.0651014726178387)
+ (0.02, 0.09131385334597009, 7.6889382933701, 2.0354078697915967)
+ (0.02, 1.1885766367375572, 19.114097281325094, 1.381133123272827)
+ (0.02, 3.9804371550787767, 23.853405760253743, 1.185532990309715)
+ (0.02, 8.84078503958126, 5.847345434942631, 1.0196090739651438)
+ (0.02, 5.582363837550646, 11.995288548626041, 0.041581327093415034)
+ (0.02, 0.643229668934614, 23.346238053688726, 0.22539644325045938)

And let's get the mean of the trajectory for each of the parameters.

 serial_out = map(p -> compute_trajectory_mean4(@SVector([1.0,0.0,0.0]),p),ps)
 
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

Now let's do this with multithreading:

 function tmap(f,ps)
   out = Vector{typeof(@SVector([1.0,0.0,0.0]))}(undef,1000)
@@ -243,26 +243,26 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 threaded_out = tmap(p -> compute_trajectory_mean4(@SVector([1.0,0.0,0.0]),p),ps)
 
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

Let's check the output:

 serial_out - threaded_out
 
@@ -296,7 +296,7 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 end
 @btime compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p)
 
-7.429 μs (1 allocation: 32 bytes)
+7.424 μs (1 allocation: 32 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -330,53 +330,53 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 
 @btime serial_out = map(p -> compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p),ps)
 
-7.420 ms (3 allocations: 23.50 KiB)
+7.419 ms (3 allocations: 23.50 KiB)
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]
 
 @btime threaded_out = tmap(p -> compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p),ps)
 
-7.420 ms (8 allocations: 24.06 KiB)
+7.421 ms (8 allocations: 24.06 KiB)
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

Hierarchical Task-Based Multithreading and Dynamic Scheduling

The major change in Julia v1.3 is that Julia's Tasks, which are traditionally its green threads interface, are now the basis of its multithreading infrastructure. This means that all independent threads are parallelized, and a new interface for multithreading will exist that works by spawning threads.

This implementation follows Go's goroutines and the classic multithreading interface of Cilk. There is a Julia-level scheduler that handles the multithreading to put different tasks on different vCPU threads. A benefit from this is hierarchical multithreading. Since Julia's tasks can spawn tasks, what can happen is a task can create tasks which create tasks which etc. In Julia (/Go/Cilk), this is then seen as a single pool of tasks which it can schedule, and thus it will still make sure only N are running at a time (as opposed to the naive implementation where the total number of running threads is equal then multiplied). This is essential for numerical performance because running multiple compute threads on a single CPU thread requires constant context switching between the threads, which will slow down the computations.

To directly use the task-based interface, simply use Threads.@spawn to spawn new tasks. For example:

 function tmap2(f,ps)
   tasks = [Threads.@spawn f(ps[i]) for i in 1:1000]
@@ -385,51 +385,51 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 threaded_out = tmap2(p -> compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p),ps)
 
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

However, if we check the timing we see:

 @btime tmap2(p -> compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p),ps)
 
-7.761 ms (6005 allocations: 562.70 KiB)
+7.741 ms (6005 allocations: 562.70 KiB)
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

Threads.@threads is built on the same multithreading infrastructure, so why is this so much slower? The reason is because Threads.@threads employs static scheduling while Threads.@spawn is using dynamic scheduling. Dynamic scheduling is the model of allowing the runtime to determine the ordering and scheduling of processes, i.e. what tasks will run run where and when. Julia's task-based multithreading system has a thread scheduler which will automatically do this for you in the background, but because this is done at runtime it will have overhead. Static scheduling is the model of pre-determining where and when tasks will run, instead of allowing this to be determined at runtime. Threads.@threads is "quasi-static" in the sense that it cuts the loop so that it spawns only as many tasks as there are threads, essentially assigning one thread for even chunks of the input data.

Does this lack of runtime overhead mean that static scheduling is "better"? No, it simply has trade-offs. Static scheduling assumes that the runtime of each block is the same. For this specific case where there are fixed number of loop iterations for the dynamical systems, we know that every compute_trajectory_mean5 costs exactly the same, and thus this will be more efficient. However, There are many cases where this might not be efficient. For example:

 function sleepmap_static()
   out = Vector{Int}(undef,24)
@@ -448,7 +448,7 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 @btime sleepmap_static()
 @btime sleepmap_spawn()
 
-30.057 s (104 allocations: 3.75 KiB)
+30.053 s (104 allocations: 3.75 KiB)
   2.401 s (220 allocations: 14.77 KiB)
 24-element Vector{Int64}:
   1
@@ -489,24 +489,24 @@ 
The Basics of Single Node Parallel Computing
 
Chris Rac
 A*B
 
 10000×10000 Matrix{Float64}:
- 2542.68  2521.97  2510.56  2497.9   …  2509.28  2525.0   2523.88  2520.4
- 2509.41  2523.24  2499.23  2466.59     2498.17  2490.19  2504.73  2494.0
- 2527.03  2511.33  2512.35  2475.63     2512.02  2515.73  2511.64  2515.72
- 2530.45  2522.96  2505.87  2477.47     2514.41  2511.84  2483.23  2504.55
- 2528.08  2517.44  2492.39  2481.61     2514.61  2518.98  2501.12  2506.62
- 2513.9   2530.09  2495.53  2463.74  …  2513.33  2488.51  2489.3   2499.18
- 2483.17  2492.63  2484.71  2445.09     2490.3   2491.62  2488.92  2477.49
- 2540.57  2535.27  2522.95  2491.98     2527.58  2505.16  2526.23  2522.58
- 2541.76  2536.15  2523.99  2492.26     2520.01  2514.86  2519.61  2519.08
- 2510.24  2515.49  2519.26  2479.86     2499.94  2503.5   2503.25  2500.0
+ 2525.87  2488.32  2499.41  2509.97  …  2511.13  2489.44  2531.07  2502.22
+ 2519.05  2496.88  2482.01  2506.51     2527.63  2493.75  2516.93  2503.97
+ 2534.89  2503.33  2495.34  2520.59     2529.65  2508.02  2536.28  2522.71
+ 2532.22  2520.62  2515.99  2518.95     2535.94  2524.97  2546.52  2512.56
+ 2514.36  2479.37  2504.63  2489.56     2507.64  2494.65  2523.54  2493.13
+ 2529.33  2500.05  2501.9   2505.24  …  2531.42  2486.38  2515.62  2512.86
+ 2540.09  2497.2   2504.75  2516.27     2521.17  2512.34  2529.63  2512.79
+ 2509.74  2477.49  2488.59  2479.55     2506.92  2485.0   2531.66  2480.12
+ 2481.75  2476.91  2490.11  2465.56     2465.47  2467.87  2509.78  2478.45
+ 2544.86  2496.0   2514.15  2517.92     2519.5   2511.37  2552.28  2527.42
     ⋮                                ⋱                             
- 2525.23  2512.21  2498.54  2488.94     2513.71  2501.97  2511.88  2520.2
- 2511.07  2491.4   2491.45  2465.4      2511.1   2497.52  2478.17  2490.51
- 2502.33  2508.4   2498.52  2465.6      2503.36  2502.37  2492.31  2501.05
- 2521.12  2509.38  2498.31  2474.29     2515.97  2501.83  2512.41  2488.79
- 2490.9   2496.69  2483.61  2441.27  …  2490.72  2471.2   2475.08  2476.2
- 2535.03  2541.2   2529.11  2497.66     2535.53  2517.47  2531.13  2519.93
- 2511.1   2510.73  2493.1   2492.95     2510.81  2499.15  2542.01  2518.67
- 2518.64  2526.74  2517.93  2488.57     2531.73  2504.55  2515.35  2513.5
- 2523.56  2519.73  2500.24  2485.91     2513.05  2535.31  2523.33  2518.76
+ 2527.82  2496.37  2506.14  2490.21     2519.83  2528.13  2550.02  2506.49
+ 2512.87  2490.24  2480.03  2490.32     2526.82  2474.85  2525.26  2493.48
+ 2494.52  2487.08  2483.83  2499.03     2504.63  2476.85  2533.39  2491.2
+ 2531.16  2499.71  2516.22  2520.35     2536.82  2507.74  2517.3   2530.53
+ 2521.62  2481.61  2484.72  2506.89  …  2514.74  2495.46  2525.36  2499.07
+ 2516.2   2491.17  2493.77  2509.39     2502.63  2491.1   2508.44  2497.63
+ 2499.9   2481.76  2506.27  2492.37     2494.85  2491.09  2518.55  2480.2
+ 2527.68  2503.12  2528.21  2511.25     2532.84  2503.63  2558.09  2533.89
+ 2540.76  2513.19  2517.55  2496.81     2522.99  2506.76  2533.4   2518.68

If you are using a computer that has N cores, then this will use N cores. Try it and look at your resource usage!

Array-Based Parallelism

The simplest form of parallelism is array-based parallelism. The idea is that you use some construction of an array whose operations are already designed to be parallel under the hood. In Julia, some examples of this are:

  • DistributedArrays (Distributed Computing)

  • Elemental

  • MPIArrays

  • CuArrays (GPUs)

This is not a Julia specific idea either.

BLAS and Standard Libraries

The basic linear algebra calls are all handled by a set of libraries which follow the same interface known as BLAS (Basic Linear Algebra Subroutines). It's divided into 3 portions:

  • BLAS1: Element-wise operations (O(n))

  • BLAS2: Matrix-vector operations (O(n^2))

  • BLAS3: Matrix-matrix operations (O(n^3))

BLAS implementations are highly optimized, like OpenBLAS and Intel MKL, so every numerical language and library essentially uses similar underlying BLAS implementations. Extensions to these, known as LAPACK, include operations like factorizations, and are included in these standard libraries. These are all multithreaded. The reason why this is a location to target is because the operation count is high enough that parallelism can be made efficient even when only targeting this level: a matrix multiplication can take on the order of seconds, minutes, hours, or even days, and these are all highly parallel operations. This means you can get away with a bunch just by parallelizing at this level, which happens to be a bottleneck for a lot scientific computing codes.

This is also commonly the level at which GPU computing occurs in machine learning libraries for reasons which we will explain later.

MPI

Well, this is a big topic and we'll address this one later!

Conclusion

The easiest forms of parallelism are:

  • Embarrassingly parallel

  • Array-level parallelism (built into linear algebra)

Exploit these when possible.

Published from parallelism_overview.jmd using Weave.jl v0.10.12 on 2024-07-26.

\ No newline at end of file diff --git a/_weave/lecture06/styles_of_parallelism/index.html b/_weave/lecture06/styles_of_parallelism/index.html index 34279380..2c0202b4 100644 --- a/_weave/lecture06/styles_of_parallelism/index.html +++ b/_weave/lecture06/styles_of_parallelism/index.html @@ -9,26 +9,26 @@

The Different Flavors of Parallelism

Chris Rackauckas< arr = [MyComplex(rand(),rand()) for i in 1:100] 
 100-element Vector{MyComplex}:
- MyComplex(0.14309417245278067, 0.9215224222944839)
- MyComplex(0.6563768312045752, 0.8044767963545393)
- MyComplex(0.5728049188895872, 0.33765746952116094)
- MyComplex(0.8240073299674354, 0.9470426273216329)
- MyComplex(0.7606061347124898, 0.9327907399185521)
- MyComplex(0.755341833759231, 0.5434958255089687)
- MyComplex(0.9731438085219921, 0.31820007095612113)
- MyComplex(0.6770182257794624, 0.3681202537043393)
- MyComplex(0.08480739105077983, 0.65477879435604)
- MyComplex(0.879044108699869, 0.5694763191704512)
+ MyComplex(0.5948216400563927, 0.7202742856092115)
+ MyComplex(0.8858224485878035, 0.04464426300983426)
+ MyComplex(0.4536724251568013, 0.48301713342468633)
+ MyComplex(0.17632669227127473, 0.966975004988031)
+ MyComplex(0.7423602389573798, 0.7381611563859961)
+ MyComplex(0.5049342978821135, 0.2713721944949681)
+ MyComplex(0.8956475760986714, 0.006535319763830483)
+ MyComplex(0.5275017128612832, 0.6089884504844824)
+ MyComplex(0.22199738442114458, 0.3133842057610531)
+ MyComplex(0.7056746294589479, 0.17085581295301622)
  â‹®
- MyComplex(0.6184831153175321, 0.8420717430705801)
- MyComplex(0.11097401639991189, 0.4733738650181497)
- MyComplex(0.9598982066766498, 0.5024876345986231)
- MyComplex(0.7704581759272878, 0.06431948804402998)
- MyComplex(0.5067467487289342, 0.37276230976872216)
- MyComplex(0.22392490395791997, 0.24997212682983505)
- MyComplex(0.7938100950862472, 0.499794534596235)
- MyComplex(0.5800027114018041, 0.5539306180949976)
- MyComplex(0.8613302068792389, 0.11099259999056299)
+ MyComplex(0.8949805161160401, 0.421323667111285)
+ MyComplex(0.026053733946824753, 0.9006485214268137)
+ MyComplex(0.8300583605559014, 0.8018415810144012)
+ MyComplex(0.12592125655230224, 0.27000881231142315)
+ MyComplex(0.3618587243109601, 0.4880572513222716)
+ MyComplex(0.3150275713704259, 0.13939673381769024)
+ MyComplex(0.03638847026177339, 0.8020351165630957)
+ MyComplex(0.08217419775238222, 0.7933908176556499)
+ MyComplex(0.03792572372188874, 0.1330438491287449)

is represented in memory as

[real1,imag1,real2,imag2,...]

while the struct of array formats are

@@ -43,18 +43,18 @@

The Different Flavors of Parallelism

Chris Rackauckas< 
-MyComplexes([0.2897241260009822, 0.48104950344076425, 0.1550991673426182, 0
-.4303053146297099, 0.9652545002585547, 0.129691784563555, 0.930721759368184
-5, 0.04236592750770252, 0.12121370563973322, 0.6554310506492863  …  0.73820
-40247731638, 0.057317858458742266, 0.20857864801382386, 0.7618557750226116,
- 0.30777823594243947, 0.19986995680131348, 0.7036870522240618, 0.5790694102
-590798, 0.14664887682564642, 0.44821427291761984], [0.46688590924729956, 0.
-11887230044632568, 0.2806224613334578, 0.43126362776131033, 0.9794620042427
-044, 0.42610522792220007, 0.3077309824326909, 0.6937145539324716, 0.8143594
-272939957, 0.7105773101090823  …  0.24229997305610995, 0.18587298385876627,
- 0.24053515975314443, 0.9719512037832595, 0.03802957417362052, 0.4362822838
-808521, 0.4088166896868569, 0.8960777149330896, 0.5386366912236279, 0.07960
-564919501523])
+MyComplexes([0.4463978501148237, 0.9534146019623043, 0.3272506411204227, 0.
+6610621659220888, 0.7518861360127663, 0.6604957283856057, 0.568257801511073
+4, 0.5249214846587075, 0.28031214215080524, 0.4019397592681623  …  0.230795
+16952268475, 0.4587073529434471, 0.38888294861698836, 0.8764601865880245, 0
+.7664927638985168, 0.5500101294037606, 0.8467864454608346, 0.75150871387712
+67, 0.9504725250659736, 0.7891716988140378], [0.2354669170781668, 0.3484246
+5928318156, 0.4847930258306149, 0.5050470303321312, 0.8805989081762119, 0.3
+6730961426487874, 0.7503851083942035, 0.7838065781271188, 0.833361952697676
+8, 0.4100985361969356  …  0.7241872639287391, 0.2685519569209651, 0.2209582
+6963895615, 0.2074266359413821, 0.6228410415260852, 0.17911567432938458, 0.
+5661141870291473, 0.9051093078571997, 0.37633923990420315, 0.29487469397695
+53])
@@ -73,7 +73,7 @@

The Different Flavors of Parallelism

Chris Rackauckas< 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:5 within `average`
-define void @julia_average_10372([2 x double]* noalias nocapture noundef no
+define void @julia_average_10360([2 x double]* noalias nocapture noundef no
 nnull sret([2 x double]) align 8 dereferenceable(16) %0, {}* noundef nonnul
 l align 16 dereferenceable(40) %1) #0 {
 top:
@@ -108,21 +108,21 @@ 
The Different Flavors of Parallelism
 
Chris Rackauckas<
             %.sub = getelementptr inbounds [4 x {}*], [4 x {}*]* %2, i64 0,
  i64 0
 ; ││││││││││ @ reduce.jl:432 within `_mapreduce`
-            store {}* inttoptr (i64 139788167070960 to {}*), {}** %.sub, al
+            store {}* inttoptr (i64 139762867029232 to {}*), {}** %.sub, al
 ign 8
             %5 = getelementptr inbounds [4 x {}*], [4 x {}*]* %2, i64 0, i6
 4 1
-            store {}* inttoptr (i64 139788214797840 to {}*), {}** %5, align
+            store {}* inttoptr (i64 139762914756112 to {}*), {}** %5, align
  8
             %6 = getelementptr inbounds [4 x {}*], [4 x {}*]* %2, i64 0, i6
 4 2
             store {}* %1, {}** %6, align 8
             %7 = getelementptr inbounds [4 x {}*], [4 x {}*]* %2, i64 0, i6
 4 3
-            store {}* inttoptr (i64 139788196084624 to {}*), {}** %7, align
+            store {}* inttoptr (i64 139762896042896 to {}*), {}** %7, align
  8
-            %8 = call nonnull {}* @ijl_invoke({}* inttoptr (i64 13978817579
-4192 to {}*), {}** nonnull %.sub, i32 4, {}* inttoptr (i64 139788383992288 
+            %8 = call nonnull {}* @ijl_invoke({}* inttoptr (i64 13976287575
+2464 to {}*), {}** nonnull %.sub, i32 4, {}* inttoptr (i64 139761459797280 
 to {}*))
             call void @llvm.trap()
             unreachable
@@ -233,7 +233,7 @@ 
The Different Flavors of Parallelism
 
Chris Rackauckas<
 ; ││││││││││└
 ; ││││││││││ @ reduce.jl:447 within `_mapreduce`
 ; ││││││││││┌ @ reduce.jl:277 within `mapreduce_impl`
-             call void @j_mapreduce_impl_10374([2 x double]* noalias nocapt
+             call void @j_mapreduce_impl_10362([2 x double]* noalias nocapt
 ure noundef nonnull sret([2 x double]) %tmpcast, {}* nonnull %1, i64 signex
 t 1, i64 signext %arraylen, i64 signext 1024)
 ; └└└└└└└└└└└
@@ -583,7 +583,7 @@ 
Next Level Up: Multithreading

 
 
 
-59.061 μs (6 allocations: 576 bytes)
+59.671 μs (6 allocations: 576 bytes)
 

 
 
@@ -596,7 +596,7 @@ 
Next Level Up: Multithreading

 
 
 
-66.033 μs (6 allocations: 576 bytes)
+66.023 μs (6 allocations: 576 bytes)
 

 
 
@@ -609,7 +609,7 @@ 
Next Level Up: Multithreading

 
 
 
-20.138 μs (6 allocations: 576 bytes)
+20.148 μs (6 allocations: 576 bytes)
 

 
 
@@ -637,17 +637,17 @@ 
Next Level Up: Multithreading

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:26 within `h`
-define void @julia_h_10465() #0 {
+define void @julia_h_10453() #0 {
 top:
-  %.promoted = load i64, i64* inttoptr (i64 139785926445312 to i64*), align
- 256
+  %.promoted = load i64, i64* inttoptr (i64 139761557327088 to i64*), align
+ 16
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:28 within `h`
   %0 = add i64 %.promoted, 10000
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:29 within `h`
 ; ┌ @ Base.jl within `setproperty!`
-   store i64 %0, i64* inttoptr (i64 139785926445312 to i64*), align 256
+   store i64 %0, i64* inttoptr (i64 139761557327088 to i64*), align 16
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:30 within `h`
@@ -673,7 +673,7 @@ 
Next Level Up: Multithreading

 
 
 
-2.785 ns (0 allocations: 0 bytes)
+3.095 ns (0 allocations: 0 bytes)
 

 
 
@@ -686,13 +686,13 @@ 
Next Level Up: Multithreading

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:3 within `h2`
-define void @julia_h2_10472() #0 {
+define void @julia_h2_10460() #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:6 within `h2`
 ; ┌ @ refvalue.jl:59 within `getindex`
 ; │┌ @ Base.jl:37 within `getproperty`
-    %0 = load i64, i64* inttoptr (i64 139787416178784 to i64*), align 32
+    %0 = load i64, i64* inttoptr (i64 139762055925984 to i64*), align 32
 ; └└
 ; ┌ @ range.jl:897 within `iterate`
 ; │┌ @ range.jl:672 within `isempty`
@@ -703,15 +703,15 @@ 
Next Level Up: Multithreading

   br i1 %1, label %L34, label %L18.preheader
 
 L18.preheader:                                    ; preds = %top
-  %.promoted = load i64, i64* inttoptr (i64 139785926445312 to i64*), align
- 256
+  %.promoted = load i64, i64* inttoptr (i64 139761557327088 to i64*), align
+ 16
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:8 within `h2`
   %2 = add i64 %.promoted, %0
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:7 within `h2`
 ; ┌ @ Base.jl within `setproperty!`
-   store i64 %2, i64* inttoptr (i64 139785926445312 to i64*), align 256
+   store i64 %2, i64* inttoptr (i64 139761557327088 to i64*), align 16
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:8 within `h2`
@@ -742,7 +742,7 @@ 
Next Level Up: Multithreading

 
 
 
-4.939 ns (0 allocations: 0 bytes)
+4.628 ns (0 allocations: 0 bytes)
 

 
 
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_12_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_12_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_12_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_12_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_13_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_13_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_13_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_13_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_14_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_14_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_14_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_14_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_15_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_15_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_15_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_15_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_16_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_16_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_16_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_16_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_19_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_19_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_19_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_19_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_20_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_20_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_20_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_20_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_7_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_7_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_7_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_7_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_8_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_8_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_8_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_8_1.png
diff --git a/_weave/lecture07/jl_hCDNAD/discretizing_odes_9_1.png b/_weave/lecture07/jl_77KvXW/discretizing_odes_9_1.png
similarity index 100%
rename from _weave/lecture07/jl_hCDNAD/discretizing_odes_9_1.png
rename to _weave/lecture07/jl_77KvXW/discretizing_odes_9_1.png
diff --git a/_weave/lecture08/automatic_differentiation.jmd b/_weave/lecture08/automatic_differentiation.jmd
index 5c0c9711..11e24ac5 100644
--- a/_weave/lecture08/automatic_differentiation.jmd
+++ b/_weave/lecture08/automatic_differentiation.jmd
@@ -585,7 +585,7 @@ $f(d) = f(d_0) + Je_1 \epsilon_1 + \ldots + Je_n \epsilon_n$
 
 computes all columns of the Jacobian simultaneously.
 
-### Array of Structs Representation
+### Struct of Arrays Representation
 
 Instead of thinking about a vector of dual numbers, thus we can instead think of
 dual numbers with vectors for the components. But if there are vectors for the
@@ -598,7 +598,7 @@ $$\Sigma = \begin{bmatrix}
         d_{11} & d_{12} & \cdots & d_{1n} \\
         d_{21} & d_{22} &  & \vdots \\
         \vdots & & \ddots & \vdots \\
-        d_{m1} & \hdots & \hdots & d_{mn}
+        d_{m1} & \cdots & \cdots & d_{mn}
     \end{bmatrix}$$
          
 $$\epsilon=[\epsilon_1,\epsilon_2,\ldots,\epsilon_m]$$
diff --git a/_weave/lecture08/automatic_differentiation/index.html b/_weave/lecture08/automatic_differentiation/index.html
index 711534a1..a76a4fda 100644
--- a/_weave/lecture08/automatic_differentiation/index.html
+++ b/_weave/lecture08/automatic_differentiation/index.html
@@ -18,9 +18,9 @@ 
Forward-Mode Automatic Differentiation (AD) via High Dimensiona
 ϵ2 = (1+ϵ) - 1
 (ϵ - ϵ2)
 
 
-ϵ = 3.873035531141372e-11
-1 + ϵ = 1.0000000000387304
-3.0527602113762817e-18
+ϵ = 6.736227852961307e-11
+1 + ϵ = 1.0000000000673623
+-5.940030844987341e-17
 
 
See how $\epsilon$ is only rebuilt at accuracy around $10^{-16}$ and thus we only keep around 6 digits of accuracy when it's generated at the size of around $10^{-10}$!
 
Finite Differencing and Numerical Stability
 
To start understanding how to compute derivatives on a computer, we start with finite differencing. For finite differencing, recall that the definition of the derivative is:
 
\[ f'(x) = \lim_{\epsilon \rightarrow 0} \frac{f(x+\epsilon)-f(x)}{\epsilon} \]
 
Finite differencing directly follows from this definition by choosing a small $\epsilon$. However, choosing a good $\epsilon$ is very difficult. If $\epsilon$ is too large than there is error since this definition is asymptotic. However, if $\epsilon$ is too small, you receive roundoff error. To understand why you would get roundoff error, recall that floating point error is relative, and can essentially store 16 digits of accuracy. So let's say we choose $\epsilon = 10^{-6}$. Then $f(x+\epsilon) - f(x)$ is roughly the same in the first 6 digits, meaning that after the subtraction there is only 10 digits of accuracy, and then dividing by $10^{-6}$ simply brings those 10 digits back up to the correct relative size.
 
 
This means that we want to choose $\epsilon$ small enough that the $\mathcal{O}(\epsilon^2)$ error of the truncation is balanced by the $O(1/\epsilon)$ roundoff error. Under some minor assumptions, one can argue that the average best point is $\sqrt(E)$, where E is machine epsilon
 
 @show eps(Float64)
 @show sqrt(eps(Float64))
@@ -95,17 +95,17 @@ 
Forward-Mode Automatic Differentiation (AD) via High Dimensiona
 add(a, b)
 @btime add($(Ref(a))[], $(Ref(b))[])
 
 
-3.396 ns (0 allocations: 0 bytes)
+3.105 ns (0 allocations: 0 bytes)
 Dual{Int64}(4, 6)
 
 
It seems like we have lost no performance.
 
 @code_native add(1, 2, 3, 4)
 
 
 .text
 	.file	"add"
-	.globl	julia_add_13194                 # -- Begin function julia_add_13194
+	.globl	julia_add_13182                 # -- Begin function julia_add_13182
 	.p2align	4, 0x90
-	.type	julia_add_13194,@function
-julia_add_13194:                        # @julia_add_13194
+	.type	julia_add_13182,@function
+julia_add_13182:                        # @julia_add_13182
 ; ┌ @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture08/automatic_diff
 erentiation.jmd:2 within `add`
 # %bb.0:                                # %top
@@ -121,7 +121,7 @@ 
Forward-Mode Automatic Differentiation (AD) via High Dimensiona
 	pop	rbp
 	ret
 .Lfunc_end0:
-	.size	julia_add_13194, .Lfunc_end0-julia_add_13194
+	.size	julia_add_13182, .Lfunc_end0-julia_add_13182
 ; â””
                                         # -- End function
 	.section	".note.GNU-stack","",@progbits
@@ -130,10 +130,10 @@ 
Forward-Mode Automatic Differentiation (AD) via High Dimensiona
 
 
 .text
 	.file	"add"
-	.globl	julia_add_13196                 # -- Begin function julia_add_13196
+	.globl	julia_add_13184                 # -- Begin function julia_add_13184
 	.p2align	4, 0x90
-	.type	julia_add_13196,@function
-julia_add_13196:                        # @julia_add_13196
+	.type	julia_add_13184,@function
+julia_add_13184:                        # @julia_add_13184
 ; ┌ @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture08/automatic_diff
 erentiation.jmd:5 within `add`
 # %bb.0:                                # %top
@@ -152,7 +152,7 @@ 
Forward-Mode Automatic Differentiation (AD) via High Dimensiona
 	pop	rbp
 	ret
 .Lfunc_end0:
-	.size	julia_add_13196, .Lfunc_end0-julia_add_13196
+	.size	julia_add_13184, .Lfunc_end0-julia_add_13184
 ; └└└
                                         # -- End function
 	.section	".note.GNU-stack","",@progbits
@@ -274,7 +274,7 @@ 
Forward-Mode Automatic Differentiation (AD) via High Dimensiona
 2-element Vector{Int64}:
  6
  2
-
 
Directional derivative and gradient of functions $f: \mathbb{R}^n \to \mathbb{R}$
 
For a function $f: \mathbb{R}^n \to \mathbb{R}$ the basic operation is the directional derivative:
 
\[ \lim_{\epsilon \to 0} \frac{f(\mathbf{x} + \epsilon \mathbf{v}) - f(\mathbf{x})}{\epsilon} = [\nabla f(\mathbf{x})] \cdot \mathbf{v}, \]
 
where $\epsilon$ is still a single dimension and $\nabla f(\mathbf{x})$ is the direction in which we calculate.
 
We can directly do this using the same simple Dual numbers as above, using the same $\epsilon$, e.g.
 
\[ f(x, y) = x^2 \sin(y) \]
 
\[ \begin{align} f(x_0 + a\epsilon, y_0 + b\epsilon) &= (x_0 + a\epsilon)^2 \sin(y_0 + b\epsilon) \\ &= x_0^2 \sin(y_0) + \epsilon[2ax_0 \sin(y_0) + x_0^2 b \cos(y_0)] + o(\epsilon) \end{align} \]
 
so we have indeed calculated $\nabla f(x_0, y_0) \cdot \mathbf{v},$ where $\mathbf{v} = (a, b)$ are the components that we put into the derivative component of the Dual numbers.
 
If we wish to calculate the directional derivative in another direction, we could repeat the calculation with a different $\mathbf{v}$. A better solution is to use another independent epsilon $\epsilon$, expanding $x = x_0 + a_1 \epsilon_1 + a_2 \epsilon_2$ and putting $\epsilon_1 \epsilon_2 = 0$.
 
In particular, if we wish to calculate the gradient itself, $\nabla f(x_0, y_0)$, we need to calculate both partial derivatives, which corresponds to two directional derivatives, in the directions $(1, 0)$ and $(0, 1)$, respectively.
 
Forward-Mode AD as jvp
 
Note that another representation of the directional derivative is $f'(x)v$, where $f'(x)$ is the Jacobian or total derivative of $f$ at $x$. To see the equivalence of this to a directional derivative, write it out in the standard basis:
 
\[ w_i = \sum_{j}^{m} J_{ij} v_{j} \]
 
Now write out what $J$ means and we see that:
 
\[ w_i = \sum_j^{m} \frac{df_i}{dx_j} v_j = \nabla f_i(x) \cdot v \]
 
The primitive action of forward-mode AD is $f'(x)v$!
 
This is also known as a Jacobian-vector product, or jvp for short.
 
We can thus represent vector calculus with multidimensional dual numbers as follows. Let $d =[x,y]$, the vector of dual numbers. We can instead represent this as:
 
\[ d = d_0 + v_1 \epsilon_1 + v_2 \epsilon_2 \]
 
where $d_0$ is the primal vector $[x_0,y_0]$ and the $v_i$ are the vectors for the dual directions. If you work out this algebra, then note that a single application of $f$ to a multidimensional dual number calculates:
 
\[ f(d) = f(d_0) + f'(d_0)v_1 \epsilon_1 + f'(d_0)v_2 \epsilon_2 \]
 
i.e. it calculates the result of $f(x,y)$ and two separate directional derivatives. Note that because the information about $f(d_0)$ is shared between the calculations, this is more efficient than doing multiple applications of $f$. And of course, this is then generalized to $m$ many directional derivatives at once by:
 
\[ d = d_0 + v_1 \epsilon_1 + v_2 \epsilon_2 + \ldots + v_m \epsilon_m \]
 
Jacobian
 
For a function $f: \mathbb{R}^n \to \mathbb{R}^m$, we reduce (conceptually, although not necessarily in code) to its component functions $f_i: \mathbb{R}^n \to \mathbb{R}$, where $f(x) = (f_1(x), f_2(x), \ldots, f_m(x))$.
 
Then
 
\[ \begin{align} f(x + \epsilon v) &= (f_1(x + \epsilon v), \ldots, f_m(x + \epsilon v)) \\ &= (f_1(x) + \epsilon[\nabla f_1(x) \cdot v], \dots, f_m(x) + \epsilon[\nabla f_m(x) \cdot v] \\ &= f(x) + [f'(x) \cdot v] \epsilon, \end{align} \]
 
To calculate the complete Jacobian, we calculate these directional derivatives in the $n$ different directions of the basis vectors, i.e. if
 
\[ d = d_0 + e_1 \epsilon_1 + \ldots + e_n \epsilon_n \]
 
for $e_i$ the $i$th basis vector, then
 
\[ f(d) = f(d_0) + Je_1 \epsilon_1 + \ldots + Je_n \epsilon_n \]
 
computes all columns of the Jacobian simultaneously.
 
Array of Structs Representation
 
Instead of thinking about a vector of dual numbers, thus we can instead think of dual numbers with vectors for the components. But if there are vectors for the components, then we can think of the grouping of dual components as a matrix. Thus define our multidimensional multi-partial dual number as:
 
\[ D_0 = [d_1,d_2,d_3,\ldots,d_n] \]
 
\[ \Sigma = \begin{bmatrix} d_{11} & d_{12} & \cdots & d_{1n} \\ d_{21} & d_{22} & & \vdots \\ \vdots & & \ddots & \vdots \\ d_{m1} & \hdots & \hdots & d_{mn} \end{bmatrix} \]
 
\[ \epsilon=[\epsilon_1,\epsilon_2,\ldots,\epsilon_m] \]
 
\[ D = D_0 + \Sigma \epsilon \]
 
where $D_0$ is a vector in $\mathbb{R}^n$, $\epsilon$ is a vector of dimensional signifiers and $\Sigma$ is a matrix in $\mathbb{R}^{n \times m}$ where $m$ is the number of concurrent differentiation dimensions. Each row of this is a dual number, but now we can use this to easily define higher dimensional primitives.
 
For example, let $f(x) = Ax$, matrix multiplication. Then, we can show with our dual number arithmetic that:
 
\[ f(D) = A*D_0 + A*\Sigma*\epsilon \]
 
is how one would compute the value of $f(D_0)$ and the derivative $f'(D_0)$ in all directions signified by the columns of $\Sigma$ simultaneously. Using multidimensional Taylor series expansions and doing the manipulations like before indeed implies that the arithmetic on this object should follow:
 
\[ f(D) = f(D_0) + f'(D_0)\Sigma \epsilon \]
 
where $f'$ is the total derivative or the Jacobian of $f$. This then allows our system to be highly efficient by allowing the definition of multidimensional functions, like linear algebra, to be primitives of multi-directional derivatives.
 
Higher derivatives
 
The above techniques can be extended to higher derivatives by adding more terms to the Taylor polynomial, e.g.
 
\[ f(a + \epsilon) = f(a) + \epsilon f'(a) + \frac{1}{2} \epsilon^2 f''(a) + o(\epsilon^2). \]
 
We treat this as a degree-2 (or degree-$n$, in general) polynomial and do polynomial arithmetic to calculate the new polynomials. The coefficients of powers of $\epsilon$ then give the higher-order derivatives.
 
For example, for a function $f: \mathbb{R}^n \to \mathbb{R}$ we have
 
\[ f(x + \epsilon v) = f(x) + \epsilon \left[ \sum_i (\partial_i f)(x) v_i \right] + \frac{1}{2}\epsilon^2 \left[ \sum_i \sum_j (\partial_{i,j} f) v_i v_j \right] \]
 
using Dual numbers with a single $\epsilon$ component. In this way we can compute coefficients of the (symmetric) Hessian matrix.
 
Application: solving nonlinear equations using the Newton method
 
As an application, we will see how to solve nonlinear equations of the form $f(x) = 0$ for functions $f: \mathbb{R}^n \to \mathbb{R}^n$.
 
Since in general we cannot do anything with nonlinearity, we try to reduce it (approximate it) with something linear. Furthermore, in general we know that it is not possible to solve nonlinear equations in closed form (even for polynomials of degree $\ge 5$), so we will need some kind of iterative method.
 
We start from an initial guess $x_0$. The idea of the Newton method is to follow the tangent line to the function $f$ at the point $x_0$ and find where it intersects the $x$-axis; this will give the next iterate $x_1$.
 
Algebraically, we want to solve $f(x_1) = 0$. Suppose that $x_1 = x_0 + \delta$ for some $\delta$ that is currently unknown and which we wish to calculate.
 
Assuming $\delta$ is small, we can expand:
 
\[ f(x_1) = f(x_0 + \delta) = f(x_0) + Df(x_0) \cdot \delta + \mathcal{O}(\| \delta \|^2). \]
 
Since we wish to solve
 
\[ f(x_0 + \delta) \simeq 0, \]
 
we put
 
\[ f(x_0) + Df(x_0) \cdot \delta = 0, \]
 
so that mathematically we have
 
\[ \delta = -[Df(x_0)]^{-1} \cdot f(x_0). \]
 
Computationally we prefer to solve the matrix equation
 
\[ J \delta = -f(x_0), \]
 
where $J := Df(x_0)$ is the Jacobian of the function; Julia uses the syntax \ ("backslash") for solving linear systems in an efficient way:
 
+
 
Directional derivative and gradient of functions $f: \mathbb{R}^n \to \mathbb{R}$
 
For a function $f: \mathbb{R}^n \to \mathbb{R}$ the basic operation is the directional derivative:
 
\[ \lim_{\epsilon \to 0} \frac{f(\mathbf{x} + \epsilon \mathbf{v}) - f(\mathbf{x})}{\epsilon} = [\nabla f(\mathbf{x})] \cdot \mathbf{v}, \]
 
where $\epsilon$ is still a single dimension and $\nabla f(\mathbf{x})$ is the direction in which we calculate.
 
We can directly do this using the same simple Dual numbers as above, using the same $\epsilon$, e.g.
 
\[ f(x, y) = x^2 \sin(y) \]
 
\[ \begin{align} f(x_0 + a\epsilon, y_0 + b\epsilon) &= (x_0 + a\epsilon)^2 \sin(y_0 + b\epsilon) \\ &= x_0^2 \sin(y_0) + \epsilon[2ax_0 \sin(y_0) + x_0^2 b \cos(y_0)] + o(\epsilon) \end{align} \]
 
so we have indeed calculated $\nabla f(x_0, y_0) \cdot \mathbf{v},$ where $\mathbf{v} = (a, b)$ are the components that we put into the derivative component of the Dual numbers.
 
If we wish to calculate the directional derivative in another direction, we could repeat the calculation with a different $\mathbf{v}$. A better solution is to use another independent epsilon $\epsilon$, expanding $x = x_0 + a_1 \epsilon_1 + a_2 \epsilon_2$ and putting $\epsilon_1 \epsilon_2 = 0$.
 
In particular, if we wish to calculate the gradient itself, $\nabla f(x_0, y_0)$, we need to calculate both partial derivatives, which corresponds to two directional derivatives, in the directions $(1, 0)$ and $(0, 1)$, respectively.
 
Forward-Mode AD as jvp
 
Note that another representation of the directional derivative is $f'(x)v$, where $f'(x)$ is the Jacobian or total derivative of $f$ at $x$. To see the equivalence of this to a directional derivative, write it out in the standard basis:
 
\[ w_i = \sum_{j}^{m} J_{ij} v_{j} \]
 
Now write out what $J$ means and we see that:
 
\[ w_i = \sum_j^{m} \frac{df_i}{dx_j} v_j = \nabla f_i(x) \cdot v \]
 
The primitive action of forward-mode AD is $f'(x)v$!
 
This is also known as a Jacobian-vector product, or jvp for short.
 
We can thus represent vector calculus with multidimensional dual numbers as follows. Let $d =[x,y]$, the vector of dual numbers. We can instead represent this as:
 
\[ d = d_0 + v_1 \epsilon_1 + v_2 \epsilon_2 \]
 
where $d_0$ is the primal vector $[x_0,y_0]$ and the $v_i$ are the vectors for the dual directions. If you work out this algebra, then note that a single application of $f$ to a multidimensional dual number calculates:
 
\[ f(d) = f(d_0) + f'(d_0)v_1 \epsilon_1 + f'(d_0)v_2 \epsilon_2 \]
 
i.e. it calculates the result of $f(x,y)$ and two separate directional derivatives. Note that because the information about $f(d_0)$ is shared between the calculations, this is more efficient than doing multiple applications of $f$. And of course, this is then generalized to $m$ many directional derivatives at once by:
 
\[ d = d_0 + v_1 \epsilon_1 + v_2 \epsilon_2 + \ldots + v_m \epsilon_m \]
 
Jacobian
 
For a function $f: \mathbb{R}^n \to \mathbb{R}^m$, we reduce (conceptually, although not necessarily in code) to its component functions $f_i: \mathbb{R}^n \to \mathbb{R}$, where $f(x) = (f_1(x), f_2(x), \ldots, f_m(x))$.
 
Then
 
\[ \begin{align} f(x + \epsilon v) &= (f_1(x + \epsilon v), \ldots, f_m(x + \epsilon v)) \\ &= (f_1(x) + \epsilon[\nabla f_1(x) \cdot v], \dots, f_m(x) + \epsilon[\nabla f_m(x) \cdot v] \\ &= f(x) + [f'(x) \cdot v] \epsilon, \end{align} \]
 
To calculate the complete Jacobian, we calculate these directional derivatives in the $n$ different directions of the basis vectors, i.e. if
 
\[ d = d_0 + e_1 \epsilon_1 + \ldots + e_n \epsilon_n \]
 
for $e_i$ the $i$th basis vector, then
 
\[ f(d) = f(d_0) + Je_1 \epsilon_1 + \ldots + Je_n \epsilon_n \]
 
computes all columns of the Jacobian simultaneously.
 
Struct of Arrays Representation
 
Instead of thinking about a vector of dual numbers, thus we can instead think of dual numbers with vectors for the components. But if there are vectors for the components, then we can think of the grouping of dual components as a matrix. Thus define our multidimensional multi-partial dual number as:
 
\[ D_0 = [d_1,d_2,d_3,\ldots,d_n] \]
 
\[ \Sigma = \begin{bmatrix} d_{11} & d_{12} & \cdots & d_{1n} \\ d_{21} & d_{22} & & \vdots \\ \vdots & & \ddots & \vdots \\ d_{m1} & \cdots & \cdots & d_{mn} \end{bmatrix} \]
 
\[ \epsilon=[\epsilon_1,\epsilon_2,\ldots,\epsilon_m] \]
 
\[ D = D_0 + \Sigma \epsilon \]
 
where $D_0$ is a vector in $\mathbb{R}^n$, $\epsilon$ is a vector of dimensional signifiers and $\Sigma$ is a matrix in $\mathbb{R}^{n \times m}$ where $m$ is the number of concurrent differentiation dimensions. Each row of this is a dual number, but now we can use this to easily define higher dimensional primitives.
 
For example, let $f(x) = Ax$, matrix multiplication. Then, we can show with our dual number arithmetic that:
 
\[ f(D) = A*D_0 + A*\Sigma*\epsilon \]
 
is how one would compute the value of $f(D_0)$ and the derivative $f'(D_0)$ in all directions signified by the columns of $\Sigma$ simultaneously. Using multidimensional Taylor series expansions and doing the manipulations like before indeed implies that the arithmetic on this object should follow:
 
\[ f(D) = f(D_0) + f'(D_0)\Sigma \epsilon \]
 
where $f'$ is the total derivative or the Jacobian of $f$. This then allows our system to be highly efficient by allowing the definition of multidimensional functions, like linear algebra, to be primitives of multi-directional derivatives.
 
Higher derivatives
 
The above techniques can be extended to higher derivatives by adding more terms to the Taylor polynomial, e.g.
 
\[ f(a + \epsilon) = f(a) + \epsilon f'(a) + \frac{1}{2} \epsilon^2 f''(a) + o(\epsilon^2). \]
 
We treat this as a degree-2 (or degree-$n$, in general) polynomial and do polynomial arithmetic to calculate the new polynomials. The coefficients of powers of $\epsilon$ then give the higher-order derivatives.
 
For example, for a function $f: \mathbb{R}^n \to \mathbb{R}$ we have
 
\[ f(x + \epsilon v) = f(x) + \epsilon \left[ \sum_i (\partial_i f)(x) v_i \right] + \frac{1}{2}\epsilon^2 \left[ \sum_i \sum_j (\partial_{i,j} f) v_i v_j \right] \]
 
using Dual numbers with a single $\epsilon$ component. In this way we can compute coefficients of the (symmetric) Hessian matrix.
 
Application: solving nonlinear equations using the Newton method
 
As an application, we will see how to solve nonlinear equations of the form $f(x) = 0$ for functions $f: \mathbb{R}^n \to \mathbb{R}^n$.
 
Since in general we cannot do anything with nonlinearity, we try to reduce it (approximate it) with something linear. Furthermore, in general we know that it is not possible to solve nonlinear equations in closed form (even for polynomials of degree $\ge 5$), so we will need some kind of iterative method.
 
We start from an initial guess $x_0$. The idea of the Newton method is to follow the tangent line to the function $f$ at the point $x_0$ and find where it intersects the $x$-axis; this will give the next iterate $x_1$.
 
Algebraically, we want to solve $f(x_1) = 0$. Suppose that $x_1 = x_0 + \delta$ for some $\delta$ that is currently unknown and which we wish to calculate.
 
Assuming $\delta$ is small, we can expand:
 
\[ f(x_1) = f(x_0 + \delta) = f(x_0) + Df(x_0) \cdot \delta + \mathcal{O}(\| \delta \|^2). \]
 
Since we wish to solve
 
\[ f(x_0 + \delta) \simeq 0, \]
 
we put
 
\[ f(x_0) + Df(x_0) \cdot \delta = 0, \]
 
so that mathematically we have
 
\[ \delta = -[Df(x_0)]^{-1} \cdot f(x_0). \]
 
Computationally we prefer to solve the matrix equation
 
\[ J \delta = -f(x_0), \]
 
where $J := Df(x_0)$ is the Jacobian of the function; Julia uses the syntax \ ("backslash") for solving linear systems in an efficient way:
 
 using ForwardDiff, StaticArrays
 
 function newton_step(f, x0)
diff --git a/_weave/lecture16/jl_AHYinz/probabilistic_programming_3_1.png b/_weave/lecture16/jl_AHYinz/probabilistic_programming_3_1.png
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diff --git a/_weave/lecture16/jl_AHYinz/probabilistic_programming_9_1.png b/_weave/lecture16/jl_GcpKod/probabilistic_programming_9_1.png
similarity index 100%
rename from _weave/lecture16/jl_AHYinz/probabilistic_programming_9_1.png
rename to _weave/lecture16/jl_GcpKod/probabilistic_programming_9_1.png
diff --git a/_weave/lecture16/probabilistic_programming/index.html b/_weave/lecture16/probabilistic_programming/index.html
index 2801b464..2454a0c7 100644
--- a/_weave/lecture16/probabilistic_programming/index.html
+++ b/_weave/lecture16/probabilistic_programming/index.html
@@ -24,7 +24,7 @@ 
From Optimization to Probabilistic Programming
 
Chris R
 prob1 = ODEProblem(lotka_volterra,u0,tspan,_θ)
 sol = solve(prob1,Tsit5())
 plot(sol)
-
  
and from which we can get an ensemble of solutions:
 
+
  
and from which we can get an ensemble of solutions:
 
 prob_func = function (prob,i,repeat)
   remake(prob,p=rand.(θ))
 end
@@ -34,21 +34,21 @@ 
From Optimization to Probabilistic Programming
 
Chris R
 
 using DiffEqBase.EnsembleAnalysis
 plot(EnsembleSummary(sol))
-
  
From just a few variables having probabilities, every variable has an induced probability: there is a probability distribution on the integrator states, the output at time t_i, etc.
 
Bayesian Estimation with Point Estimates: Bayes' Rule, Maximum Likelihood, and MAP
 
Recall from our previous studies that the difficult part of modeling is not necessarily the forward modeling approach, rather it's the incorporation of data or the estimation problem that is difficult. When your variables are now random distributions, how do you "fit" them?
 
The answer comes from Bayes' rule, which is the following. Assume you had a prior distribution $p(\theta)$ for the probability that $X$ is a given value $\theta$. Then the posterior probability distribution, $p(\theta|D)$, or the distribution which is updated to include data, is given by:
 
\[ p(\theta|D) = \frac{p(D|\theta)p(\theta)}{\int_\Omega p(D|\theta)p(\theta)d\theta} \]
 
The scaling factor on the denominator is simply a constant to make the distribution integrate 1 (so that the resulting function is a probability distribution!). The numerator is simply the prior distribution multiplied by the likelihood of seeing the data given the value of the random variable. The prior distribution must be given but notice that the likelihood has another name: the likelihood is the model.
 
The reason why it's the same thing is because the model is what tells you the expected outcomes given a value of the random variable, and your data is on an expected outcome! However, the likelihood encodes a little bit more information in that it again is a distribution and not a point estimate. We need to make a choice for our measurement distribution on our model's results.
 
Quick Question: Why is this referred to as measurement noise? Why is it not process noise?
 
A common choice for the measurement distribution is the Normal distribution. This comes from the Central Limit Theorem (CLT) which essentially states that, given enough interacting mechanisms, the average values of things "tend to become normally distributed". The true statement of the CLT is much more complex, but that is a decent working definition for practical use. The normal distribution is defined by two parameters, $\mu$ and $\sigma$, and is given by the following function:
 
\[ f(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2\sigma^2}\right) \]
 
This is a bell curve centered at $\mu$ with a variance of $\sigma$. Our best guess for the output, i.e. the model's prediction, should be the average measurement, meaning that $\mu$ is the result from the simulator. $\sigma$ is a parameter for how much measurement error we expect (some intuition on $\sigma$ will come soon).
 
Let's return to thinking about the ODE example. In this case, we have $\theta$ as a vector of random variables. This means that $u(t;\theta)$ is a random variable for the ODE $u'= ...$'s solution at a given point in time $t$. If we have a measurement at a time $t_i$ and assume our measurement noise is normally distributed with some constant measurement noise $\sigma$, then the likelihood of our data would be $f(x_i;u(t_i;\theta),\sigma)$ at each data point $(t_i,x_i)$. From probability we know that seeing the composition of events is given by the multiplication of probabilities, so the probability of seeing the full dataset given observations $D = (t_i,x_i)$ along the timeseries is:
 
\[ p(D|\theta) = \prod_i f(x_i;u(t_i;\theta),\sigma) \]
 
This can be read as: solve the model with the given parameters, and the probability of having seen the measurement is thus given by a product of normal distribution calculations. Note that in many cases the product is not numerically stable (and grows exponentially), and so the likelihood is transformed to the log-likelihood. To get this expression, we take the log of both sides and notice that the product becomes a summation, and thus:
 
\[ \begin{align} \log p(D|\theta) &= \sum_i \log f(x_i;u(t_i;\theta),\sigma)\\ &= \frac{N}{\log(\sqrt{2\pi}\sigma)} + \frac{1}{2\sigma^2} \sum_i -(x_i - u(t_i; \theta))^2 \end{align} \]
 
Notice that maximizing this log-likelihood is equivalent to minimizing the L2 norm of the solution against the data!. Thus we can see a few things:
 
     
  1. Previous parameter estimation by minimizing a norm against data can be seen as maximum likelihood with some measurement distribution. L2 norm corresponds to assuming measurement noise is normally distributed and all of the measurements have the same error variance.
     
  2. By the same derivation, having different error variances with normally distributed errors is equivalent to doing weighted L2 estimation.
     
 
This reformulation (generalization?) to likelihoods of probability distributions is known as maximum likelihood estimation (MLE), but is equivalent to our previous forms of parameter estimation using point estimates against data. However, this calculation is ignoring Bayes' rule, and is thus not finding the parameters which have the highest probability. To do that, we need to go back to Bayes' rule which states that:
 
\[ \log p(\theta|D) = \log p(D|\theta) + \log p(\theta) - C \]
 
Thus, maximizing the log-likelihood is "almost" the same as finding the most probable parameters, except that we need to add weights given $\log p(\theta)$ from our prior distribution! If we assume our prior distribution is flat, like a uniform distribution, then we have a non-informative prior and the maximum posterior point matches that of the maximum likelihood estimation. However, this formulation allows us to get point estimates in a way that takes into account prior knowledge, and is call maximum a posteriori estimation (MAP).
 
Bayesian Estimation of Posterior Distributions with Monte Carlo
 
The previous discussion still solely focused on getting point estimates for the most probable parameters. However, what if we wanted to find the distributions of the parameters, i.e. the full $p(D|\theta)$? Outside of very few small models, this cannot be done analytically and is thus the basic problem of probabilistic programming. There are two general approaches:
 
     
  1. Sampling-based approaches. Sample parameters $\theta_i$ in such a manner that the array $[\theta_i]$ converges to an array sampled from the true distribution, and thus with enough samples one can capture the distribution numerically.
     
  2. Variational inference. Find some way to represent the probability distribution and push forward the distributions at every step of the program.
     
 
Recovering Distributions from Sampled Points
 
It's clear from above that if you have a distribution, like Normal(5,1), that you can sample from the distribution to get an array of values which follow the distribution. However, in order for the following sampling approaches to make sense, we need to see how to recover a distribution from discrete samples. So let's say you had a bunch of normally distributed points:
 
+
  
From just a few variables having probabilities, every variable has an induced probability: there is a probability distribution on the integrator states, the output at time t_i, etc.
 
Bayesian Estimation with Point Estimates: Bayes' Rule, Maximum Likelihood, and MAP
 
Recall from our previous studies that the difficult part of modeling is not necessarily the forward modeling approach, rather it's the incorporation of data or the estimation problem that is difficult. When your variables are now random distributions, how do you "fit" them?
 
The answer comes from Bayes' rule, which is the following. Assume you had a prior distribution $p(\theta)$ for the probability that $X$ is a given value $\theta$. Then the posterior probability distribution, $p(\theta|D)$, or the distribution which is updated to include data, is given by:
 
\[ p(\theta|D) = \frac{p(D|\theta)p(\theta)}{\int_\Omega p(D|\theta)p(\theta)d\theta} \]
 
The scaling factor on the denominator is simply a constant to make the distribution integrate 1 (so that the resulting function is a probability distribution!). The numerator is simply the prior distribution multiplied by the likelihood of seeing the data given the value of the random variable. The prior distribution must be given but notice that the likelihood has another name: the likelihood is the model.
 
The reason why it's the same thing is because the model is what tells you the expected outcomes given a value of the random variable, and your data is on an expected outcome! However, the likelihood encodes a little bit more information in that it again is a distribution and not a point estimate. We need to make a choice for our measurement distribution on our model's results.
 
Quick Question: Why is this referred to as measurement noise? Why is it not process noise?
 
A common choice for the measurement distribution is the Normal distribution. This comes from the Central Limit Theorem (CLT) which essentially states that, given enough interacting mechanisms, the average values of things "tend to become normally distributed". The true statement of the CLT is much more complex, but that is a decent working definition for practical use. The normal distribution is defined by two parameters, $\mu$ and $\sigma$, and is given by the following function:
 
\[ f(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2\sigma^2}\right) \]
 
This is a bell curve centered at $\mu$ with a variance of $\sigma$. Our best guess for the output, i.e. the model's prediction, should be the average measurement, meaning that $\mu$ is the result from the simulator. $\sigma$ is a parameter for how much measurement error we expect (some intuition on $\sigma$ will come soon).
 
Let's return to thinking about the ODE example. In this case, we have $\theta$ as a vector of random variables. This means that $u(t;\theta)$ is a random variable for the ODE $u'= ...$'s solution at a given point in time $t$. If we have a measurement at a time $t_i$ and assume our measurement noise is normally distributed with some constant measurement noise $\sigma$, then the likelihood of our data would be $f(x_i;u(t_i;\theta),\sigma)$ at each data point $(t_i,x_i)$. From probability we know that seeing the composition of events is given by the multiplication of probabilities, so the probability of seeing the full dataset given observations $D = (t_i,x_i)$ along the timeseries is:
 
\[ p(D|\theta) = \prod_i f(x_i;u(t_i;\theta),\sigma) \]
 
This can be read as: solve the model with the given parameters, and the probability of having seen the measurement is thus given by a product of normal distribution calculations. Note that in many cases the product is not numerically stable (and grows exponentially), and so the likelihood is transformed to the log-likelihood. To get this expression, we take the log of both sides and notice that the product becomes a summation, and thus:
 
\[ \begin{align} \log p(D|\theta) &= \sum_i \log f(x_i;u(t_i;\theta),\sigma)\\ &= \frac{N}{\log(\sqrt{2\pi}\sigma)} + \frac{1}{2\sigma^2} \sum_i -(x_i - u(t_i; \theta))^2 \end{align} \]
 
Notice that maximizing this log-likelihood is equivalent to minimizing the L2 norm of the solution against the data!. Thus we can see a few things:
 
     
  1. Previous parameter estimation by minimizing a norm against data can be seen as maximum likelihood with some measurement distribution. L2 norm corresponds to assuming measurement noise is normally distributed and all of the measurements have the same error variance.
     
  2. By the same derivation, having different error variances with normally distributed errors is equivalent to doing weighted L2 estimation.
     
 
This reformulation (generalization?) to likelihoods of probability distributions is known as maximum likelihood estimation (MLE), but is equivalent to our previous forms of parameter estimation using point estimates against data. However, this calculation is ignoring Bayes' rule, and is thus not finding the parameters which have the highest probability. To do that, we need to go back to Bayes' rule which states that:
 
\[ \log p(\theta|D) = \log p(D|\theta) + \log p(\theta) - C \]
 
Thus, maximizing the log-likelihood is "almost" the same as finding the most probable parameters, except that we need to add weights given $\log p(\theta)$ from our prior distribution! If we assume our prior distribution is flat, like a uniform distribution, then we have a non-informative prior and the maximum posterior point matches that of the maximum likelihood estimation. However, this formulation allows us to get point estimates in a way that takes into account prior knowledge, and is call maximum a posteriori estimation (MAP).
 
Bayesian Estimation of Posterior Distributions with Monte Carlo
 
The previous discussion still solely focused on getting point estimates for the most probable parameters. However, what if we wanted to find the distributions of the parameters, i.e. the full $p(D|\theta)$? Outside of very few small models, this cannot be done analytically and is thus the basic problem of probabilistic programming. There are two general approaches:
 
     
  1. Sampling-based approaches. Sample parameters $\theta_i$ in such a manner that the array $[\theta_i]$ converges to an array sampled from the true distribution, and thus with enough samples one can capture the distribution numerically.
     
  2. Variational inference. Find some way to represent the probability distribution and push forward the distributions at every step of the program.
     
 
Recovering Distributions from Sampled Points
 
It's clear from above that if you have a distribution, like Normal(5,1), that you can sample from the distribution to get an array of values which follow the distribution. However, in order for the following sampling approaches to make sense, we need to see how to recover a distribution from discrete samples. So let's say you had a bunch of normally distributed points:
 
 X = Normal(5,1)
 x = [rand(X) for i in 1:100]
 scatter(x,[1 for i in 1:100])
-
  
Notice that there are more points in the areas of higher probability. Thus the density of sampled points gives us an estimate for the probability of having points in a given area. We can then count the number of points in a bin and divide by the total number of points in order to get the probability of being in a specific region. This is depicted by a histogram:
 
+
  
Notice that there are more points in the areas of higher probability. Thus the density of sampled points gives us an estimate for the probability of having points in a given area. We can then count the number of points in a bin and divide by the total number of points in order to get the probability of being in a specific region. This is depicted by a histogram:
 
 histogram(x)
-
  
and we see this converges when we get more points:
 
+
  
and we see this converges when we get more points:
 
 histogram([rand(X) for i in 1:10000],normed=true)
 using StatsPlots
 plot!(X,lw=5)
-
  
A continuous form of this is the kernel density estimate, which is essentially a smoothed binning approach.
 
+
  
A continuous form of this is the kernel density estimate, which is essentially a smoothed binning approach.
 
 using KernelDensity
 plot(kde([rand(X) for i in 1:10000]),lw=5)
 plot!(X,lw=5)
-
  
Thus, for the sampling-based approaches, we simply need to arrive at an array which is sampled according to the distribution that we want to estimate, and from that array we can recover the distribution.
 
Sampling Distributions with the Metropolis-Hastings Algorithm
 
The Metropolis-Hastings algorithm is the simplest form of Markov Chain Monte Carlo (MCMC) which gives a way of sampling the $\theta$ distribution. To see how this algorithm works, let's understand the ratio between two points in the posterior probability. If we have $x_i$ and $x_j$, the ratio of the two probabilities would be given by:
 
\[ \frac{p(x_i|D)}{p(x_j|D)} = \frac{p(D|x_i)p(x_i)}{p(D|x_j)p(x_j)} \]
 
(notice that the integration constant cancels). This motivates the idea that all we have to do is ensure we only go to a point $x_j$ from $x_i$ with probability difference that matches that ratio, and over time if we do this between "all points" we will have the right number of "each point" in the distribution (quotes because it's continuous). With a bit more rigour we arrive at the following algorithm:
 
     
  1. Starting at $x_i$, take $x_{i+1}$ from a sampling algorithm $g(x_{i+1}|x_i)$.
     
  2. Calculate $A = \min\left(1,\frac{p(D|x_{i+1})p(x_{i+1})g(x_i|x_{i+1})}{p(D|x_i)p(x_i)g(x_{i+1}|x_i)}\right)$. Notice that if we require $g$ to be symmetric, then this simplifies to the probability ratio $A = \min\left(1,\frac{p(D|x_{i+1})p(x_{i+1})}{p(D|x_i)p(x_i)}\right)$
     
  3. Use a random number to accept the step with a probability $A$. Go back to step 1, incrementing $i$ if accepted, otherwise just repeat.
     
 
I.e, we just walk around the space biasing the acceptance of a step by the factor $\frac{p(x_i|D)}{p(x_j|D)}$ and sooner or later we will have spent the right amount of time in each area, giving the correct distribution.
 
(This can be rigorously proven, and those details are left out.)
 
The Cost of Bayesian Estimation
 
Let's take a quick moment to understand the high cost of Bayesian posterior estimations. While before we were getting point estimates, now we are trying to recover a full probability distribution, and each accept/reject probability calculation requires evaluating the likelihood at some point. Remember, the likelihood is generated by our simulator, and thus every evaluation here is an ODE solver call or a neural network forward pass! This means that to get good distributions, we are solving the ODE hundreds of thousands of times, i.e. even more than when doing parameter estimation! This is something to keep in mind.
 
However, notice that this process is trivially parallelizable. We can just have parallel chains on going, i.e. start 16 processes all doing Metropolis-Hastings, and in the end they are all sampling from the same distribution, so the final array can simply be the pooled results of each chain.
 
Hamiltonian Monte Carlo
 
Metropolis-Hastings is easy to motivate and implement. However, it does not do well in high dimensional spaces because it searches in all directions. For example, it's common for the sampling distribution $g$ to be a multivariable distribution (i.e. normal in all directions). However, high dimensional objects commonly sit on low dimensional manifolds (known as the manifold hypothesis). If that's the case, the most probable set of parameters is something that is low dimensional. For example, parameters may compensate for one another, and so $\theta_1^2 + \theta_2^2 + \theta_3^2 = 1$ might be the manifold on which all of the most probable choices for $\theta$ lie, in which case we need sample on the sphere instead of all of $\mathbb{R}^3$.
 
However, it's quick to see that this will give Metropolis-Hastings some trouble, since it will use a normal distribution around the current point, and thus even if we start on the sphere, it will have a high chance of trying a point not on the sphere in the next round! This can be depicted as:
 
 
Recall that every single rejection is still evaluating the likelihood (since it's calculating an acceptance probability, finding it near zero, rejecting and starting again), and every likelihood call is calling our simulator, and so this is sllllllooooooooooooooowwwwwwwww in high dimensions!
 
What we need to do instead is ensure that we walk along the path of high probability. What we want to do is thus build a vector field that matches our high probability regions
 
 
and follow said vector field (following a vector field is solving what kind of equation?). The first idea one might have is to use the gradient. However, while this idea has the right intentions, the issue is that the gradient of the probability will average out all of the possible probabilities, and will thus flow towards the mode of the distribution:
 
 
To overcome this issue, we look to physical systems and see that a satellite orbiting a planet always nicely stays on some manifold instead of following the gradient:
 
 
The reason why it does is because it has momentum. Recall from basic physics that one way to describe a physical system is through Hamiltonian mechanics, where $H(x,p)$ is the energy associated with the state $(x,p)$ (normally $x$ is location and $p$ is momentum). Due to conservation of energy, the solution of the dynamical equations leads to $H(x,p)$ being constant, and thus the dynamics follow the level sets of $H$. From the Hamiltonian the dynamics of the system are:
 
\[ \begin{align} \frac{dx}{dt} &= \frac{dH}{dp}\\ &= -\frac{dH}{dx} \end{align} \]
 
Here we want our Hamiltonian to be our posterior probability, so that way we stay on the manifold of high probability. This means:
 
\[ H(x,p) = - \log \pi(x,p) \]
 
where $\pi(x,p) = \pi(p|x)\pi(x)$ (where I am now using $pi$ for probability since $p$ is momentum!). So to lift from a probability over parameters to one that includes momentum, we simply need to choose a conditional distribution $\pi(p|x)$. This would mean that
 
\[ \begin{align} H(x,p) &= -log \pi(p|x) - \log \pi(x)\\ &= K(p,x) + V(x) \end{align} \]
 
where $K$ is the kinetic energy and $V$ is the potential. Thus the potential energy is directly given by the posterior calculation, and the kinetic energy is thus a choice that is used to build the correct Hamiltonian. Hamiltonian Monte Carlo methods then dig into good ways to choose the kinetic energy function. This is done at the start (along with the choice of ODE solver time step) in such a way that it maximizes acceptance probabilities.
 
Connections to Differentiable Programming
 
\[ -\frac{dH}{dx} \]
 
requires calculating the gradient of the likelihood function with respect to the parameters, so we are once again using the gradient of our simulator! This means that all of our previous discussion on automatic differentiation and differentiable programming applies to the Hamiltonian Monte Carlo context.
 
There's another thread to follow that transformations of probability distributions are pushforwards of the Jacobian transformations (given the transformation of an integral formula), and this is used when doing variational inference.
 
Symplectic and Geometric Integration
 
One way to integrate the system of ODEs which result from the Hamiltonian system is to convert it to a system of first order ODEs and solve it directly. However, this loses information and can result in drift. This is demonstrated by looking at the long time solution of the pendulum:
 
+
  
Thus, for the sampling-based approaches, we simply need to arrive at an array which is sampled according to the distribution that we want to estimate, and from that array we can recover the distribution.
 
Sampling Distributions with the Metropolis-Hastings Algorithm
 
The Metropolis-Hastings algorithm is the simplest form of Markov Chain Monte Carlo (MCMC) which gives a way of sampling the $\theta$ distribution. To see how this algorithm works, let's understand the ratio between two points in the posterior probability. If we have $x_i$ and $x_j$, the ratio of the two probabilities would be given by:
 
\[ \frac{p(x_i|D)}{p(x_j|D)} = \frac{p(D|x_i)p(x_i)}{p(D|x_j)p(x_j)} \]
 
(notice that the integration constant cancels). This motivates the idea that all we have to do is ensure we only go to a point $x_j$ from $x_i$ with probability difference that matches that ratio, and over time if we do this between "all points" we will have the right number of "each point" in the distribution (quotes because it's continuous). With a bit more rigour we arrive at the following algorithm:
 
     
  1. Starting at $x_i$, take $x_{i+1}$ from a sampling algorithm $g(x_{i+1}|x_i)$.
     
  2. Calculate $A = \min\left(1,\frac{p(D|x_{i+1})p(x_{i+1})g(x_i|x_{i+1})}{p(D|x_i)p(x_i)g(x_{i+1}|x_i)}\right)$. Notice that if we require $g$ to be symmetric, then this simplifies to the probability ratio $A = \min\left(1,\frac{p(D|x_{i+1})p(x_{i+1})}{p(D|x_i)p(x_i)}\right)$
     
  3. Use a random number to accept the step with a probability $A$. Go back to step 1, incrementing $i$ if accepted, otherwise just repeat.
     
 
I.e, we just walk around the space biasing the acceptance of a step by the factor $\frac{p(x_i|D)}{p(x_j|D)}$ and sooner or later we will have spent the right amount of time in each area, giving the correct distribution.
 
(This can be rigorously proven, and those details are left out.)
 
The Cost of Bayesian Estimation
 
Let's take a quick moment to understand the high cost of Bayesian posterior estimations. While before we were getting point estimates, now we are trying to recover a full probability distribution, and each accept/reject probability calculation requires evaluating the likelihood at some point. Remember, the likelihood is generated by our simulator, and thus every evaluation here is an ODE solver call or a neural network forward pass! This means that to get good distributions, we are solving the ODE hundreds of thousands of times, i.e. even more than when doing parameter estimation! This is something to keep in mind.
 
However, notice that this process is trivially parallelizable. We can just have parallel chains on going, i.e. start 16 processes all doing Metropolis-Hastings, and in the end they are all sampling from the same distribution, so the final array can simply be the pooled results of each chain.
 
Hamiltonian Monte Carlo
 
Metropolis-Hastings is easy to motivate and implement. However, it does not do well in high dimensional spaces because it searches in all directions. For example, it's common for the sampling distribution $g$ to be a multivariable distribution (i.e. normal in all directions). However, high dimensional objects commonly sit on low dimensional manifolds (known as the manifold hypothesis). If that's the case, the most probable set of parameters is something that is low dimensional. For example, parameters may compensate for one another, and so $\theta_1^2 + \theta_2^2 + \theta_3^2 = 1$ might be the manifold on which all of the most probable choices for $\theta$ lie, in which case we need sample on the sphere instead of all of $\mathbb{R}^3$.
 
However, it's quick to see that this will give Metropolis-Hastings some trouble, since it will use a normal distribution around the current point, and thus even if we start on the sphere, it will have a high chance of trying a point not on the sphere in the next round! This can be depicted as:
 
 
Recall that every single rejection is still evaluating the likelihood (since it's calculating an acceptance probability, finding it near zero, rejecting and starting again), and every likelihood call is calling our simulator, and so this is sllllllooooooooooooooowwwwwwwww in high dimensions!
 
What we need to do instead is ensure that we walk along the path of high probability. What we want to do is thus build a vector field that matches our high probability regions
 
 
and follow said vector field (following a vector field is solving what kind of equation?). The first idea one might have is to use the gradient. However, while this idea has the right intentions, the issue is that the gradient of the probability will average out all of the possible probabilities, and will thus flow towards the mode of the distribution:
 
 
To overcome this issue, we look to physical systems and see that a satellite orbiting a planet always nicely stays on some manifold instead of following the gradient:
 
 
The reason why it does is because it has momentum. Recall from basic physics that one way to describe a physical system is through Hamiltonian mechanics, where $H(x,p)$ is the energy associated with the state $(x,p)$ (normally $x$ is location and $p$ is momentum). Due to conservation of energy, the solution of the dynamical equations leads to $H(x,p)$ being constant, and thus the dynamics follow the level sets of $H$. From the Hamiltonian the dynamics of the system are:
 
\[ \begin{align} \frac{dx}{dt} &= \frac{dH}{dp}\\ &= -\frac{dH}{dx} \end{align} \]
 
Here we want our Hamiltonian to be our posterior probability, so that way we stay on the manifold of high probability. This means:
 
\[ H(x,p) = - \log \pi(x,p) \]
 
where $\pi(x,p) = \pi(p|x)\pi(x)$ (where I am now using $pi$ for probability since $p$ is momentum!). So to lift from a probability over parameters to one that includes momentum, we simply need to choose a conditional distribution $\pi(p|x)$. This would mean that
 
\[ \begin{align} H(x,p) &= -log \pi(p|x) - \log \pi(x)\\ &= K(p,x) + V(x) \end{align} \]
 
where $K$ is the kinetic energy and $V$ is the potential. Thus the potential energy is directly given by the posterior calculation, and the kinetic energy is thus a choice that is used to build the correct Hamiltonian. Hamiltonian Monte Carlo methods then dig into good ways to choose the kinetic energy function. This is done at the start (along with the choice of ODE solver time step) in such a way that it maximizes acceptance probabilities.
 
Connections to Differentiable Programming
 
\[ -\frac{dH}{dx} \]
 
requires calculating the gradient of the likelihood function with respect to the parameters, so we are once again using the gradient of our simulator! This means that all of our previous discussion on automatic differentiation and differentiable programming applies to the Hamiltonian Monte Carlo context.
 
There's another thread to follow that transformations of probability distributions are pushforwards of the Jacobian transformations (given the transformation of an integral formula), and this is used when doing variational inference.
 
Symplectic and Geometric Integration
 
One way to integrate the system of ODEs which result from the Hamiltonian system is to convert it to a system of first order ODEs and solve it directly. However, this loses information and can result in drift. This is demonstrated by looking at the long time solution of the pendulum:
 
 using ParameterizedFunctions
 u0 = [1.,0.]
 harmonic! = @ode_def HarmonicOscillator begin
diff --git a/_weave/lecture17/global_sensitivity/index.html b/_weave/lecture17/global_sensitivity/index.html
index 6913e00c..8a81b84c 100644
--- a/_weave/lecture17/global_sensitivity/index.html
+++ b/_weave/lecture17/global_sensitivity/index.html
@@ -9,4 +9,4 @@ 
Global Sensitivity Analysis
 
Chris Rackauckas
 

 using LatinHypercubeSampling
 p = LHCoptim(120,2,1000)
 scatter(p[1][:,1],p[1][:,2])
-
  
For a reference library with many different quasi-Monte Carlo samplers, check out QuasiMonteCarlo.jl.
 
Fourier Amplitude Sensitivity Sampling (FAST) and eFAST
 
The FAST method is a change to the Sobol method to allow for faster convergence. First transform the variables $x_i$ onto the space $[0,1]$. Then, instead of the linear decomposition, one decomposes into a Fourier basis:
 
\[ f(x_i,x_2,\ldots,x_n) = \sum_{m_1 = -\infty}^{\infty} \ldots \sum_{m_n = -\infty}^{\infty} C_{m_1m_2\ldots m_n}\exp\left(2\pi i (m_1 x_1 + \ldots + m_n x_n)\right) \]
 
where
 
\[ C_{m_1m_2\ldots m_n} = \int_0^1 \ldots \int_0^1 f(x_i,x_2,\ldots,x_n) \exp\left(-2\pi i (m_1 x_1 + \ldots + m_n x_n)\right) \]
 
The ANOVA like decomposition is thus
 
\[ f_0 = C_{0\ldots 0} \]
 
\[ f_j = \sum_{m_j \neq 0} C_{0\ldots 0 m_j 0 \ldots 0} \exp (2\pi i m_j x_j) \]
 
\[ f_{jk} = \sum_{m_j \neq 0} \sum_{m_k \neq 0} C_{0\ldots 0 m_j 0 \ldots m_k 0 \ldots 0} \exp \left(2\pi i (m_j x_j + m_k x_k)\right) \]
 
The first order conditional variance is thus:
 
\[ V_j = \int_0^1 f_j^2 (x_j) dx_j = \sum_{m_j \neq 0} |C_{0\ldots 0 m_j 0 \ldots 0}|^2 \]
 
or
 
\[ V_j = 2\sum_{m_j = 1}^\infty \left(A_{m_j}^2 + B_{m_j}^2 \right) \]
 
where $C_{0\ldots 0 m_j 0 \ldots 0} = A_{m_j} + i B_{m_j}$. By Fourier series we know this to be:
 
\[ A_{m_j} = \int_0^1 \ldots \int_0^1 f(x)\cos(2\pi m_j x_j)dx \]
 
\[ B_{m_j} = \int_0^1 \ldots \int_0^1 f(x)\sin(2\pi m_j x_j)dx \]
 
Implementation via the Ergodic Theorem
 
Define
 
\[ X_j(s) = \frac{1}{2\pi} (\omega_j s \mod 2\pi) \]
 
By the ergodic theorem, if $\omega_j$ are irrational numbers, then the dynamical system will never repeat values and thus it will create a solution that is dense in the plane (Let's prove a bit later). As an animation:
 
 
(here, $\omega_1 = \pi$ and $\omega_2 = 7$)
 
This means that:
 
\[ A_{m_j} = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(x)\cos(m_j \omega_j s)ds \]
 
\[ B_{m_j} = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(x)\sin(m_j \omega_j s)ds \]
 
i.e. the multidimensional integral can be approximated by the integral over a single line.
 
One can satisfy this approximately to get a simpler form for the integral. Using $\omega_i$ as integers, the integral is periodic and so only integrating over $2\pi$ is required. This would mean that:
 
\[ A_{m_j} \approx \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\cos(m_j \omega_j s)ds \]
 
\[ B_{m_j} \approx \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\sin(m_j \omega_j s)ds \]
 
It's only approximate since the sequence cannot be dense. For example, with $\omega_1 = 11$ and $\omega_2 = 7$:
 
 
A higher period thus gives a better fill of the space and thus a better approximation, but may require a more points. However, this transformation makes the true integrals simple one dimensional quadratures which can be efficiently computed.
 
To get the total index from this method, one can calculate the total contribution of the complementary set, i.e. $V_{c_i} = \sum_{j \neq i} V_j$ and then
 
\[ S_{T_i} = 1 - S_{c_i} \]
 
Note that this then is a fast measure for the total contribution of variable $i$, including all higher-order nonlinear interactions, all from one-dimensional integrals! (This extension is called extended FAST or eFAST)
 
Proof of the Ergodic Theorem
 
Look at the map $x_{n+1} = x_n + \alpha (\text{mod} 1)$, where $\alpha$ is irrational. This is the irrational rotation map that corresponds to our problem. We wish to prove that in any interval $I$, there is a point of our orbit in this interval.
 
First let's prove a useful result: our points get arbitrarily close. Assume that for some finite $\epsilon$ that no two points are $\epsilon$ apart. This means that we at most have spacings of $\epsilon$ between the points, and thus we have at most $\frac{2\pi}{\epsilon}$ points (rounded up). This means our orbit is periodic. This means that there is a $p$ such that
 
\[ x_{n+p} = x_n \]
 
which means that $p \alpha = 1$ or $p = \frac{1}{\alpha}$ which is a contradiction since $\alpha$ is irrational.
 
Thus for every $\epsilon$ there are two points which are $\epsilon$ apart. Now take any arbitrary $I$. Let $\epsilon < d/2$ where $d$ is the length of the interval. We have just shown that there are two points $\epsilon$ apart, so there is a point that is $x_{n+m}$ and $x_{n+k}$ which are $<\epsilon$ apart. Assuming WLOG $m>k$, this means that $m-k$ rotations takes one from $x_{n+k}$ to $x_{n+m}$, and so $m-k$ rotations is a rotation by $\epsilon$. If we do $\frac{1}{\epsilon}$ rounded up rotations, we will then cover the space with intervals of length epsilon, each with one point of the orbit in it. Since $\epsilon < d/2$, one of those intervals is completely encapsulated in $I$, which means there is at least one point in our orbit that is in $I$.
 
Thus for every interval we have at least one point in our orbit that lies in it, proving that the rotation map with irrational $\alpha$ is dense. Note that during the proof we essentially showed as well that if $\alpha$ is rational, then the map is periodic based on the denominator of the map in its reduced form.
 
A Quick Note on Parallelism
 
Very quick note: all of these are hyper parallel since it does the same calculation per parameter or trajectory, and each calculation is long. For quasi-Monte Carlo, after generating "good enough" trajectories, one can evaluate the model at all points in parallel, and then simply do the GSA index measurement. For FAST, one can do each quadrature in parallel.
 
 
 Published from global_sensitivity.jmd using Weave.jl v0.10.12 on 2024-07-26. 
 

\ No newline at end of file
+
  
For a reference library with many different quasi-Monte Carlo samplers, check out QuasiMonteCarlo.jl.
 
Fourier Amplitude Sensitivity Sampling (FAST) and eFAST
 
The FAST method is a change to the Sobol method to allow for faster convergence. First transform the variables $x_i$ onto the space $[0,1]$. Then, instead of the linear decomposition, one decomposes into a Fourier basis:
 
\[ f(x_i,x_2,\ldots,x_n) = \sum_{m_1 = -\infty}^{\infty} \ldots \sum_{m_n = -\infty}^{\infty} C_{m_1m_2\ldots m_n}\exp\left(2\pi i (m_1 x_1 + \ldots + m_n x_n)\right) \]
 
where
 
\[ C_{m_1m_2\ldots m_n} = \int_0^1 \ldots \int_0^1 f(x_i,x_2,\ldots,x_n) \exp\left(-2\pi i (m_1 x_1 + \ldots + m_n x_n)\right) \]
 
The ANOVA like decomposition is thus
 
\[ f_0 = C_{0\ldots 0} \]
 
\[ f_j = \sum_{m_j \neq 0} C_{0\ldots 0 m_j 0 \ldots 0} \exp (2\pi i m_j x_j) \]
 
\[ f_{jk} = \sum_{m_j \neq 0} \sum_{m_k \neq 0} C_{0\ldots 0 m_j 0 \ldots m_k 0 \ldots 0} \exp \left(2\pi i (m_j x_j + m_k x_k)\right) \]
 
The first order conditional variance is thus:
 
\[ V_j = \int_0^1 f_j^2 (x_j) dx_j = \sum_{m_j \neq 0} |C_{0\ldots 0 m_j 0 \ldots 0}|^2 \]
 
or
 
\[ V_j = 2\sum_{m_j = 1}^\infty \left(A_{m_j}^2 + B_{m_j}^2 \right) \]
 
where $C_{0\ldots 0 m_j 0 \ldots 0} = A_{m_j} + i B_{m_j}$. By Fourier series we know this to be:
 
\[ A_{m_j} = \int_0^1 \ldots \int_0^1 f(x)\cos(2\pi m_j x_j)dx \]
 
\[ B_{m_j} = \int_0^1 \ldots \int_0^1 f(x)\sin(2\pi m_j x_j)dx \]
 
Implementation via the Ergodic Theorem
 
Define
 
\[ X_j(s) = \frac{1}{2\pi} (\omega_j s \mod 2\pi) \]
 
By the ergodic theorem, if $\omega_j$ are irrational numbers, then the dynamical system will never repeat values and thus it will create a solution that is dense in the plane (Let's prove a bit later). As an animation:
 
 
(here, $\omega_1 = \pi$ and $\omega_2 = 7$)
 
This means that:
 
\[ A_{m_j} = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(x)\cos(m_j \omega_j s)ds \]
 
\[ B_{m_j} = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(x)\sin(m_j \omega_j s)ds \]
 
i.e. the multidimensional integral can be approximated by the integral over a single line.
 
One can satisfy this approximately to get a simpler form for the integral. Using $\omega_i$ as integers, the integral is periodic and so only integrating over $2\pi$ is required. This would mean that:
 
\[ A_{m_j} \approx \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\cos(m_j \omega_j s)ds \]
 
\[ B_{m_j} \approx \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\sin(m_j \omega_j s)ds \]
 
It's only approximate since the sequence cannot be dense. For example, with $\omega_1 = 11$ and $\omega_2 = 7$:
 
 
A higher period thus gives a better fill of the space and thus a better approximation, but may require a more points. However, this transformation makes the true integrals simple one dimensional quadratures which can be efficiently computed.
 
To get the total index from this method, one can calculate the total contribution of the complementary set, i.e. $V_{c_i} = \sum_{j \neq i} V_j$ and then
 
\[ S_{T_i} = 1 - S_{c_i} \]
 
Note that this then is a fast measure for the total contribution of variable $i$, including all higher-order nonlinear interactions, all from one-dimensional integrals! (This extension is called extended FAST or eFAST)
 
Proof of the Ergodic Theorem
 
Look at the map $x_{n+1} = x_n + \alpha (\text{mod} 1)$, where $\alpha$ is irrational. This is the irrational rotation map that corresponds to our problem. We wish to prove that in any interval $I$, there is a point of our orbit in this interval.
 
First let's prove a useful result: our points get arbitrarily close. Assume that for some finite $\epsilon$ that no two points are $\epsilon$ apart. This means that we at most have spacings of $\epsilon$ between the points, and thus we have at most $\frac{2\pi}{\epsilon}$ points (rounded up). This means our orbit is periodic. This means that there is a $p$ such that
 
\[ x_{n+p} = x_n \]
 
which means that $p \alpha = 1$ or $p = \frac{1}{\alpha}$ which is a contradiction since $\alpha$ is irrational.
 
Thus for every $\epsilon$ there are two points which are $\epsilon$ apart. Now take any arbitrary $I$. Let $\epsilon < d/2$ where $d$ is the length of the interval. We have just shown that there are two points $\epsilon$ apart, so there is a point that is $x_{n+m}$ and $x_{n+k}$ which are $<\epsilon$ apart. Assuming WLOG $m>k$, this means that $m-k$ rotations takes one from $x_{n+k}$ to $x_{n+m}$, and so $m-k$ rotations is a rotation by $\epsilon$. If we do $\frac{1}{\epsilon}$ rounded up rotations, we will then cover the space with intervals of length epsilon, each with one point of the orbit in it. Since $\epsilon < d/2$, one of those intervals is completely encapsulated in $I$, which means there is at least one point in our orbit that is in $I$.
 
Thus for every interval we have at least one point in our orbit that lies in it, proving that the rotation map with irrational $\alpha$ is dense. Note that during the proof we essentially showed as well that if $\alpha$ is rational, then the map is periodic based on the denominator of the map in its reduced form.
 
A Quick Note on Parallelism
 
Very quick note: all of these are hyper parallel since it does the same calculation per parameter or trajectory, and each calculation is long. For quasi-Monte Carlo, after generating "good enough" trajectories, one can evaluate the model at all points in parallel, and then simply do the GSA index measurement. For FAST, one can do each quadrature in parallel.
 
 
 Published from global_sensitivity.jmd using Weave.jl v0.10.12 on 2024-07-26. 
 

\ No newline at end of file
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diff --git a/_weave/lecture18/jl_OKpj4T/code_profiling_5_1.png b/_weave/lecture18/jl_bQVm2t/code_profiling_5_1.png
similarity index 100%
rename from _weave/lecture18/jl_OKpj4T/code_profiling_5_1.png
rename to _weave/lecture18/jl_bQVm2t/code_profiling_5_1.png
diff --git a/_weave/lecture19/jl_gJdXoT/uncertainty_programming_1_1.png b/_weave/lecture19/jl_1YHT0C/uncertainty_programming_1_1.png
similarity index 100%
rename from _weave/lecture19/jl_gJdXoT/uncertainty_programming_1_1.png
rename to _weave/lecture19/jl_1YHT0C/uncertainty_programming_1_1.png
diff --git a/notes/02-Optimizing_Serial_Code/index.html b/notes/02-Optimizing_Serial_Code/index.html
index 13f96603..81cedc22 100644
--- a/notes/02-Optimizing_Serial_Code/index.html
+++ b/notes/02-Optimizing_Serial_Code/index.html
@@ -10,7 +10,7 @@
 end
 @btime inner_rows!(C,A,B)
 
-7.654 μs (0 allocations: 0 bytes)
+7.587 μs (0 allocations: 0 bytes)
 
 function inner_cols!(C,A,B)
   for j in 1:100, i in 1:100
@@ -19,7 +19,7 @@
 end
 @btime inner_cols!(C,A,B)
 
-2.414 μs (0 allocations: 0 bytes)
+2.386 μs (0 allocations: 0 bytes)

Lower Level View: The Stack and the Heap

Locally, the stack is composed of a stack and a heap. The stack requires a static allocation: it is ordered. Because it's ordered, it is very clear where things are in the stack, and therefore accesses are very quick (think instantaneous). However, because this is static, it requires that the size of the variables is known at compile time (to determine all of the variable locations). Since that is not possible with all variables, there exists the heap. The heap is essentially a stack of pointers to objects in memory. When heap variables are needed, their values are pulled up the cache chain and accessed.

Heap Allocations and Speed

Heap allocations are costly because they involve this pointer indirection, so stack allocation should be done when sensible (it's not helpful for really large arrays, but for small values like scalars it's essential!)

 function inner_alloc!(C,A,B)
   for j in 1:100, i in 1:100
@@ -29,7 +29,7 @@
 end
 @btime inner_alloc!(C,A,B)
 
-234.700 μs (10000 allocations: 625.00 KiB)
+234.498 μs (10000 allocations: 625.00 KiB)
 
 function inner_noalloc!(C,A,B)
   for j in 1:100, i in 1:100
@@ -39,7 +39,7 @@
 end
 @btime inner_noalloc!(C,A,B)
 
-2.413 μs (0 allocations: 0 bytes)
+2.386 μs (0 allocations: 0 bytes)

Why does the array here get heap-allocated? It isn't able to prove/guarantee at compile-time that the array's size will always be a given value, and thus it allocates it to the heap. @btime tells us this allocation occurred and shows us the total heap memory that was taken. Meanwhile, the size of a Float64 number is known at compile-time (64-bits), and so this is stored onto the stack and given a specific location that the compiler will be able to directly address.

Note that one can use the StaticArrays.jl library to get statically-sized arrays and thus arrays which are stack-allocated:

 using StaticArrays
 function static_inner_alloc!(C,A,B)
@@ -50,7 +50,7 @@
 end
 @btime static_inner_alloc!(C,A,B)
 
-2.411 μs (0 allocations: 0 bytes)
+2.388 μs (0 allocations: 0 bytes)

Mutation to Avoid Heap Allocations

Many times you do need to write into an array, so how can you write into an array without performing a heap allocation? The answer is mutation. Mutation is changing the values of an already existing array. In that case, no free memory has to be found to put the array (and no memory has to be freed by the garbage collector).

In Julia, functions which mutate the first value are conventionally noted by a !. See the difference between these two equivalent functions:

 function inner_noalloc!(C,A,B)
   for j in 1:100, i in 1:100
@@ -60,7 +60,7 @@
 end
 @btime inner_noalloc!(C,A,B)
 
-2.413 μs (0 allocations: 0 bytes)
+2.385 μs (0 allocations: 0 bytes)
 
 function inner_alloc(A,B)
   C = similar(A)
@@ -71,7 +71,7 @@
 end
 @btime inner_alloc(A,B)
 
-6.807 μs (2 allocations: 78.17 KiB)
+6.803 μs (2 allocations: 78.17 KiB)

To use this algorithm effectively, the ! algorithm assumes that the caller already has allocated the output array to put as the output argument. If that is not true, then one would need to manually allocate. The goal of that interface is to give the caller control over the allocations to allow them to manually reduce the total number of heap allocations and thus increase the speed.

Julia's Broadcasting Mechanism

Wouldn't it be nice to not have to write the loop there? In many high level languages this is simply called vectorization. In Julia, we will call it array vectorization to distinguish it from the SIMD vectorization which is common in lower level languages like C, Fortran, and Julia.

In Julia, if you use . on an operator it will transform it to the broadcasted form. Broadcast is lazy: it will build up an entire .'d expression and then call broadcast! on composed expression. This is customizable and documented in detail. However, to a first approximation we can think of the broadcast mechanism as a mechanism for building fused expressions. For example, the Julia code:

 A .+ B .+ C;

under the hood lowers to something like:

@@ -86,29 +86,29 @@
 end
 @btime unfused(A,B,C);
 
-6.622 μs (4 allocations: 156.34 KiB)
+9.294 μs (4 allocations: 156.34 KiB)
 
 fused(A,B,C) = A .+ B .+ C
 @btime fused(A,B,C);
 
-4.630 μs (2 allocations: 78.17 KiB)
+4.433 μs (2 allocations: 78.17 KiB)

Note that we can also fuse the output by using .=. This is essentially the vectorized version of a ! function:

 D = similar(A)
 fused!(D,A,B,C) = (D .= A .+ B .+ C)
 @btime fused!(D,A,B,C);
 
-3.436 μs (0 allocations: 0 bytes)
+3.441 μs (0 allocations: 0 bytes)

Note on Broadcasting Function Calls

Julia allows for broadcasting the call () operator as well. .() will call the function element-wise on all arguments, so sin.(A) will be the elementwise sine function. This will fuse Julia like the other operators.

Note on Vectorization and Speed

In articles on MATLAB, Python, R, etc., this is where you will be told to vectorize your code. Notice from above that this isn't a performance difference between writing loops and using vectorized broadcasts. This is not abnormal! The reason why you are told to vectorize code in these other languages is because they have a high per-operation overhead (which will be discussed further down). This means that every call, like +, is costly in these languages. To get around this issue and make the language usable, someone wrote and compiled the loop for the C/Fortran function that does the broadcasted form (see numpy's Github repo). Thus A .+ B's MATLAB/Python/R equivalents are calling a single C function to generally avoid the cost of function calls and thus are faster.

But this is not an intrinsic property of vectorization. Vectorization isn't "fast" in these languages, it's just close to the correct speed. The reason vectorization is recommended is because looping is slow in these languages. Because looping isn't slow in Julia (or C, C++, Fortran, etc.), loops and vectorization generally have the same speed. So use the one that works best for your code without a care about performance.

(As a small side effect, these high level languages tend to allocate a lot of temporary variables since the individual C kernels are written for specific numbers of inputs and thus don't naturally fuse. Julia's broadcast mechanism is just generating and JIT compiling Julia functions on the fly, and thus it can accommodate the combinatorial explosion in the amount of choices just by only compiling the combinations that are necessary for a specific code)

Heap Allocations from Slicing

It's important to note that slices in Julia produce copies instead of views. Thus for example:

 A[50,50]
 
-0.17778218519466293
+0.4378374168708473

allocates a new output. This is for safety, since if it pointed to the same array then writing to it would change the original array. We can demonstrate this by asking for a view instead of a copy.

 @show A[1]
 E = @view A[1:5,1:5]
 E[1] = 2.0
 @show A[1]
 
-A[1] = 0.5530951950733606
+A[1] = 0.37233258859606955
 A[1] = 2.0
 2.0

However, this means that @view A[1:5,1:5] did not allocate an array (it does allocate a pointer if the escape analysis is unable to prove that it can be elided. This means that in small loops there will be no allocation, while if the view is returned from a function for example it will allocate the pointer, ~80 bytes, but not the memory of the array. This means that it is O(1) in cost but with a relatively small constant).

Asymptotic Cost of Heap Allocations

Heap allocations have to locate and prepare a space in RAM that is proportional to the amount of memory that is calculated, which means that the cost of a heap allocation for an array is O(n), with a large constant. As RAM begins to fill up, this cost dramatically increases. If you run out of RAM, your computer may begin to use swap, which is essentially RAM simulated on your hard drive. Generally when you hit swap your performance is so dead that you may think that your computation froze, but if you check your resource use you will notice that it's actually just filled the RAM and starting to use the swap.

But think of it as O(n) with a large constant factor. This means that for operations which only touch the data once, heap allocations can dominate the computational cost:

@@ -130,7 +130,7 @@
 plot(ns,alloc,label="=",xscale=:log10,yscale=:log10,legend=:bottomright,
      title="Micro-optimizations matter for BLAS1")
 plot!(ns,noalloc,label=".=")
-

However, when the computation takes O(n^3), like in matrix multiplications, the high constant factor only comes into play when the matrices are sufficiently small:

+

However, when the computation takes O(n^3), like in matrix multiplications, the high constant factor only comes into play when the matrices are sufficiently small:

 using LinearAlgebra, BenchmarkTools
 function alloc_timer(n)
     A = rand(n,n)
@@ -149,7 +149,7 @@
 plot(ns,alloc,label="*",xscale=:log10,yscale=:log10,legend=:bottomright,
      title="Micro-optimizations only matter for small matmuls")
 plot!(ns,noalloc,label="mul!")
-

Though using a mutating form is never bad and always is a little bit better.

Optimizing Memory Use Summary

  • Avoid cache misses by reusing values

  • Iterate along columns

  • Avoid heap allocations in inner loops

  • Heap allocations occur when the size of things is not proven at compile-time

  • Use fused broadcasts (with mutated outputs) to avoid heap allocations

  • Array vectorization confers no special benefit in Julia because Julia loops are as fast as C or Fortran

  • Use views instead of slices when applicable

  • Avoiding heap allocations is most necessary for O(n) algorithms or algorithms with small arrays

  • Use StaticArrays.jl to avoid heap allocations of small arrays in inner loops

Julia's Type Inference and the Compiler

Many people think Julia is fast because it is JIT compiled. That is simply not true (we've already shown examples where Julia code isn't fast, but it's always JIT compiled!). Instead, the reason why Julia is fast is because the combination of two ideas:

  • Type inference

  • Type specialization in functions

These two features naturally give rise to Julia's core design feature: multiple dispatch. Let's break down these pieces.

Type Inference

At the core level of the computer, everything has a type. Some languages are more explicit about said types, while others try to hide the types from the user. A type tells the compiler how to to store and interpret the memory of a value. For example, if the compiled code knows that the value in the register is supposed to be interpreted as a 64-bit floating point number, then it understands that slab of memory like:

Importantly, it will know what to do for function calls. If the code tells it to add two floating point numbers, it will send them as inputs to the Floating Point Unit (FPU) which will give the output.

If the types are not known, then... ? So one cannot actually compute until the types are known, since otherwise it's impossible to interpret the memory. In languages like C, the programmer has to declare the types of variables in the program:

void add(double *a, double *b, double *c, size_t n){
+

Though using a mutating form is never bad and always is a little bit better.

Optimizing Memory Use Summary

  • Avoid cache misses by reusing values

  • Iterate along columns

  • Avoid heap allocations in inner loops

  • Heap allocations occur when the size of things is not proven at compile-time

  • Use fused broadcasts (with mutated outputs) to avoid heap allocations

  • Array vectorization confers no special benefit in Julia because Julia loops are as fast as C or Fortran

  • Use views instead of slices when applicable

  • Avoiding heap allocations is most necessary for O(n) algorithms or algorithms with small arrays

  • Use StaticArrays.jl to avoid heap allocations of small arrays in inner loops

Julia's Type Inference and the Compiler

Many people think Julia is fast because it is JIT compiled. That is simply not true (we've already shown examples where Julia code isn't fast, but it's always JIT compiled!). Instead, the reason why Julia is fast is because the combination of two ideas:

  • Type inference

  • Type specialization in functions

These two features naturally give rise to Julia's core design feature: multiple dispatch. Let's break down these pieces.

Type Inference

At the core level of the computer, everything has a type. Some languages are more explicit about said types, while others try to hide the types from the user. A type tells the compiler how to to store and interpret the memory of a value. For example, if the compiled code knows that the value in the register is supposed to be interpreted as a 64-bit floating point number, then it understands that slab of memory like:

Importantly, it will know what to do for function calls. If the code tells it to add two floating point numbers, it will send them as inputs to the Floating Point Unit (FPU) which will give the output.

If the types are not known, then... ? So one cannot actually compute until the types are known, since otherwise it's impossible to interpret the memory. In languages like C, the programmer has to declare the types of variables in the program:

void add(double *a, double *b, double *c, size_t n){
   size_t i;
   for(i = 0; i < n; ++i) {
     c[i] = a[i] + b[i];
@@ -172,7 +172,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `f`
-define i64 @julia_f_3082(i64 signext %0, i64 signext %1) #0 {
+define i64 @julia_f_3070(i64 signext %0, i64 signext %1) #0 {
 top:
 ; ┌ @ int.jl:87 within `+`
    %2 = add i64 %1, %0
@@ -184,7 +184,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `f`
-define double @julia_f_3084(double %0, double %1) #0 {
+define double @julia_f_3072(double %0, double %1) #0 {
 top:
 ; ┌ @ float.jl:409 within `+`
    %2 = fadd double %0, %1
@@ -204,7 +204,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define i64 @julia_g_3086(i64 signext %0, i64 signext %1) #0 {
+define i64 @julia_g_3074(i64 signext %0, i64 signext %1) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 6 within `g`
@@ -249,7 +249,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `f`
-define double @julia_f_3545(double %0, i64 signext %1) #0 {
+define double @julia_f_3533(double %0, i64 signext %1) #0 {
 top:
 ; ┌ @ promotion.jl:422 within `+`
 ; │┌ @ promotion.jl:393 within `promote`
@@ -290,7 +290,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define double @julia_g_3548(double %0, i64 signext %1) #0 {
+define double @julia_g_3536(double %0, i64 signext %1) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 5 within `g`
@@ -381,7 +381,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `ff`
-define double @julia_ff_3667(double %0, i64 signext %1) #0 {
+define double @julia_ff_3655(double %0, i64 signext %1) #0 {
 top:
 ; ┌ @ promotion.jl:422 within `+`
 ; │┌ @ promotion.jl:393 within `promote`
@@ -537,7 +537,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define void @julia_g_3806([2 x double]* noalias nocapture noundef nonnull s
+define void @julia_g_3794([2 x double]* noalias nocapture noundef nonnull s
 ret([2 x double]) align 8 dereferenceable(16) %0, [2 x double]* nocapture n
 oundef nonnull readonly align 8 dereferenceable(16) %1, [2 x double]* nocap
 ture noundef nonnull readonly align 8 dereferenceable(16) %2) #0 {
@@ -657,7 +657,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define [2 x float] @julia_g_3818([2 x float]* nocapture noundef nonnull rea
+define [2 x float] @julia_g_3806([2 x float]* nocapture noundef nonnull rea
 donly align 4 dereferenceable(8) %0, [2 x float]* nocapture noundef nonnull
  readonly align 4 dereferenceable(8) %1) #0 {
 top:
@@ -763,7 +763,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `g`
-define void @julia_g_3837([2 x {}*]* noalias nocapture noundef nonnull sret
+define void @julia_g_3825([2 x {}*]* noalias nocapture noundef nonnull sret
 ([2 x {}*]) align 8 dereferenceable(16) %0, [2 x {}*]* nocapture noundef no
 nnull readonly align 8 dereferenceable(16) %1, [2 x {}*]* nocapture noundef
  nonnull readonly align 8 dereferenceable(16) %2) #0 {
@@ -796,25 +796,25 @@
    store {}** %14, {}*** %13, align 8
    %15 = bitcast {}*** %tls_pgcstack to {}***
    store {}** %gcframe2.sub, {}*** %15, align 8
-   call void @"j_+_3839"([2 x {}*]* noalias nocapture noundef nonnull sret(
-[2 x {}*]) %9, [2 x {}*]* nocapture nonnull readonly %1, i64 signext 4)
+   call void @"j_+_3827"([2 x {}*]* noalias nocapture noundef nonnull sret(
+[2 x {}*]) %7, [2 x {}*]* nocapture nonnull readonly %1, i64 signext 4)
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 6 within `g`
 ; ┌ @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd
 :2 within `f`
-   call void @"j_+_3840"([2 x {}*]* noalias nocapture noundef nonnull sret(
-[2 x {}*]) %5, i64 signext 2, [2 x {}*]* nocapture readonly %9)
+   call void @"j_+_3828"([2 x {}*]* noalias nocapture noundef nonnull sret(
+[2 x {}*]) %5, i64 signext 2, [2 x {}*]* nocapture readonly %7)
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 7 within `g`
 ; ┌ @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd
 :2 within `f`
-   call void @"j_+_3841"([2 x {}*]* noalias nocapture noundef nonnull sret(
-[2 x {}*]) %7, [2 x {}*]* nocapture readonly %5, [2 x {}*]* nocapture nonnu
+   call void @"j_+_3829"([2 x {}*]* noalias nocapture noundef nonnull sret(
+[2 x {}*]) %9, [2 x {}*]* nocapture readonly %5, [2 x {}*]* nocapture nonnu
 ll readonly %2)
    %16 = bitcast [2 x {}*]* %0 to i8*
-   %17 = bitcast {}** %6 to i8*
+   %17 = bitcast {}** %8 to i8*
    call void @llvm.memcpy.p0i8.p0i8.i64(i8* noundef nonnull align 8 derefer
 enceable(16) %16, i8* noundef nonnull align 16 dereferenceable(16) %17, i64
  16, i1 false)
@@ -844,28 +844,28 @@
 b = MyComplex(2.0,1.0)
 @btime g(a,b)
 
-20.439 ns (1 allocation: 32 bytes)
+20.388 ns (1 allocation: 32 bytes)
 MyComplex(9.0, 2.0)
 
 a = MyParameterizedComplex(1.0,1.0)
 b = MyParameterizedComplex(2.0,1.0)
 @btime g(a,b)
 
-21.304 ns (1 allocation: 32 bytes)
+21.142 ns (1 allocation: 32 bytes)
 MyParameterizedComplex{Float64}(9.0, 2.0)
 
 a = MySlowComplex(1.0,1.0)
 b = MySlowComplex(2.0,1.0)
 @btime g(a,b)
 
-121.043 ns (5 allocations: 96 bytes)
+113.240 ns (5 allocations: 96 bytes)
 MySlowComplex(9.0, 2.0)
 
 a = MySlowComplex2(1.0,1.0)
 b = MySlowComplex2(2.0,1.0)
 @btime g(a,b)
 
-853.314 ns (14 allocations: 288 bytes)
+519.620 ns (14 allocations: 288 bytes)
 MySlowComplex2(9.0, 2.0)

Note on Julia

Note that, because of these type specialization, value types, etc. properties, the number types, even ones such as Int, Float64, and Complex, are all themselves implemented in pure Julia! Thus even basic pieces can be implemented in Julia with full performance, given one uses the features correctly.

Note on isbits

Note that a type which is mutable struct will not be isbits. This means that mutable structs will be a pointer to a heap allocated object, unless it's shortlived and the compiler can erase its construction. Also, note that isbits compiles down to bit operations from pure Julia, which means that these types can directly compile to GPU kernels through CUDAnative without modification.

Function Barriers

Since functions automatically specialize on their input types in Julia, we can use this to our advantage in order to make an inner loop fully inferred. For example, take the code from above but with a loop:

 function r(x)
@@ -880,7 +880,7 @@
 end
 @btime r(x)
 
-5.756 μs (300 allocations: 4.69 KiB)
+5.878 μs (300 allocations: 4.69 KiB)
 604.0

In here, the loop variables are not inferred and thus this is really slow. However, we can force a function call in the middle to end up with specialization and in the inner loop be stable:

 s(x) = _s(x[1],x[2])
@@ -896,7 +896,7 @@
 end
 @btime s(x)
 
-299.899 ns (1 allocation: 16 bytes)
+296.107 ns (1 allocation: 16 bytes)
 604.0

Notice that this algorithm still doesn't infer:

 @code_warntype s(x)
@@ -928,7 +928,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `fff`
-define i64 @julia_fff_3931(i64 signext %0) #0 {
+define i64 @julia_fff_3919(i64 signext %0) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 8 within `fff`
@@ -942,7 +942,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `fff`
-define double @julia_fff_3933(double %0) #0 {
+define double @julia_fff_3921(double %0) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 8 within `fff`
@@ -957,7 +957,7 @@
   C[i,j] = A[i,j] + B[i,j]
 end
 
-769.551 μs (30000 allocations: 468.75 KiB)
+792.921 μs (30000 allocations: 468.75 KiB)

This is very slow because the types of A, B, and C cannot be inferred. Why can't they be inferred? Well, at any time in the dynamic REPL scope I can do something like C = "haha now a string!", and thus it cannot specialize on the types currently existing in the REPL (since asynchronous changes could also occur), and therefore it defaults back to doing a type check at every single function which slows it down. Moral of the story, Julia functions are fast but its global scope is too dynamic to be optimized.

Summary

  • Julia is not fast because of its JIT, it's fast because of function specialization and type inference

  • Type stable functions allow inference to fully occur

  • Multiple dispatch works within the function specialization mechanism to create overhead-free compile time controls

  • Julia will specialize the generic functions

  • Making sure values are concretely typed in inner loops is essential for performance

Overheads of Individual Operations

Now let's dig even a little deeper. Everything the processor does has a cost. A great chart to keep in mind is this classic one. A few things should immediately jump out to you:

  • Simple arithmetic, like floating point additions, are super cheap. ~1 clock cycle, or a few nanoseconds.

  • Processors do branch prediction on if statements. If the code goes down the predicted route, the if statement costs ~1-2 clock cycles. If it goes down the wrong route, then it will take ~10-20 clock cycles. This means that predictable branches, like ones with clear patterns or usually the same output, are much cheaper (almost free) than unpredictable branches.

  • Function calls are expensive: 15-60 clock cycles!

  • RAM reads are very expensive, with lower caches less expensive.

Bounds Checking

Let's check the LLVM IR on one of our earlier loops:

 function inner_noalloc!(C,A,B)
   for j in 1:100, i in 1:100
@@ -969,7 +969,7 @@
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `inner_noalloc!`
-define nonnull {}* @"japi1_inner_noalloc!_3942"({}* %function, {}** noalias
+define nonnull {}* @"japi1_inner_noalloc!_3930"({}* %function, {}** noalias
  nocapture noundef readonly %args, i32 %nargs) #0 {
 top:
   %stackargs = alloca {}**, align 8
@@ -1997,7 +1997,7 @@
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 6 within `inner_noalloc!`
-  ret {}* inttoptr (i64 139788390752264 to {}*)
+  ret {}* inttoptr (i64 139763090710536 to {}*)
 
 oob:                                              ; preds = %L5.us.us.us.po
 stloop, %L2.split.us.split, %L2.split.us.split.us.split, %L5.us.us.us.postl
@@ -2098,17 +2098,17 @@
 end
 @btime inner_noalloc!(C,A,B)
 
-2.411 μs (0 allocations: 0 bytes)
+2.386 μs (0 allocations: 0 bytes)
 
 @btime inner_noalloc_ib!(C,A,B)
 
-2.347 μs (0 allocations: 0 bytes)
+2.342 μs (0 allocations: 0 bytes)

SIMD

Now let's inspect the LLVM IR again:

 @code_llvm inner_noalloc_ib!(C,A,B)
 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `inner_noalloc_ib!`
-define nonnull {}* @"japi1_inner_noalloc_ib!_3978"({}* %function, {}** noal
+define nonnull {}* @"japi1_inner_noalloc_ib!_3966"({}* %function, {}** noal
 ias nocapture noundef readonly %args, i32 %nargs) #0 {
 top:
   %stackargs = alloca {}**, align 8
@@ -4013,7 +4013,7 @@
   br i1 %.not.not24, label %L36, label %L2
 
 L36:                                              ; preds = %L25
-  ret {}* inttoptr (i64 139788390752264 to {}*)
+  ret {}* inttoptr (i64 139763090710536 to {}*)
 }

If you look closely, you will see things like:

%wide.load24 = load <4 x double>, <4 x double> addrspac(13)* %46, align 8
 ; â””
@@ -4022,7 +4022,7 @@
 @code_llvm fma(2.0,5.0,3.0)
 
 ;  @ floatfuncs.jl:439 within `fma`
-define double @julia_fma_3979(double %0, double %1, double %2) #0 {
+define double @julia_fma_3967(double %0, double %1, double %2) #0 {
 common.ret:
 ; ┌ @ floatfuncs.jl:434 within `fma_llvm`
    %3 = call double @llvm.fma.f64(double %0, double %1, double %2)
@@ -4070,7 +4070,7 @@ 
Inlining

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 4 within `qinline`
-define double @julia_qinline_3982(double %0, double %1) #0 {
+define double @julia_qinline_3970(double %0, double %1) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 7 within `qinline`
@@ -4107,17 +4107,17 @@ 
Inlining

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 11 within `qnoinline`
-define double @julia_qnoinline_3984(double %0, double %1) #0 {
+define double @julia_qnoinline_3972(double %0, double %1) #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 14 within `qnoinline`
-  %2 = call double @j_fnoinline_3986(double %0, i64 signext 4)
+  %2 = call double @j_fnoinline_3974(double %0, i64 signext 4)
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 15 within `qnoinline`
-  %3 = call double @j_fnoinline_3987(i64 signext 2, double %2)
+  %3 = call double @j_fnoinline_3975(i64 signext 2, double %2)
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 16 within `qnoinline`
-  %4 = call double @j_fnoinline_3988(double %3, double %1)
+  %4 = call double @j_fnoinline_3976(double %3, double %1)
   ret double %4
 }
 

@@ -4136,7 +4136,7 @@ 
Inlining

 
 
 
-21.605 ns (1 allocation: 16 bytes)
+20.982 ns (1 allocation: 16 bytes)
 9.0
 

 
@@ -4148,7 +4148,7 @@ 
Inlining

 
 
 
-25.319 ns (1 allocation: 16 bytes)
+25.017 ns (1 allocation: 16 bytes)
 9.0
 

 
@@ -4193,7 +4193,7 @@ 
Note on Benchmarking

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture02/optimizing.jmd:
 2 within `cheat`
-define double @julia_cheat_4016() #0 {
+define double @julia_cheat_4004() #0 {
 top:
   ret double 9.000000e+00
 }
diff --git a/notes/03-Introduction_to_Scientific_Machine_Learning_through_Physics-Informed_Neural_Networks/index.html b/notes/03-Introduction_to_Scientific_Machine_Learning_through_Physics-Informed_Neural_Networks/index.html
index 6abbe5df..3f9c965b 100644
--- a/notes/03-Introduction_to_Scientific_Machine_Learning_through_Physics-Informed_Neural_Networks/index.html
+++ b/notes/03-Introduction_to_Scientific_Machine_Learning_through_Physics-Informed_Neural_Networks/index.html
@@ -19,11 +19,11 @@
 simpleNN(rand(10))
 
 
 5-element Vector{Float64}:
-  6.191491803887638
- 10.961350935082876
- -0.086794697902561
-  4.329029452777256
- -0.4819728343992912
+ -1.1205287975184102
+ -6.95758884176028
+  4.900880949623628
+  3.7834125354726122
+  4.191471397681439
 
 
This is our direct definition of a neural network. Notice that we choose to use tanh as our activation function between the layers.
 
Defining Neural Networks with Flux.jl
 
One of the main deep learning libraries in Julia is Flux.jl. Flux is an interesting library for scientific machine learning because it is built on top of language-wide automatic differentiation libraries, giving rise to a programming paradigm known as differentiable programming, which means that one can write a program in a manner that it has easily accessible fast derivatives. However, due to being built on a differentiable programming base, the underlying functionality is simply standard Julia code,
 
To learn how to use the library, consult the documentation. A Google search will bring up the Flux.jl Github repository. From there, the blue link on the README brings you to the package documentation. This is common through Julia so it's a good habit to learn!
 
In the documentation you will find that the way a neural network is defined is through a Chain of layers. A Dense layer is the kind we defined above, which is given by an input size, an output size, and an activation function. For example, the following recreates the neural network that we had above:
 
 using Flux
 NN2 = Chain(Dense(10 => 32,tanh),
@@ -32,11 +32,11 @@
 NN2(rand(10))
 
 
 5-element Vector{Float32}:
- -0.40341762
- -0.32073504
- -0.60663885
-  0.6273459
-  0.3418348
+ -0.41757214
+  0.3268494
+  0.6551256
+  0.3942203
+  0.5987754
 
 
Notice that Flux.jl as a library is written in pure Julia, which means that every piece of this syntax is just sugar over some Julia code that we can specialize ourselves (this is the advantage of having a language fast enough for the implementation of the library and the use of the library!)
 
For example, the activation function is just a scalar Julia function. If we wanted to replace it by something like the quadratic function, we can just use an anonymous function to define the scalar function we would like to use:
 
 NN3 = Chain(Dense(10 => 32,x->x^2),
             Dense(32 => 32,x->max(0,x)),
@@ -44,11 +44,11 @@
 NN3(rand(10))
 
 
 5-element Vector{Float32}:
- -0.18383282
- -0.030633058
- -0.021082453
- -0.08232622
- -0.22971418
+ -0.02532567
+ -0.14441662
+ -0.051778786
+ -0.38974416
+ -0.10805663
 
 
The second activation function there is what's known as a relu. A relu can be good to use because it's an exceptionally fast operation and satisfies a form of the universal approximation theorem (UAT). However, a downside is that its derivative is not continuous, which could impact the numerical properties of some algorithms, and thus it's widely used throughout standard machine learning but we'll see reasons why it may be disadvantageous in some cases in scientific machine learning.
 
Digging into the Construction of a Neural Network Library
 
Again, as mentioned before, this neural network NN2 is simply a function:
 
 simpleNN(x) = W[3]*tanh.(W[2]*tanh.(W[1]*x + b[1]) + b[2]) + b[3]
 
 
@@ -96,26 +96,26 @@
 denselayer_f(rand(32))
 
 
 32-element Vector{Float32}:
-  0.015360817
- -0.46573928
-  0.84118056
-  0.6889729
- -0.04318807
- -0.486555
-  0.7456628
- -0.1116474
- -0.8562411
-  0.14605747
+ -0.5618108
+ -0.6442508
+  0.4827489
+ -0.2672533
+ -0.87355
+ -0.030600643
+  0.333129
+ -0.16809873
+ -0.03701947
+  0.38146716
   â‹®
-  0.07789154
-  0.35291886
-  0.27881867
- -0.68542886
- -0.3935992
-  0.657173
- -0.18403019
-  0.063687466
- -0.15492688
+ -0.39834422
+  0.8219398
+ -0.55398756
+ -0.031825528
+  0.17183039
+  0.72180593
+ -0.7143836
+  0.49238148
+ -0.7401128
 
 
So okay, Dense objects are just functions that have weight and bias matrices inside of them. Now what does Chain do?
 
 @which Chain(1,2,3)
 
 Chain(xs...) in Flux at /home/runner/.julia/packages/Flux/Wz6D4/src/layers/basic.jl:39 
Again, for our explanations here we will look at the slightly simpler code From and earlier version of the Flux package:
 
@@ -169,52 +169,52 @@
 loss() = sum(abs2,sum(abs2,NN(rand(10)).-1) for i in 1:100)
 loss()
 
 
-4269.761f0
+2630.6902f0
 
 
This loss function takes 100 random points in $[0,1]^{10}$ and then computes the output of the neural network minus 1 on each of the values, and sums up the squared values (abs2). Why the squared values? This means that every computed loss value is positive, and so we know that by decreasing the loss this means that, on average our neural network outputs are closer to 1. What are the weights? Since we're using the Flux callable struct style from above, the weights are those inside of the NN chain object, which we can inspect:
 
 NN[1].weight # The W matrix of the first layer
 
 
 32×10 Matrix{Float32}:
-  0.210141   -0.0318616   0.0161381  …   0.320312   -0.331868    0.187002
- -0.102589    0.208266   -0.18678        0.259299   -0.0850212   0.325602
-  0.0954246   0.29503     0.155045      -0.13743    -0.226485   -0.352321
-  0.094165    0.147697   -0.0402125      0.376543   -0.0652325  -0.0910839
- -0.0960281  -0.239555    0.0736242      0.179233    0.0747602  -0.231401
-  0.346408   -0.229251    0.247542   …  -0.227708   -0.342039   -0.3638
- -0.0874939  -0.146548    0.162742      -0.194633   -0.280423   -0.374489
-  0.340297    0.264646    0.097446       0.312401   -0.212878    0.307074
-  0.365767   -0.366916    0.261565      -0.0138854  -0.105345   -0.265025
-  0.228151    0.247341   -0.135668       0.247047    0.140386   -0.19907
-  ⋮                                  ⋱                          
-  0.186276    0.115302   -0.303067      -0.0957572  -0.202029    0.199585
- -0.0958866   0.139531    0.182455      -0.0669233  -0.190238    0.280298
- -0.108114    0.258266    0.052903   …   0.191436   -0.12086    -0.186751
-  0.2797     -0.0260807   0.256827       0.363836   -0.308047    0.104556
- -0.132173   -0.372458    0.122993       0.102172   -0.186959    0.195341
- -0.0277494  -0.15691     0.231842      -0.35223    -0.130247   -0.256816
- -0.309665   -0.313657   -0.345432       0.338976    0.302093    0.373044
- -0.368304    0.298401   -0.0528614  …  -0.196604    0.361721    0.234842
-  0.147451    0.324855   -0.175287       0.0836575   0.331776   -0.0320324
+ -0.0912051   0.171404   -0.225056   …   0.26575    -0.2033       0.19557
+ -0.023353   -0.193015    0.33744        0.135803    0.305452     0.330386
+ -0.21381     0.222574    0.146357       0.302459   -0.204567     0.177094
+ -0.236288   -0.0437021   0.149399      -0.322455    0.0413544    0.376834
+  0.0779307  -0.363982   -0.256043       0.150865    0.310071     0.141492
+  0.310811   -0.0969612  -0.1377     …   0.377674   -0.372038     0.237749
+  0.155738   -0.272288    0.148163      -0.175247    0.0568673   -0.302354
+ -0.338095    0.375844   -0.0522764     -0.0950616   0.241941     0.177569
+ -0.313516    0.0510053  -0.283776      -0.262828    0.127621     0.0381291
+ -0.238924   -0.019392   -0.345777       0.123743   -0.171073     0.286924
+  ⋮                                  ⋱                           
+  0.361331   -0.300791    0.340877       0.051346    0.250142     0.236606
+  0.256587   -0.144472   -0.250453       0.0694984  -0.00493134   0.242588
+  0.321232    0.133965    0.080042   …   0.0232834  -0.357839     0.175813
+  0.29538     0.0922859   0.246678      -0.102874   -0.237856    -0.340453
+  0.344922    0.306977    0.155636      -0.132782    0.127812    -0.0488943
+ -0.0161293  -0.25159    -0.220966       0.190733    0.0313222   -0.100159
+  0.0903141   0.251727    0.341801       0.127054   -0.243489    -0.0498583
+  0.0220798  -0.314075   -0.0345888  …  -0.0257494   0.109001    -0.0628161
+ -0.321938    0.144522   -0.0387606      0.17195     0.200624    -0.272704
 
 
Now let's grab all of the parameters together:
 
 p = Flux.params(NN)
 
 
-Params([Float32[0.21014117 -0.031861562 … -0.33186755 0.18700212; -0.102589
-46 0.20826595 … -0.0850212 0.32560205; … ; -0.36830446 0.29840133 … 0.36172
-083 0.23484248; 0.14745119 0.3248552 … 0.33177575 -0.03203242], Float32[0.0
-, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 0.0, 
-0.0, 0.0, 0.0, 0.0, 0.0], Float32[-0.18695994 -0.04698782 … -0.055100985 -0
-.1341884; 0.0074982448 0.11674296 … -0.22039233 -0.05518888; … ; -0.0007238
-363 0.2559794 … -0.29302862 -0.30272353; 0.087237954 0.087170936 … 0.295242
-6 -0.19100738], Float32[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …
-  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Float32[0.029854774 -0
-.06820957 … 0.09176909 -0.11000199; 0.2896224 0.2912958 … 0.034562368 -0.17
-458537; … ; 0.07649842 0.26569244 … -0.07407499 0.37595725; -0.28656915 0.0
-42809594 … -0.24949594 -0.33653733], Float32[0.0, 0.0, 0.0, 0.0, 0.0]])
+Params([Float32[-0.091205075 0.17140365 … -0.20329987 0.19557032; -0.023352
+979 -0.19301474 … 0.3054521 0.33038554; … ; 0.022079762 -0.31407467 … 0.109
+00141 -0.062816136; -0.3219381 0.14452209 … 0.20062439 -0.27270395], Float3
+2[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 
+0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Float32[0.019839598 -0.2168316 … -0.2577663 
+0.30027536; -0.19628644 0.07071809 … -0.117501654 0.20883556; … ; 0.0368188
+55 -0.012596854 … -0.24249776 -0.19441248; 0.2561112 -0.19482055 … -0.30460
+06 -0.06886764], Float32[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  
+…  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Float32[-0.12173528 -
+0.3812579 … -0.3371982 -0.27681643; 0.33429277 0.32999685 … -0.3093447 -0.2
+4638811; … ; 0.38407874 -0.25517553 … -0.011652403 0.027645448; -0.18082629
+ 0.22756484 … 0.22212335 0.053659596], Float32[0.0, 0.0, 0.0, 0.0, 0.0]])
 
 
That's a helper function on Chain which recursively gathers all of the defining parameters. Let's now find the optimal values p which cause the neural network to be the constant 1 function:
 
 Flux.train!(loss, p, Iterators.repeated((), 10000), ADAM(0.1))
 
 
Now let's check the loss:
 
 loss()
 
 
-3.377496f-5
+3.805218f-5
 
 
This means that NN(x) is now a very good function approximator to f(x) = ones(5)!
 
So Why Machine Learning? Why Neural Networks?
 
All we did was find parameters that made NN(x) act like a function f(x). How does that relate to machine learning? Well, in any case where one is acting on data (x,y), the idea is to assume that there exists some underlying mathematical model f(x) = y. If we had perfect knowledge of what f is, then from only the information of x we can then predict what y would be. The inference problem is to then figure out what function f should be. Therefore, machine learning on data is simply this problem of finding an approximator to some unknown function!
 
So why neural networks? Neural networks satisfy two properties. The first of which is known as the Universal Approximation Theorem (UAT), which in simple non-mathematical language means that, for any ϵ of accuracy, if your neural network is large enough (has enough layers, the weight matrices are large enough), then it can approximate any (nice) function f within that ϵ. Therefore, we can reduce the problem of finding missing functions, the problem of machine learning, to a problem of finding the weights of neural networks, which is a well-defined mathematical optimization problem.
 
Why neural networks specifically? That's a fairly good question, since there are many other functions with this property. For example, you will have learned from analysis that $a_0 + a_1 x + a_2 x^2 + \ldots$ arbitrary polynomials can be used to approximate any analytic function (this is the Taylor series). Similarly, a Fourier series
 
\[ f(x) = a_0 + \sum_k b_k \cos(kx) + c_k \sin(kx) \]
 
can approximate any continuous function f (and discontinuous functions also can have convergence, etc. these are the details of a harmonic analysis course).
 
That's all for one dimension. How about two dimensional functions? It turns out it's not difficult to prove that tensor products of universal approximators will give higher dimensional universal approximators. So for example, tensoring together two polynomials:
 
\[ a_0 + a_1 x + a_2 y + a_3 x y + a_4 x^2 y + a_5 x y^2 + a_6 x^2 y^2 + \ldots \]
 
will give a two-dimensional function approximator. But notice how we have to resolve every combination of terms. This means that if we used n coefficients in each dimension d, the total number of coefficients to build a d-dimensional universal approximator from one-dimensional objects would need $n^d$ coefficients. This exponential growth is known as the curse of dimensionality.
 
The second property of neural networks that makes them applicable to machine learning is that they overcome the curse of dimensionality. The proofs in this area can be a little difficult to parse, but what they boil down to is proving in many cases that the growth of neural networks to sufficiently approximate a d-dimensional function grows as a polynomial of d, rather than exponential. This means that there's some dimensional cutoff where for $d>cutoff$ it is more efficient to use a neural network. This can be problem-specific, but generally it tends to be the case at least by 8 or 10 dimensions.
 
Neural networks have a few other properties to consider as well:
 
     
  1. The assumptions of the neural network can be encoded into the neural architectures. A neural network where the last layer has an activation function x->x^2 is a neural network where all outputs are positive. This means that if you want to find a positive function, you can make the optimization easier by enforcing this constraint. A lot of other constraints can be enforced, like tanh activation functions can make the neural network be a smooth (all derivatives finite) function, or other activations can cause finite numbers of learnable discontinuities.
     
  2. Generating higher dimensional forms from one dimensional forms does not have good symmetry. For example, the two-dimensional tensor Fourier basis does not have a good way to represent $sin(xy)$. This property of the approximator is called (non)isotropy and more detail can be found in this wonderful talk about function approximation for multidimensional integration (cubature). Neural networks are naturally not aligned to a basis.
     
  3. Neural networks are "easy" to compute. There's good software for them, GPU-acceleration, and all other kinds of tooling that make them particularly simple to use.
     
  4. There are proofs that in many scenarios for neural networks the local minima are the global minima, meaning that local optimization is sufficient for training a neural network. Global optimization (which we will cover later in the course) is much more expensive than local methods like gradient descent, and thus this can be a good property to abuse for faster computation.
     
 
From Machine Learning to Scientific Machine Learning: Structure and Science
 
This understanding of a neural network and their libraries directly bridges to the understanding of scientific machine learning and the computation done in the field. In scientific machine learning, neural networks and machine learning are used as the basis to solve problems in scientific computing. Scientific computing, as a discipline also known as Computational Science, is a field of study which focuses on scientific simulation, using tools such as differential equations to investigate physical, biological, and other phenomena.
 
What we wish to do in scientific machine learning is use these properties of neural networks to improve the way that we investigate our scientific models.
 
Aside: Why Differential Equations?
 
Why do differential equations come up so often in as the model in the scientific context? This is a deep question with quite a simple answer. Essentially, all scientific experiments always have to test how things change. For example, you take a system now, you change it, and your measurement is how the changes you made caused changes in the system. This boils down to gather information about how, for some arbitrary system $y = f(x)$, how $\Delta x$ is related to $\Delta y$. Thus what you learn from scientific experiments, what is codified as scientific laws, is not "the answer", but the answer to how things change. This process of writing down equations by describing how they change precisely gives differential equations.
 
Solving ODEs with Neural Networks: The Physics-Informed Neural Network
 
Now let's get to our first true SciML application: solving ordinary differential equations with neural networks. The process of solving a differential equation with a neural network, or using a differential equation as a regularizer in the loss function, is known as a physics-informed neural network, since this allows for physical equations to guide the training of the neural network in circumstances where data might be lacking.
 
Background: A Method for Solving Ordinary Differential Equations with Neural Networks
 
This is a result first due to Lagaris et. al from 1998. The idea is to solve differential equations using neural networks by representing the solution by a neural network and training the resulting network to satisfy the conditions required by the differential equation.
 
Let's say we want to solve a system of ordinary differential equations
 
\[ u' = f(u,t) \]
 
with $t \in [0,1]$ and a known initial condition $u(0)=u_0$. To solve this, we approximate the solution by a neural network:
 
\[ NN(t) \approx u(t) \]
 
If $NN(t)$ was the true solution, then it would hold that $NN'(t) = f(NN(t),t)$ for all $t$. Thus we turn this condition into our loss function. This motivates the loss function:
 
\[ L(p) = \sum_i \left(\frac{dNN(t_i)}{dt} - f(NN(t_i),t_i) \right)^2 \]
 
The choice of $t_i$ could be done in many ways: it can be random, it can be a grid, etc. Anyways, when this loss function is minimized (gradients computed with standard reverse-mode automatic differentiation), then we have that $\frac{dNN(t_i)}{dt} \approx f(NN(t_i),t_i)$ and thus $NN(t)$ approximately solves the differential equation.
 
Note that we still have to handle the initial condition. One simple way to do this is to add an initial condition term to the cost function. This would look like:
 
\[ L(p) = (NN(0) - u_0)^2 + \sum_i \left(\frac{dNN(t_i)}{dt} - f(NN(t_i),t_i) \right)^2 \]
 
While that would work, it can be more efficient to encode the initial condition into the function itself so that it's trivially satisfied for any possible set of parameters. For example, instead of directly using a neural network, we can use:
 
\[ g(t) = u_0 + tNN(t) \]
 
as our solution. Notice that $g(t)$ is thus a universal approximator for all continuous functions such that $g(0)=u_0$ (this is a property one should prove!). Since $g(t)$ will always satisfy the initial condition, we can train $g(t)$ to satisfy the derivative function then it will automatically be a solution to the derivative function. In this sense, we can use the loss function:
 
\[ L(p) = \sum_i \left(\frac{dg(t_i)}{dt} - f(g(t_i),t_i) \right)^2 \]
 
where $p$ are the parameters that define $g$, which in turn are the parameters which define the neural network $NN$ that define $g$. Thus this reduces down, once again, to simply finding weights which minimize a loss function!
 
Coding Up the Method
 
Now let's implement this method with Flux. Let's define a neural network to be the NN(t) above. To make the problem easier, let's look at the ODE:
 
\[ u' = \cos 2\pi t \]
 
and approximate it with the neural network from a scalar to a scalar:
 
 using Flux
 NNODE = Chain(x -> [x], # Take in a scalar and transform it into an array
@@ -223,7 +223,7 @@
            first) # Take first value, i.e. return a scalar
 NNODE(1.0)
 
 
-0.10085204f0
+-0.23149511f0
 
 
Instead of directly approximating the neural network, we will use the transformed equation that is forced to satisfy the boundary conditions. Using u0=1.0, we have the function:
 
 g(t) = t*NNODE(t) + 1f0
 
 
@@ -247,23 +247,23 @@
 display(loss())
 Flux.train!(loss, Flux.params(NNODE), data, opt; cb=cb)
 
 
-0.5089291687042273
-0.4660090972283474
-0.37798210626977025
-0.18461971004026578
-0.04371968207719241
-0.019536930111555596
-0.014619263249474197
-0.012650337532635803
-0.011729013939159596
-0.011283318492406378
-0.010946978192004615
+0.5708075887278983
+0.4843225444558577
+0.431599836626712
+0.26459017110776495
+0.05310078897817342
+0.011030989241838655
+0.007069401923025258
+0.00634987145212996
+0.00601922323173909
+0.0057577987424845736
+0.005500448268481607
 
 
How well did this do? Well if we take the integral of both sides of our differential equation, we see it's fairly trivial:
 
\[ \int g' = g = \int \cos 2\pi t = C + \frac{\sin 2\pi t}{2\pi} \]
 
where we defined $C = 1$. Let's take a bunch of (input,output) pairs from the neural network and plot it against the analytical solution to the differential equation:
 
 using Plots
 t = 0:0.001:1.0
 plot(t,g.(t),label="NN")
 plot!(t,1.0 .+ sin.(2Ï€.*t)/2Ï€, label = "True Solution")
-
  
We see that it matches very well, and we can keep improving this fit by increasing the size of the neural network, using more training points, and training for more iterations.
 
Example: Harmonic Oscillator Informed Training
 
Using this idea, differential equations encoding physical laws can be utilized inside of loss functions for terms which we have some basis to believe should approximately follow some physical system. Let's investigate this last step by looking at how to inform the training of a neural network using the harmonic oscillator.
 
Let's assume that we are taking measurements of (position,force) in some real one-dimensional spring pushing and pulling against a wall.
 
 
But instead of the simple spring, let's assume we had a more complex spring, for example, let's say $F(x) = -kx + 0.1sin(x)$ where this extra term is due to some deformities in the metal (assume mass=1). Then by Newton's law of motion we have a second order ordinary differential equation:
 
\[ x'' = -kx + 0.1 \sin(x) \]
 
We can use the DifferentialEquations.jl package to solve this differential equation and see what this system looks like:
 
+
  
We see that it matches very well, and we can keep improving this fit by increasing the size of the neural network, using more training points, and training for more iterations.
 
Example: Harmonic Oscillator Informed Training
 
Using this idea, differential equations encoding physical laws can be utilized inside of loss functions for terms which we have some basis to believe should approximately follow some physical system. Let's investigate this last step by looking at how to inform the training of a neural network using the harmonic oscillator.
 
Let's assume that we are taking measurements of (position,force) in some real one-dimensional spring pushing and pulling against a wall.
 
 
But instead of the simple spring, let's assume we had a more complex spring, for example, let's say $F(x) = -kx + 0.1sin(x)$ where this extra term is due to some deformities in the metal (assume mass=1). Then by Newton's law of motion we have a second order ordinary differential equation:
 
\[ x'' = -kx + 0.1 \sin(x) \]
 
We can use the DifferentialEquations.jl package to solve this differential equation and see what this system looks like:
 
 using DifferentialEquations
 k = 1.0
 force(dx,x,k,t) = -k*x + 0.1sin(x)
@@ -300,7 +300,7 @@
 loss() = sum(abs2,NNForce(position_data[i]) - force_data[i] for i in 1:length(position_data))
 loss()
 
 
-0.0021075816164434196
+0.001639101475417397
 
 
Our random parameters do not do so well, so let's train!
 
 opt = Flux.Descent(0.01)
 data = Iterators.repeated((), 5000)
@@ -314,24 +314,24 @@
 display(loss())
 Flux.train!(loss, Flux.params(NNForce), data, opt; cb=cb)
 
 
-0.0021075816164434196
-0.0016626665734058707
-0.0014129151050661442
-0.0012004879578227366
-0.001019683666213778
-0.00086574214000748
-0.0007346664655364099
-0.0006230813332454321
-0.0005281206353467606
-0.0004473482765489638
-0.00037868342158744984
+0.001639101475417397
+0.001281078565346881
+0.001078228101263855
+0.000907144491383387
+0.000762834587654969
+0.0006411230663627027
+0.0005385050286698408
+0.00045202585782455135
+0.0003791885175632164
+0.00031788120953395697
+0.0002663162119004075
 
 
The neural network almost exactly matched the dataset, but how well did it actually learn the real force function? Let's plot it to see:
 
 learned_force_plot = NNForce.(positions_plot)
 
 plot(plot_t,force_plot,xlabel="t",label="True Force")
 plot!(plot_t,learned_force_plot,label="Predicted Force")
 scatter!(t,force_data,label="Force Measurements")
-
  
Ouch. The problem is that a neural network can approximate any function, so it approximated a function that fits the data, but not the correct function. We somehow need to have more data... but where can we get more data?
 
Well, even a first year undergrad in physics will know Hooke's law, which is that the idealized spring should satisfy $F(x) = -kx$. This is a decent assumption for the evolution of the system:
 
+
  
Ouch. The problem is that a neural network can approximate any function, so it approximated a function that fits the data, but not the correct function. We somehow need to have more data... but where can we get more data?
 
Well, even a first year undergrad in physics will know Hooke's law, which is that the idealized spring should satisfy $F(x) = -kx$. This is a decent assumption for the evolution of the system:
 
 force2(dx,x,k,t) = -k*x
 prob_simplified = SecondOrderODEProblem(force2,1.0,0.0,(0.0,10.0),k)
 sol_simplified = solve(prob_simplified)
@@ -342,7 +342,7 @@
 loss_ode() = sum(abs2,NNForce(x) - (-k*x) for x in random_positions)
 loss_ode()
 
 
-10.993295820275433
+8.460564225345507
 
 
If this term is zero, then $F(x) = -kx$, which is approximately true. So now let's put these together:
 
 λ = 0.1
 composed_loss() = loss() + λ*loss_ode()
@@ -367,15 +367,15 @@
 plot!(plot_t,learned_force_plot,label="Predicted Force")
 scatter!(t,force_data,label="Force Measurements")
 
 
-1.0997082654491308
-0.0005850661060708156
-0.0005522938565740613
-0.0005232864079531569
-0.0004973263251085195
-0.00047389858231739466
-0.000452611856433718
-0.0004331665468480585
-0.00041531905615030537
-0.00039886843159506777
-0.000383654997635387
-
  
And there we go: we have used knowledge of physics to help inform our neural network training process!
 
Conclusion
 
In this lecture we motivated machine learning not as a process of predicting from data but as a process for learning arbitrary nonlinear functions. Neural networks were just one choice of possible function. We then demonstrated how differential equations could be solved using this function approximation technique and then put together these two domains, solving differential equations and approximating data, into a single process to allow for physical knowledge to be embedded into the training process of a neural network, thus arriving at a physics-informed neural network. This is just one method in scientific machine learning which we will be exploring in more detail, demonstrating how we can utilize scientific knowledge to improve fits and allow for data-efficient machine learning.
 
 
 Published from sciml.jmd using Weave.jl v0.10.12 on 2024-07-26. 
 
 
  
   
\ No newline at end of file
+0.8463227387464513
+0.000210408027752647
+0.00020519506792570903
+0.00020026862808211442
+0.00019559928791649267
+0.00019116211020453466
+0.00018693271539265902
+0.00018289622008194647
+0.00017903277909909004
+0.00017533236251209575
+0.0001717819891828354
+
  
And there we go: we have used knowledge of physics to help inform our neural network training process!
 
Conclusion
 
In this lecture we motivated machine learning not as a process of predicting from data but as a process for learning arbitrary nonlinear functions. Neural networks were just one choice of possible function. We then demonstrated how differential equations could be solved using this function approximation technique and then put together these two domains, solving differential equations and approximating data, into a single process to allow for physical knowledge to be embedded into the training process of a neural network, thus arriving at a physics-informed neural network. This is just one method in scientific machine learning which we will be exploring in more detail, demonstrating how we can utilize scientific knowledge to improve fits and allow for data-efficient machine learning.
 
 
 Published from sciml.jmd using Weave.jl v0.10.12 on 2024-07-26. 
 
 
  
   
\ No newline at end of file
diff --git a/notes/04-How_Loops_Work-An_Introduction_to_Discrete_Dynamics/index.html b/notes/04-How_Loops_Work-An_Introduction_to_Discrete_Dynamics/index.html
index 69e459e4..66a012d1 100644
--- a/notes/04-How_Loops_Work-An_Introduction_to_Discrete_Dynamics/index.html
+++ b/notes/04-How_Loops_Work-An_Introduction_to_Discrete_Dynamics/index.html
@@ -113,7 +113,7 @@
 end
 @time solve_system_save(lorenz,[1.0,0.0,0.0],p,1000)
 
-0.000050 seconds (1.00 k allocations: 86.062 KiB)
+0.000054 seconds (1.00 k allocations: 86.062 KiB)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -138,7 +138,7 @@
 
 @time solve_system_save_push(lorenz,[1.0,0.0,0.0],p,1000)
 
-0.012133 seconds (4.50 k allocations: 336.016 KiB, 99.40% compilation tim
+0.011324 seconds (4.50 k allocations: 336.016 KiB, 99.43% compilation tim
 e)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
@@ -164,7 +164,7 @@

The first time Julia compiles the function, and the second is a straight call.

 @time solve_system_save_push(lorenz,[1.0,0.0,0.0],p,1000)
 
-0.000057 seconds (1.01 k allocations: 99.984 KiB)
+0.000056 seconds (1.01 k allocations: 99.984 KiB)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -190,7 +190,7 @@
 using BenchmarkTools
 @btime solve_system_save(lorenz,[1.0,0.0,0.0],p,1000)
 
-38.372 μs (1001 allocations: 86.06 KiB)
+27.531 μs (1001 allocations: 86.06 KiB)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -215,7 +215,7 @@
 
 @btime solve_system_save_push(lorenz,[1.0,0.0,0.0],p,1000)
 
-44.223 μs (1006 allocations: 99.98 KiB)
+33.333 μs (1006 allocations: 99.98 KiB)
 1000-element Vector{Vector{Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -248,7 +248,7 @@
 end
 @btime solve_system_save_matrix(lorenz,[1.0,0.0,0.0],p,1000)
 
-69.190 μs (2001 allocations: 179.66 KiB)
+66.845 μs (2001 allocations: 179.66 KiB)
 3×1000 Matrix{Float64}:
  1.0  0.8   0.752    0.80096   0.920338   …   1.98201    1.67886    1.4744
  0.0  0.56  0.9968   1.39785   1.81805        0.466287   0.656559   0.85300
@@ -265,7 +265,7 @@
 end
 @btime solve_system_save_matrix_view(lorenz,[1.0,0.0,0.0],p,1000)
 
-34.535 μs (1002 allocations: 101.61 KiB)
+34.956 μs (1002 allocations: 101.61 KiB)
 3×1000 Matrix{Float64}:
  1.0  0.8   0.752    0.80096   0.920338   …   1.98201    1.67886    1.4744
  0.0  0.56  0.9968   1.39785   1.81805        0.466287   0.656559   0.85300
@@ -282,7 +282,7 @@
 end
 @btime solve_system_save_matrix_resize(lorenz,[1.0,0.0,0.0],p,1000)
 
-1.198 ms (2318 allocations: 11.65 MiB)
+1.043 ms (2318 allocations: 11.65 MiB)
 3×1000 Matrix{Float64}:
  1.0  0.8   0.752    0.80096   0.920338   …   1.98201    1.67886    1.4744
  0.0  0.56  0.9968   1.39785   1.81805        0.466287   0.656559   0.85300
@@ -388,7 +388,7 @@

which would compute f and then take the values of du and update u with them, but that's 3 extra operations than required, whereas u,du = du,u will change u to be a pointer to the updated memory and now du is an "empty" cache array that we can refill (this decreases the computational cost by ~33%). Let's see what the cost is with this newest version:

 @btime solve_system(lorenz,[1.0,0.0,0.0],p,1000)
 
-37.210 μs (1000 allocations: 78.12 KiB)
+25.919 μs (1000 allocations: 78.12 KiB)
 3-element Vector{Float64}:
   1.4744010677851374
   0.8530017039412324
@@ -396,7 +396,7 @@
 
 @btime solve_system_mutate(lorenz,[1.0,0.0,0.0],p,1000)
 
-7.301 μs (3 allocations: 240 bytes)
+7.291 μs (3 allocations: 240 bytes)
 3-element Vector{Float64}:
   1.4744010677851374
   0.8530017039412324
@@ -445,7 +445,7 @@
 
 @btime solve_system_save(lorenz,@SVector[1.0,0.0,0.0],p,1000)
 
-8.766 μs (2 allocations: 23.48 KiB)
+8.850 μs (2 allocations: 23.48 KiB)
 1000-element Vector{SVector{3, Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -511,7 +511,7 @@
 
 @btime solve_system_save(lorenz,@SVector[1.0,0.0,0.0],p,1000)
 
-6.723 μs (2 allocations: 23.48 KiB)
+6.721 μs (2 allocations: 23.48 KiB)
 1000-element Vector{SVector{3, Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
@@ -536,7 +536,7 @@

And we can get down to non-allocating for the loop:

 @btime solve_system(lorenz,@SVector([1.0,0.0,0.0]),p,1000)
 
-6.502 μs (1 allocation: 32 bytes)
+6.496 μs (1 allocation: 32 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
   1.4744010677851374
   0.8530017039412324
@@ -552,7 +552,7 @@
 u = Vector{typeof(@SVector([1.0,0.0,0.0]))}(undef,1000)
 @btime solve_system_save!(u,lorenz,@SVector([1.0,0.0,0.0]),p,1000)
 
-6.498 μs (0 allocations: 0 bytes)
+6.500 μs (0 allocations: 0 bytes)
 1000-element Vector{SVector{3, Float64}}:
  [1.0, 0.0, 0.0]
  [0.8, 0.56, 0.0]
diff --git a/notes/05-The_Basics_of_Single_Node_Parallel_Computing/index.html b/notes/05-The_Basics_of_Single_Node_Parallel_Computing/index.html
index 18131d95..159fb83d 100644
--- a/notes/05-The_Basics_of_Single_Node_Parallel_Computing/index.html
+++ b/notes/05-The_Basics_of_Single_Node_Parallel_Computing/index.html
@@ -127,7 +127,7 @@
 u = [Vector{Float64}(undef,3) for i in 1:1000]
 @btime solve_system_save_iip!(u,lorenz_mt!,[1.0,0.0,0.0],p,1000);
 
-1.202 ms (5995 allocations: 608.84 KiB)
+1.203 ms (5995 allocations: 608.84 KiB)

Parallelism doesn't always make things faster. There are two costs associated with this code. For one, we had to go to the slower heap+mutation version, so its implementation starting point is slower. But secondly, and more importantly, the cost of spinning a new thread is non-negligible. In fact, here we can see that it even needs to make a small allocation for the new context. The total cost is on the order of It's on the order of 50ns: not huge, but something to take note of. So what we've done is taken almost free calculations and made them ~50ns by making each in a different thread, instead of just having it be one thread with one call stack.

The moral of the story is that you need to make sure that there's enough work per thread in order to effectively accelerate a program with parallelism.

Data-Parallel Problems

So not every setup is amenable to parallelism. Dynamical systems are notorious for being quite difficult to parallelize because the dependency of the future time step on the previous time step is clear, meaning that one cannot easily "parallelize through time" (though it is possible, which we will study later).

However, one common way that these systems are generally parallelized is in their inputs. The following questions allow for independent simulations:

  • What steady state does an input u0 go to for some list/region of initial conditions?

  • How does the solution very when I use a different p?

The problem has a few descriptions. For one, it's called an embarrassingly parallel problem since the problem can remain largely intact to solve the parallelism problem. To solve this, we can use the exact same solve_system_save_iip!, and just change how we are calling it. Secondly, this is called a data parallel problem, since it parallelized by splitting up the input data (here, the possible u0 or ps) and acting on them independently.

Multithreaded Parameter Searches

Now let's multithread our parameter search. Let's say we wanted to compute the mean of the values in the trajectory. For a single input pair, we can compute that like:

 using Statistics
 function compute_trajectory_mean(u0,p)
@@ -137,7 +137,7 @@
 end
 @btime compute_trajectory_mean(@SVector([1.0,0.0,0.0]),p)
 
-7.647 μs (3 allocations: 23.52 KiB)
+7.612 μs (3 allocations: 23.52 KiB)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -151,7 +151,7 @@
 end
 @btime compute_trajectory_mean2(@SVector([1.0,0.0,0.0]),p)
 
-7.464 μs (3 allocations: 112 bytes)
+7.469 μs (3 allocations: 112 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -165,7 +165,7 @@
 end
 @btime compute_trajectory_mean3(@SVector([1.0,0.0,0.0]),p)
 
-7.424 μs (1 allocation: 32 bytes)
+7.421 μs (1 allocation: 32 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -178,7 +178,7 @@
 compute_trajectory_mean4(u0,p) = _compute_trajectory_mean4(_u_cache,u0,p)
 @btime compute_trajectory_mean4(@SVector([1.0,0.0,0.0]),p)
 
-7.429 μs (1 allocation: 32 bytes)
+7.426 μs (1 allocation: 32 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -187,50 +187,50 @@
 ps = [(0.02,10.0,28.0,8/3) .* (1.0,rand(3)...) for i in 1:1000]
 
 1000-element Vector{NTuple{4, Float64}}:
- (0.02, 9.106049390932355, 3.303812946184277, 2.0289376601917164)
- (0.02, 4.735606575537758, 24.169178529667747, 1.3480009969264561)
- (0.02, 1.8782542650718392, 22.557130006439053, 2.0243816234869705)
- (0.02, 5.302261087000523, 27.07006533567516, 2.655762043438801)
- (0.02, 5.670338250773302, 4.441306677748524, 0.9560932489779355)
- (0.02, 1.4007298019106085, 8.924613440852292, 1.7845149466502728)
- (0.02, 4.210072135010345, 12.885641724584875, 1.4106354338111666)
- (0.02, 6.178913314158388, 14.077198942044106, 2.6422469838602893)
- (0.02, 9.022764003735679, 5.959401451034244, 0.2915953150851287)
- (0.02, 7.637994682331543, 2.280140073955992, 1.3174704247968336)
+ (0.02, 2.0494785010995353, 12.87474091643628, 2.221379168745697)
+ (0.02, 3.0141933992052494, 5.011093405297812, 0.9795778033401599)
+ (0.02, 6.6087512736768526, 22.554785148961784, 0.3707891781853476)
+ (0.02, 0.5160432877281274, 2.1442560677379037, 1.7411393467364724)
+ (0.02, 8.558522987629267, 18.545203958656234, 1.9346962153448815)
+ (0.02, 9.54542624443799, 14.894432021569349, 2.2001967050663778)
+ (0.02, 7.164909712680697, 14.058087153752206, 1.9721359183413034)
+ (0.02, 4.503455496371159, 15.638037179709096, 0.5478848102231814)
+ (0.02, 5.0067292534598415, 23.498092029722358, 1.8321033765018702)
+ (0.02, 2.043018481497864, 24.8343511074286, 0.40227226208229894)
  â‹®
- (0.02, 1.4295983611840468, 22.723025540961917, 0.5603759621647108)
- (0.02, 3.5691263501247104, 11.614453902298626, 0.9865390285286525)
- (0.02, 6.347641844244725, 11.287551194973615, 2.170255707975067)
- (0.02, 4.3184493713301455, 24.303906061742534, 0.09612994921103102)
- (0.02, 6.080688519020651, 7.21028271370196, 0.8896348073525427)
- (0.02, 6.3902301771886725, 13.890419308445491, 0.5118246742927562)
- (0.02, 3.0250459886141687, 16.614712639906436, 2.5380729556522454)
- (0.02, 1.4197562136945086, 12.915115904241887, 2.1974841648464785)
- (0.02, 2.760108862922477, 8.843016809441602, 0.6191363970184215)
+ (0.02, 9.459925043278687, 14.913762836267876, 1.946411348397799)
+ (0.02, 3.7058223528064884, 9.956749354099607, 2.5430008163316744)
+ (0.02, 5.6913112833876225, 12.407156482636111, 1.0651014726178387)
+ (0.02, 0.09131385334597009, 7.6889382933701, 2.0354078697915967)
+ (0.02, 1.1885766367375572, 19.114097281325094, 1.381133123272827)
+ (0.02, 3.9804371550787767, 23.853405760253743, 1.185532990309715)
+ (0.02, 8.84078503958126, 5.847345434942631, 1.0196090739651438)
+ (0.02, 5.582363837550646, 11.995288548626041, 0.041581327093415034)
+ (0.02, 0.643229668934614, 23.346238053688726, 0.22539644325045938)

And let's get the mean of the trajectory for each of the parameters.

 serial_out = map(p -> compute_trajectory_mean4(@SVector([1.0,0.0,0.0]),p),ps)
 
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

Now let's do this with multithreading:

 function tmap(f,ps)
   out = Vector{typeof(@SVector([1.0,0.0,0.0]))}(undef,1000)
@@ -243,26 +243,26 @@
 threaded_out = tmap(p -> compute_trajectory_mean4(@SVector([1.0,0.0,0.0]),p),ps)
 
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

Let's check the output:

 serial_out - threaded_out
 
@@ -296,7 +296,7 @@
 end
 @btime compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p)
 
-7.429 μs (1 allocation: 32 bytes)
+7.424 μs (1 allocation: 32 bytes)
 3-element SVector{3, Float64} with indices SOneTo(3):
  -0.31149962346484683
  -0.3097490174897651
@@ -330,53 +330,53 @@
 
 @btime serial_out = map(p -> compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p),ps)
 
-7.420 ms (3 allocations: 23.50 KiB)
+7.419 ms (3 allocations: 23.50 KiB)
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]
 
 @btime threaded_out = tmap(p -> compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p),ps)
 
-7.420 ms (8 allocations: 24.06 KiB)
+7.421 ms (8 allocations: 24.06 KiB)
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

Hierarchical Task-Based Multithreading and Dynamic Scheduling

The major change in Julia v1.3 is that Julia's Tasks, which are traditionally its green threads interface, are now the basis of its multithreading infrastructure. This means that all independent threads are parallelized, and a new interface for multithreading will exist that works by spawning threads.

This implementation follows Go's goroutines and the classic multithreading interface of Cilk. There is a Julia-level scheduler that handles the multithreading to put different tasks on different vCPU threads. A benefit from this is hierarchical multithreading. Since Julia's tasks can spawn tasks, what can happen is a task can create tasks which create tasks which etc. In Julia (/Go/Cilk), this is then seen as a single pool of tasks which it can schedule, and thus it will still make sure only N are running at a time (as opposed to the naive implementation where the total number of running threads is equal then multiplied). This is essential for numerical performance because running multiple compute threads on a single CPU thread requires constant context switching between the threads, which will slow down the computations.

To directly use the task-based interface, simply use Threads.@spawn to spawn new tasks. For example:

 function tmap2(f,ps)
   tasks = [Threads.@spawn f(ps[i]) for i in 1:1000]
@@ -385,51 +385,51 @@
 threaded_out = tmap2(p -> compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p),ps)
 
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

However, if we check the timing we see:

 @btime tmap2(p -> compute_trajectory_mean5(@SVector([1.0,0.0,0.0]),p),ps)
 
-7.761 ms (6005 allocations: 562.70 KiB)
+7.741 ms (6005 allocations: 562.70 KiB)
 1000-element Vector{SVector{3, Float64}}:
- [2.0820907627308065, 2.088471204219544, 2.155235431072218]
- [1.5595505244183643, 1.5099684640627067, 22.095262274445847]
- [6.126638249655104, 6.273549674375277, 19.649196180507175]
- [0.2695734398328459, 0.36474536430110527, 22.702246606047403]
- [1.751827841218394, 1.7589379670019873, 3.2596952947386217]
- [3.6717748780283044, 3.770314179696095, 7.617740491703323]
- [-3.2187205975639945, -3.2130512092288646, 11.079628189747254]
- [-1.3436994956146562, -1.3487865435138944, 10.286513589875154]
- [-0.40026948095723325, -0.41137422951603414, 4.484011162661078]
- [1.2403493173367404, 1.2423043385425232, 1.163725766067324]
+ [4.999057172451228, 5.0999726149344795, 11.40861028611877]
+ [1.9095218957936158, 1.9255606407677823, 3.7801161867320436]
+ [0.018098089685998762, 0.006162305680758088, 22.30859678284725]
+ [1.3360619709669204, 1.3759318173846775, 1.0332497381762005]
+ [-0.39175341662834273, -0.3765521475273949, 16.121629585090233]
+ [-1.525844575711785, -1.5169080412234797, 12.195050923062764]
+ [-2.377134690797774, -2.384556348363591, 11.71714223980902]
+ [0.6927113721729414, 0.6709618126493168, 14.094390880667179]
+ [-0.7904811808266985, -0.8894107857525068, 20.100852612157897]
+ [0.33547871420519304, 0.4166653156824929, 23.17401896843689]
  â‹®
- [0.10984575880851026, 0.18303614258211953, 18.881095404828727]
- [-2.767005081412208, -2.825454726504213, 10.443233539110137]
- [-1.4824600192283712, -1.473288968668467, 8.213790730589803]
- [0.3373356921990089, 0.32575332045401934, 28.199920884030337]
- [-1.7375406489852403, -1.762221249211257, 5.848737373322062]
- [0.14296262183546848, 0.08144207197628298, 11.996268909515498]
- [-4.8803407614198, -5.012594562811099, 13.34548086739526]
- [4.994649988098529, 5.139638001158158, 11.50546102974038]
- [-1.7255017279562215, -1.7763243048645443, 7.6751484257085965]
+ [0.22875070399599703, 0.22677752744244983, 12.353008641915155]
+ [4.600804025864118, 4.651652635956364, 8.529173090072675]
+ [0.22214974573101073, 0.222892247834588, 10.251319400659881]
+ [2.8649842402742403, 4.284471959008289, 5.7544606666198845]
+ [4.867928591376662, 5.0364221417730635, 17.3381202260852]
+ [0.6735461553758312, 0.7447967652917319, 21.503598600914458]
+ [2.0801143623157454, 2.086828195134251, 4.571613185440295]
+ [0.1667586921903792, 0.15780191304537522, 15.403259875326595]
+ [-1.7202717446442446, -1.9734556539954884, 21.377374239524237]

Threads.@threads is built on the same multithreading infrastructure, so why is this so much slower? The reason is because Threads.@threads employs static scheduling while Threads.@spawn is using dynamic scheduling. Dynamic scheduling is the model of allowing the runtime to determine the ordering and scheduling of processes, i.e. what tasks will run run where and when. Julia's task-based multithreading system has a thread scheduler which will automatically do this for you in the background, but because this is done at runtime it will have overhead. Static scheduling is the model of pre-determining where and when tasks will run, instead of allowing this to be determined at runtime. Threads.@threads is "quasi-static" in the sense that it cuts the loop so that it spawns only as many tasks as there are threads, essentially assigning one thread for even chunks of the input data.

Does this lack of runtime overhead mean that static scheduling is "better"? No, it simply has trade-offs. Static scheduling assumes that the runtime of each block is the same. For this specific case where there are fixed number of loop iterations for the dynamical systems, we know that every compute_trajectory_mean5 costs exactly the same, and thus this will be more efficient. However, There are many cases where this might not be efficient. For example:

 function sleepmap_static()
   out = Vector{Int}(undef,24)
@@ -448,7 +448,7 @@
 @btime sleepmap_static()
 @btime sleepmap_spawn()
 
-30.057 s (104 allocations: 3.75 KiB)
+30.053 s (104 allocations: 3.75 KiB)
   2.401 s (220 allocations: 14.77 KiB)
 24-element Vector{Int64}:
   1
@@ -489,24 +489,24 @@
 A*B
 
 10000×10000 Matrix{Float64}:
- 2542.68  2521.97  2510.56  2497.9   …  2509.28  2525.0   2523.88  2520.4
- 2509.41  2523.24  2499.23  2466.59     2498.17  2490.19  2504.73  2494.0
- 2527.03  2511.33  2512.35  2475.63     2512.02  2515.73  2511.64  2515.72
- 2530.45  2522.96  2505.87  2477.47     2514.41  2511.84  2483.23  2504.55
- 2528.08  2517.44  2492.39  2481.61     2514.61  2518.98  2501.12  2506.62
- 2513.9   2530.09  2495.53  2463.74  …  2513.33  2488.51  2489.3   2499.18
- 2483.17  2492.63  2484.71  2445.09     2490.3   2491.62  2488.92  2477.49
- 2540.57  2535.27  2522.95  2491.98     2527.58  2505.16  2526.23  2522.58
- 2541.76  2536.15  2523.99  2492.26     2520.01  2514.86  2519.61  2519.08
- 2510.24  2515.49  2519.26  2479.86     2499.94  2503.5   2503.25  2500.0
+ 2525.87  2488.32  2499.41  2509.97  …  2511.13  2489.44  2531.07  2502.22
+ 2519.05  2496.88  2482.01  2506.51     2527.63  2493.75  2516.93  2503.97
+ 2534.89  2503.33  2495.34  2520.59     2529.65  2508.02  2536.28  2522.71
+ 2532.22  2520.62  2515.99  2518.95     2535.94  2524.97  2546.52  2512.56
+ 2514.36  2479.37  2504.63  2489.56     2507.64  2494.65  2523.54  2493.13
+ 2529.33  2500.05  2501.9   2505.24  …  2531.42  2486.38  2515.62  2512.86
+ 2540.09  2497.2   2504.75  2516.27     2521.17  2512.34  2529.63  2512.79
+ 2509.74  2477.49  2488.59  2479.55     2506.92  2485.0   2531.66  2480.12
+ 2481.75  2476.91  2490.11  2465.56     2465.47  2467.87  2509.78  2478.45
+ 2544.86  2496.0   2514.15  2517.92     2519.5   2511.37  2552.28  2527.42
     ⋮                                ⋱                             
- 2525.23  2512.21  2498.54  2488.94     2513.71  2501.97  2511.88  2520.2
- 2511.07  2491.4   2491.45  2465.4      2511.1   2497.52  2478.17  2490.51
- 2502.33  2508.4   2498.52  2465.6      2503.36  2502.37  2492.31  2501.05
- 2521.12  2509.38  2498.31  2474.29     2515.97  2501.83  2512.41  2488.79
- 2490.9   2496.69  2483.61  2441.27  …  2490.72  2471.2   2475.08  2476.2
- 2535.03  2541.2   2529.11  2497.66     2535.53  2517.47  2531.13  2519.93
- 2511.1   2510.73  2493.1   2492.95     2510.81  2499.15  2542.01  2518.67
- 2518.64  2526.74  2517.93  2488.57     2531.73  2504.55  2515.35  2513.5
- 2523.56  2519.73  2500.24  2485.91     2513.05  2535.31  2523.33  2518.76
+ 2527.82  2496.37  2506.14  2490.21     2519.83  2528.13  2550.02  2506.49
+ 2512.87  2490.24  2480.03  2490.32     2526.82  2474.85  2525.26  2493.48
+ 2494.52  2487.08  2483.83  2499.03     2504.63  2476.85  2533.39  2491.2
+ 2531.16  2499.71  2516.22  2520.35     2536.82  2507.74  2517.3   2530.53
+ 2521.62  2481.61  2484.72  2506.89  …  2514.74  2495.46  2525.36  2499.07
+ 2516.2   2491.17  2493.77  2509.39     2502.63  2491.1   2508.44  2497.63
+ 2499.9   2481.76  2506.27  2492.37     2494.85  2491.09  2518.55  2480.2
+ 2527.68  2503.12  2528.21  2511.25     2532.84  2503.63  2558.09  2533.89
+ 2540.76  2513.19  2517.55  2496.81     2522.99  2506.76  2533.4   2518.68

If you are using a computer that has N cores, then this will use N cores. Try it and look at your resource usage!

Array-Based Parallelism

The simplest form of parallelism is array-based parallelism. The idea is that you use some construction of an array whose operations are already designed to be parallel under the hood. In Julia, some examples of this are:

  • DistributedArrays (Distributed Computing)

  • Elemental

  • MPIArrays

  • CuArrays (GPUs)

This is not a Julia specific idea either.

BLAS and Standard Libraries

The basic linear algebra calls are all handled by a set of libraries which follow the same interface known as BLAS (Basic Linear Algebra Subroutines). It's divided into 3 portions:

  • BLAS1: Element-wise operations (O(n))

  • BLAS2: Matrix-vector operations (O(n^2))

  • BLAS3: Matrix-matrix operations (O(n^3))

BLAS implementations are highly optimized, like OpenBLAS and Intel MKL, so every numerical language and library essentially uses similar underlying BLAS implementations. Extensions to these, known as LAPACK, include operations like factorizations, and are included in these standard libraries. These are all multithreaded. The reason why this is a location to target is because the operation count is high enough that parallelism can be made efficient even when only targeting this level: a matrix multiplication can take on the order of seconds, minutes, hours, or even days, and these are all highly parallel operations. This means you can get away with a bunch just by parallelizing at this level, which happens to be a bottleneck for a lot scientific computing codes.

This is also commonly the level at which GPU computing occurs in machine learning libraries for reasons which we will explain later.

MPI

Well, this is a big topic and we'll address this one later!

Conclusion

The easiest forms of parallelism are:

  • Embarrassingly parallel

  • Array-level parallelism (built into linear algebra)

Exploit these when possible.

Published from parallelism_overview.jmd using Weave.jl v0.10.12 on 2024-07-26.

\ No newline at end of file diff --git a/notes/06-The_Different_Flavors_of_Parallelism/index.html b/notes/06-The_Different_Flavors_of_Parallelism/index.html index cd16297d..1ecaba87 100644 --- a/notes/06-The_Different_Flavors_of_Parallelism/index.html +++ b/notes/06-The_Different_Flavors_of_Parallelism/index.html @@ -9,26 +9,26 @@ arr = [MyComplex(rand(),rand()) for i in 1:100] 
 100-element Vector{MyComplex}:
- MyComplex(0.14309417245278067, 0.9215224222944839)
- MyComplex(0.6563768312045752, 0.8044767963545393)
- MyComplex(0.5728049188895872, 0.33765746952116094)
- MyComplex(0.8240073299674354, 0.9470426273216329)
- MyComplex(0.7606061347124898, 0.9327907399185521)
- MyComplex(0.755341833759231, 0.5434958255089687)
- MyComplex(0.9731438085219921, 0.31820007095612113)
- MyComplex(0.6770182257794624, 0.3681202537043393)
- MyComplex(0.08480739105077983, 0.65477879435604)
- MyComplex(0.879044108699869, 0.5694763191704512)
+ MyComplex(0.5948216400563927, 0.7202742856092115)
+ MyComplex(0.8858224485878035, 0.04464426300983426)
+ MyComplex(0.4536724251568013, 0.48301713342468633)
+ MyComplex(0.17632669227127473, 0.966975004988031)
+ MyComplex(0.7423602389573798, 0.7381611563859961)
+ MyComplex(0.5049342978821135, 0.2713721944949681)
+ MyComplex(0.8956475760986714, 0.006535319763830483)
+ MyComplex(0.5275017128612832, 0.6089884504844824)
+ MyComplex(0.22199738442114458, 0.3133842057610531)
+ MyComplex(0.7056746294589479, 0.17085581295301622)
  â‹®
- MyComplex(0.6184831153175321, 0.8420717430705801)
- MyComplex(0.11097401639991189, 0.4733738650181497)
- MyComplex(0.9598982066766498, 0.5024876345986231)
- MyComplex(0.7704581759272878, 0.06431948804402998)
- MyComplex(0.5067467487289342, 0.37276230976872216)
- MyComplex(0.22392490395791997, 0.24997212682983505)
- MyComplex(0.7938100950862472, 0.499794534596235)
- MyComplex(0.5800027114018041, 0.5539306180949976)
- MyComplex(0.8613302068792389, 0.11099259999056299)
+ MyComplex(0.8949805161160401, 0.421323667111285)
+ MyComplex(0.026053733946824753, 0.9006485214268137)
+ MyComplex(0.8300583605559014, 0.8018415810144012)
+ MyComplex(0.12592125655230224, 0.27000881231142315)
+ MyComplex(0.3618587243109601, 0.4880572513222716)
+ MyComplex(0.3150275713704259, 0.13939673381769024)
+ MyComplex(0.03638847026177339, 0.8020351165630957)
+ MyComplex(0.08217419775238222, 0.7933908176556499)
+ MyComplex(0.03792572372188874, 0.1330438491287449)

is represented in memory as

[real1,imag1,real2,imag2,...]

while the struct of array formats are

@@ -43,18 +43,18 @@ 
-MyComplexes([0.2897241260009822, 0.48104950344076425, 0.1550991673426182, 0
-.4303053146297099, 0.9652545002585547, 0.129691784563555, 0.930721759368184
-5, 0.04236592750770252, 0.12121370563973322, 0.6554310506492863  …  0.73820
-40247731638, 0.057317858458742266, 0.20857864801382386, 0.7618557750226116,
- 0.30777823594243947, 0.19986995680131348, 0.7036870522240618, 0.5790694102
-590798, 0.14664887682564642, 0.44821427291761984], [0.46688590924729956, 0.
-11887230044632568, 0.2806224613334578, 0.43126362776131033, 0.9794620042427
-044, 0.42610522792220007, 0.3077309824326909, 0.6937145539324716, 0.8143594
-272939957, 0.7105773101090823  …  0.24229997305610995, 0.18587298385876627,
- 0.24053515975314443, 0.9719512037832595, 0.03802957417362052, 0.4362822838
-808521, 0.4088166896868569, 0.8960777149330896, 0.5386366912236279, 0.07960
-564919501523])
+MyComplexes([0.4463978501148237, 0.9534146019623043, 0.3272506411204227, 0.
+6610621659220888, 0.7518861360127663, 0.6604957283856057, 0.568257801511073
+4, 0.5249214846587075, 0.28031214215080524, 0.4019397592681623  …  0.230795
+16952268475, 0.4587073529434471, 0.38888294861698836, 0.8764601865880245, 0
+.7664927638985168, 0.5500101294037606, 0.8467864454608346, 0.75150871387712
+67, 0.9504725250659736, 0.7891716988140378], [0.2354669170781668, 0.3484246
+5928318156, 0.4847930258306149, 0.5050470303321312, 0.8805989081762119, 0.3
+6730961426487874, 0.7503851083942035, 0.7838065781271188, 0.833361952697676
+8, 0.4100985361969356  …  0.7241872639287391, 0.2685519569209651, 0.2209582
+6963895615, 0.2074266359413821, 0.6228410415260852, 0.17911567432938458, 0.
+5661141870291473, 0.9051093078571997, 0.37633923990420315, 0.29487469397695
+53])
@@ -73,7 +73,7 @@ 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:5 within `average`
-define void @julia_average_10372([2 x double]* noalias nocapture noundef no
+define void @julia_average_10360([2 x double]* noalias nocapture noundef no
 nnull sret([2 x double]) align 8 dereferenceable(16) %0, {}* noundef nonnul
 l align 16 dereferenceable(40) %1) #0 {
 top:
@@ -108,21 +108,21 @@
             %.sub = getelementptr inbounds [4 x {}*], [4 x {}*]* %2, i64 0,
  i64 0
 ; ││││││││││ @ reduce.jl:432 within `_mapreduce`
-            store {}* inttoptr (i64 139788167070960 to {}*), {}** %.sub, al
+            store {}* inttoptr (i64 139762867029232 to {}*), {}** %.sub, al
 ign 8
             %5 = getelementptr inbounds [4 x {}*], [4 x {}*]* %2, i64 0, i6
 4 1
-            store {}* inttoptr (i64 139788214797840 to {}*), {}** %5, align
+            store {}* inttoptr (i64 139762914756112 to {}*), {}** %5, align
  8
             %6 = getelementptr inbounds [4 x {}*], [4 x {}*]* %2, i64 0, i6
 4 2
             store {}* %1, {}** %6, align 8
             %7 = getelementptr inbounds [4 x {}*], [4 x {}*]* %2, i64 0, i6
 4 3
-            store {}* inttoptr (i64 139788196084624 to {}*), {}** %7, align
+            store {}* inttoptr (i64 139762896042896 to {}*), {}** %7, align
  8
-            %8 = call nonnull {}* @ijl_invoke({}* inttoptr (i64 13978817579
-4192 to {}*), {}** nonnull %.sub, i32 4, {}* inttoptr (i64 139788383992288 
+            %8 = call nonnull {}* @ijl_invoke({}* inttoptr (i64 13976287575
+2464 to {}*), {}** nonnull %.sub, i32 4, {}* inttoptr (i64 139761459797280 
 to {}*))
             call void @llvm.trap()
             unreachable
@@ -233,7 +233,7 @@
 ; ││││││││││└
 ; ││││││││││ @ reduce.jl:447 within `_mapreduce`
 ; ││││││││││┌ @ reduce.jl:277 within `mapreduce_impl`
-             call void @j_mapreduce_impl_10374([2 x double]* noalias nocapt
+             call void @j_mapreduce_impl_10362([2 x double]* noalias nocapt
 ure noundef nonnull sret([2 x double]) %tmpcast, {}* nonnull %1, i64 signex
 t 1, i64 signext %arraylen, i64 signext 1024)
 ; └└└└└└└└└└└
@@ -583,7 +583,7 @@ 
Next Level Up: Multithreading

 
 
 
-59.061 μs (6 allocations: 576 bytes)
+59.671 μs (6 allocations: 576 bytes)
 

 
 
@@ -596,7 +596,7 @@ 
Next Level Up: Multithreading

 
 
 
-66.033 μs (6 allocations: 576 bytes)
+66.023 μs (6 allocations: 576 bytes)
 

 
 
@@ -609,7 +609,7 @@ 
Next Level Up: Multithreading

 
 
 
-20.138 μs (6 allocations: 576 bytes)
+20.148 μs (6 allocations: 576 bytes)
 

 
 
@@ -637,17 +637,17 @@ 
Next Level Up: Multithreading

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:26 within `h`
-define void @julia_h_10465() #0 {
+define void @julia_h_10453() #0 {
 top:
-  %.promoted = load i64, i64* inttoptr (i64 139785926445312 to i64*), align
- 256
+  %.promoted = load i64, i64* inttoptr (i64 139761557327088 to i64*), align
+ 16
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:28 within `h`
   %0 = add i64 %.promoted, 10000
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:29 within `h`
 ; ┌ @ Base.jl within `setproperty!`
-   store i64 %0, i64* inttoptr (i64 139785926445312 to i64*), align 256
+   store i64 %0, i64* inttoptr (i64 139761557327088 to i64*), align 16
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:30 within `h`
@@ -673,7 +673,7 @@ 
Next Level Up: Multithreading

 
 
 
-2.785 ns (0 allocations: 0 bytes)
+3.095 ns (0 allocations: 0 bytes)
 

 
 
@@ -686,13 +686,13 @@ 
Next Level Up: Multithreading

 
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:3 within `h2`
-define void @julia_h2_10472() #0 {
+define void @julia_h2_10460() #0 {
 top:
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:6 within `h2`
 ; ┌ @ refvalue.jl:59 within `getindex`
 ; │┌ @ Base.jl:37 within `getproperty`
-    %0 = load i64, i64* inttoptr (i64 139787416178784 to i64*), align 32
+    %0 = load i64, i64* inttoptr (i64 139762055925984 to i64*), align 32
 ; └└
 ; ┌ @ range.jl:897 within `iterate`
 ; │┌ @ range.jl:672 within `isempty`
@@ -703,15 +703,15 @@ 
Next Level Up: Multithreading

   br i1 %1, label %L34, label %L18.preheader
 
 L18.preheader:                                    ; preds = %top
-  %.promoted = load i64, i64* inttoptr (i64 139785926445312 to i64*), align
- 256
+  %.promoted = load i64, i64* inttoptr (i64 139761557327088 to i64*), align
+ 16
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:8 within `h2`
   %2 = add i64 %.promoted, %0
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:7 within `h2`
 ; ┌ @ Base.jl within `setproperty!`
-   store i64 %2, i64* inttoptr (i64 139785926445312 to i64*), align 256
+   store i64 %2, i64* inttoptr (i64 139761557327088 to i64*), align 16
 ; â””
 ;  @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture06/styles_of_paral
 lelism.jmd:8 within `h2`
@@ -742,7 +742,7 @@ 
Next Level Up: Multithreading

 
 
 
-4.939 ns (0 allocations: 0 bytes)
+4.628 ns (0 allocations: 0 bytes)
 

 
 
diff --git a/notes/08-Forward-Mode_Automatic_Differentiation_(AD)_via_High_Dimensional_Algebras/index.html b/notes/08-Forward-Mode_Automatic_Differentiation_(AD)_via_High_Dimensional_Algebras/index.html
index c845e817..144ac434 100644
--- a/notes/08-Forward-Mode_Automatic_Differentiation_(AD)_via_High_Dimensional_Algebras/index.html
+++ b/notes/08-Forward-Mode_Automatic_Differentiation_(AD)_via_High_Dimensional_Algebras/index.html
@@ -18,9 +18,9 @@
 ϵ2 = (1+ϵ) - 1
 (ϵ - ϵ2)
 
 
-ϵ = 3.873035531141372e-11
-1 + ϵ = 1.0000000000387304
-3.0527602113762817e-18
+ϵ = 6.736227852961307e-11
+1 + ϵ = 1.0000000000673623
+-5.940030844987341e-17
 
 
See how $\epsilon$ is only rebuilt at accuracy around $10^{-16}$ and thus we only keep around 6 digits of accuracy when it's generated at the size of around $10^{-10}$!
 
Finite Differencing and Numerical Stability
 
To start understanding how to compute derivatives on a computer, we start with finite differencing. For finite differencing, recall that the definition of the derivative is:
 
\[ f'(x) = \lim_{\epsilon \rightarrow 0} \frac{f(x+\epsilon)-f(x)}{\epsilon} \]
 
Finite differencing directly follows from this definition by choosing a small $\epsilon$. However, choosing a good $\epsilon$ is very difficult. If $\epsilon$ is too large than there is error since this definition is asymptotic. However, if $\epsilon$ is too small, you receive roundoff error. To understand why you would get roundoff error, recall that floating point error is relative, and can essentially store 16 digits of accuracy. So let's say we choose $\epsilon = 10^{-6}$. Then $f(x+\epsilon) - f(x)$ is roughly the same in the first 6 digits, meaning that after the subtraction there is only 10 digits of accuracy, and then dividing by $10^{-6}$ simply brings those 10 digits back up to the correct relative size.
 
 
This means that we want to choose $\epsilon$ small enough that the $\mathcal{O}(\epsilon^2)$ error of the truncation is balanced by the $O(1/\epsilon)$ roundoff error. Under some minor assumptions, one can argue that the average best point is $\sqrt(E)$, where E is machine epsilon
 
 @show eps(Float64)
 @show sqrt(eps(Float64))
@@ -95,17 +95,17 @@
 add(a, b)
 @btime add($(Ref(a))[], $(Ref(b))[])
 
 
-3.396 ns (0 allocations: 0 bytes)
+3.105 ns (0 allocations: 0 bytes)
 Dual{Int64}(4, 6)
 
 
It seems like we have lost no performance.
 
 @code_native add(1, 2, 3, 4)
 
 
 .text
 	.file	"add"
-	.globl	julia_add_13194                 # -- Begin function julia_add_13194
+	.globl	julia_add_13182                 # -- Begin function julia_add_13182
 	.p2align	4, 0x90
-	.type	julia_add_13194,@function
-julia_add_13194:                        # @julia_add_13194
+	.type	julia_add_13182,@function
+julia_add_13182:                        # @julia_add_13182
 ; ┌ @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture08/automatic_diff
 erentiation.jmd:2 within `add`
 # %bb.0:                                # %top
@@ -121,7 +121,7 @@
 	pop	rbp
 	ret
 .Lfunc_end0:
-	.size	julia_add_13194, .Lfunc_end0-julia_add_13194
+	.size	julia_add_13182, .Lfunc_end0-julia_add_13182
 ; â””
                                         # -- End function
 	.section	".note.GNU-stack","",@progbits
@@ -130,10 +130,10 @@
 
 
 .text
 	.file	"add"
-	.globl	julia_add_13196                 # -- Begin function julia_add_13196
+	.globl	julia_add_13184                 # -- Begin function julia_add_13184
 	.p2align	4, 0x90
-	.type	julia_add_13196,@function
-julia_add_13196:                        # @julia_add_13196
+	.type	julia_add_13184,@function
+julia_add_13184:                        # @julia_add_13184
 ; ┌ @ /home/runner/work/SciMLBook/SciMLBook/_weave/lecture08/automatic_diff
 erentiation.jmd:5 within `add`
 # %bb.0:                                # %top
@@ -152,7 +152,7 @@
 	pop	rbp
 	ret
 .Lfunc_end0:
-	.size	julia_add_13196, .Lfunc_end0-julia_add_13196
+	.size	julia_add_13184, .Lfunc_end0-julia_add_13184
 ; └└└
                                         # -- End function
 	.section	".note.GNU-stack","",@progbits
@@ -274,7 +274,7 @@
 2-element Vector{Int64}:
  6
  2
-
 
Directional derivative and gradient of functions $f: \mathbb{R}^n \to \mathbb{R}$
 
For a function $f: \mathbb{R}^n \to \mathbb{R}$ the basic operation is the directional derivative:
 
\[ \lim_{\epsilon \to 0} \frac{f(\mathbf{x} + \epsilon \mathbf{v}) - f(\mathbf{x})}{\epsilon} = [\nabla f(\mathbf{x})] \cdot \mathbf{v}, \]
 
where $\epsilon$ is still a single dimension and $\nabla f(\mathbf{x})$ is the direction in which we calculate.
 
We can directly do this using the same simple Dual numbers as above, using the same $\epsilon$, e.g.
 
\[ f(x, y) = x^2 \sin(y) \]
 
\[ \begin{align} f(x_0 + a\epsilon, y_0 + b\epsilon) &= (x_0 + a\epsilon)^2 \sin(y_0 + b\epsilon) \\ &= x_0^2 \sin(y_0) + \epsilon[2ax_0 \sin(y_0) + x_0^2 b \cos(y_0)] + o(\epsilon) \end{align} \]
 
so we have indeed calculated $\nabla f(x_0, y_0) \cdot \mathbf{v},$ where $\mathbf{v} = (a, b)$ are the components that we put into the derivative component of the Dual numbers.
 
If we wish to calculate the directional derivative in another direction, we could repeat the calculation with a different $\mathbf{v}$. A better solution is to use another independent epsilon $\epsilon$, expanding $x = x_0 + a_1 \epsilon_1 + a_2 \epsilon_2$ and putting $\epsilon_1 \epsilon_2 = 0$.
 
In particular, if we wish to calculate the gradient itself, $\nabla f(x_0, y_0)$, we need to calculate both partial derivatives, which corresponds to two directional derivatives, in the directions $(1, 0)$ and $(0, 1)$, respectively.
 
Forward-Mode AD as jvp
 
Note that another representation of the directional derivative is $f'(x)v$, where $f'(x)$ is the Jacobian or total derivative of $f$ at $x$. To see the equivalence of this to a directional derivative, write it out in the standard basis:
 
\[ w_i = \sum_{j}^{m} J_{ij} v_{j} \]
 
Now write out what $J$ means and we see that:
 
\[ w_i = \sum_j^{m} \frac{df_i}{dx_j} v_j = \nabla f_i(x) \cdot v \]
 
The primitive action of forward-mode AD is $f'(x)v$!
 
This is also known as a Jacobian-vector product, or jvp for short.
 
We can thus represent vector calculus with multidimensional dual numbers as follows. Let $d =[x,y]$, the vector of dual numbers. We can instead represent this as:
 
\[ d = d_0 + v_1 \epsilon_1 + v_2 \epsilon_2 \]
 
where $d_0$ is the primal vector $[x_0,y_0]$ and the $v_i$ are the vectors for the dual directions. If you work out this algebra, then note that a single application of $f$ to a multidimensional dual number calculates:
 
\[ f(d) = f(d_0) + f'(d_0)v_1 \epsilon_1 + f'(d_0)v_2 \epsilon_2 \]
 
i.e. it calculates the result of $f(x,y)$ and two separate directional derivatives. Note that because the information about $f(d_0)$ is shared between the calculations, this is more efficient than doing multiple applications of $f$. And of course, this is then generalized to $m$ many directional derivatives at once by:
 
\[ d = d_0 + v_1 \epsilon_1 + v_2 \epsilon_2 + \ldots + v_m \epsilon_m \]
 
Jacobian
 
For a function $f: \mathbb{R}^n \to \mathbb{R}^m$, we reduce (conceptually, although not necessarily in code) to its component functions $f_i: \mathbb{R}^n \to \mathbb{R}$, where $f(x) = (f_1(x), f_2(x), \ldots, f_m(x))$.
 
Then
 
\[ \begin{align} f(x + \epsilon v) &= (f_1(x + \epsilon v), \ldots, f_m(x + \epsilon v)) \\ &= (f_1(x) + \epsilon[\nabla f_1(x) \cdot v], \dots, f_m(x) + \epsilon[\nabla f_m(x) \cdot v] \\ &= f(x) + [f'(x) \cdot v] \epsilon, \end{align} \]
 
To calculate the complete Jacobian, we calculate these directional derivatives in the $n$ different directions of the basis vectors, i.e. if
 
\[ d = d_0 + e_1 \epsilon_1 + \ldots + e_n \epsilon_n \]
 
for $e_i$ the $i$th basis vector, then
 
\[ f(d) = f(d_0) + Je_1 \epsilon_1 + \ldots + Je_n \epsilon_n \]
 
computes all columns of the Jacobian simultaneously.
 
Array of Structs Representation
 
Instead of thinking about a vector of dual numbers, thus we can instead think of dual numbers with vectors for the components. But if there are vectors for the components, then we can think of the grouping of dual components as a matrix. Thus define our multidimensional multi-partial dual number as:
 
\[ D_0 = [d_1,d_2,d_3,\ldots,d_n] \]
 
\[ \Sigma = \begin{bmatrix} d_{11} & d_{12} & \cdots & d_{1n} \\ d_{21} & d_{22} & & \vdots \\ \vdots & & \ddots & \vdots \\ d_{m1} & \hdots & \hdots & d_{mn} \end{bmatrix} \]
 
\[ \epsilon=[\epsilon_1,\epsilon_2,\ldots,\epsilon_m] \]
 
\[ D = D_0 + \Sigma \epsilon \]
 
where $D_0$ is a vector in $\mathbb{R}^n$, $\epsilon$ is a vector of dimensional signifiers and $\Sigma$ is a matrix in $\mathbb{R}^{n \times m}$ where $m$ is the number of concurrent differentiation dimensions. Each row of this is a dual number, but now we can use this to easily define higher dimensional primitives.
 
For example, let $f(x) = Ax$, matrix multiplication. Then, we can show with our dual number arithmetic that:
 
\[ f(D) = A*D_0 + A*\Sigma*\epsilon \]
 
is how one would compute the value of $f(D_0)$ and the derivative $f'(D_0)$ in all directions signified by the columns of $\Sigma$ simultaneously. Using multidimensional Taylor series expansions and doing the manipulations like before indeed implies that the arithmetic on this object should follow:
 
\[ f(D) = f(D_0) + f'(D_0)\Sigma \epsilon \]
 
where $f'$ is the total derivative or the Jacobian of $f$. This then allows our system to be highly efficient by allowing the definition of multidimensional functions, like linear algebra, to be primitives of multi-directional derivatives.
 
Higher derivatives
 
The above techniques can be extended to higher derivatives by adding more terms to the Taylor polynomial, e.g.
 
\[ f(a + \epsilon) = f(a) + \epsilon f'(a) + \frac{1}{2} \epsilon^2 f''(a) + o(\epsilon^2). \]
 
We treat this as a degree-2 (or degree-$n$, in general) polynomial and do polynomial arithmetic to calculate the new polynomials. The coefficients of powers of $\epsilon$ then give the higher-order derivatives.
 
For example, for a function $f: \mathbb{R}^n \to \mathbb{R}$ we have
 
\[ f(x + \epsilon v) = f(x) + \epsilon \left[ \sum_i (\partial_i f)(x) v_i \right] + \frac{1}{2}\epsilon^2 \left[ \sum_i \sum_j (\partial_{i,j} f) v_i v_j \right] \]
 
using Dual numbers with a single $\epsilon$ component. In this way we can compute coefficients of the (symmetric) Hessian matrix.
 
Application: solving nonlinear equations using the Newton method
 
As an application, we will see how to solve nonlinear equations of the form $f(x) = 0$ for functions $f: \mathbb{R}^n \to \mathbb{R}^n$.
 
Since in general we cannot do anything with nonlinearity, we try to reduce it (approximate it) with something linear. Furthermore, in general we know that it is not possible to solve nonlinear equations in closed form (even for polynomials of degree $\ge 5$), so we will need some kind of iterative method.
 
We start from an initial guess $x_0$. The idea of the Newton method is to follow the tangent line to the function $f$ at the point $x_0$ and find where it intersects the $x$-axis; this will give the next iterate $x_1$.
 
Algebraically, we want to solve $f(x_1) = 0$. Suppose that $x_1 = x_0 + \delta$ for some $\delta$ that is currently unknown and which we wish to calculate.
 
Assuming $\delta$ is small, we can expand:
 
\[ f(x_1) = f(x_0 + \delta) = f(x_0) + Df(x_0) \cdot \delta + \mathcal{O}(\| \delta \|^2). \]
 
Since we wish to solve
 
\[ f(x_0 + \delta) \simeq 0, \]
 
we put
 
\[ f(x_0) + Df(x_0) \cdot \delta = 0, \]
 
so that mathematically we have
 
\[ \delta = -[Df(x_0)]^{-1} \cdot f(x_0). \]
 
Computationally we prefer to solve the matrix equation
 
\[ J \delta = -f(x_0), \]
 
where $J := Df(x_0)$ is the Jacobian of the function; Julia uses the syntax \ ("backslash") for solving linear systems in an efficient way:
 
+
 
Directional derivative and gradient of functions $f: \mathbb{R}^n \to \mathbb{R}$
 
For a function $f: \mathbb{R}^n \to \mathbb{R}$ the basic operation is the directional derivative:
 
\[ \lim_{\epsilon \to 0} \frac{f(\mathbf{x} + \epsilon \mathbf{v}) - f(\mathbf{x})}{\epsilon} = [\nabla f(\mathbf{x})] \cdot \mathbf{v}, \]
 
where $\epsilon$ is still a single dimension and $\nabla f(\mathbf{x})$ is the direction in which we calculate.
 
We can directly do this using the same simple Dual numbers as above, using the same $\epsilon$, e.g.
 
\[ f(x, y) = x^2 \sin(y) \]
 
\[ \begin{align} f(x_0 + a\epsilon, y_0 + b\epsilon) &= (x_0 + a\epsilon)^2 \sin(y_0 + b\epsilon) \\ &= x_0^2 \sin(y_0) + \epsilon[2ax_0 \sin(y_0) + x_0^2 b \cos(y_0)] + o(\epsilon) \end{align} \]
 
so we have indeed calculated $\nabla f(x_0, y_0) \cdot \mathbf{v},$ where $\mathbf{v} = (a, b)$ are the components that we put into the derivative component of the Dual numbers.
 
If we wish to calculate the directional derivative in another direction, we could repeat the calculation with a different $\mathbf{v}$. A better solution is to use another independent epsilon $\epsilon$, expanding $x = x_0 + a_1 \epsilon_1 + a_2 \epsilon_2$ and putting $\epsilon_1 \epsilon_2 = 0$.
 
In particular, if we wish to calculate the gradient itself, $\nabla f(x_0, y_0)$, we need to calculate both partial derivatives, which corresponds to two directional derivatives, in the directions $(1, 0)$ and $(0, 1)$, respectively.
 
Forward-Mode AD as jvp
 
Note that another representation of the directional derivative is $f'(x)v$, where $f'(x)$ is the Jacobian or total derivative of $f$ at $x$. To see the equivalence of this to a directional derivative, write it out in the standard basis:
 
\[ w_i = \sum_{j}^{m} J_{ij} v_{j} \]
 
Now write out what $J$ means and we see that:
 
\[ w_i = \sum_j^{m} \frac{df_i}{dx_j} v_j = \nabla f_i(x) \cdot v \]
 
The primitive action of forward-mode AD is $f'(x)v$!
 
This is also known as a Jacobian-vector product, or jvp for short.
 
We can thus represent vector calculus with multidimensional dual numbers as follows. Let $d =[x,y]$, the vector of dual numbers. We can instead represent this as:
 
\[ d = d_0 + v_1 \epsilon_1 + v_2 \epsilon_2 \]
 
where $d_0$ is the primal vector $[x_0,y_0]$ and the $v_i$ are the vectors for the dual directions. If you work out this algebra, then note that a single application of $f$ to a multidimensional dual number calculates:
 
\[ f(d) = f(d_0) + f'(d_0)v_1 \epsilon_1 + f'(d_0)v_2 \epsilon_2 \]
 
i.e. it calculates the result of $f(x,y)$ and two separate directional derivatives. Note that because the information about $f(d_0)$ is shared between the calculations, this is more efficient than doing multiple applications of $f$. And of course, this is then generalized to $m$ many directional derivatives at once by:
 
\[ d = d_0 + v_1 \epsilon_1 + v_2 \epsilon_2 + \ldots + v_m \epsilon_m \]
 
Jacobian
 
For a function $f: \mathbb{R}^n \to \mathbb{R}^m$, we reduce (conceptually, although not necessarily in code) to its component functions $f_i: \mathbb{R}^n \to \mathbb{R}$, where $f(x) = (f_1(x), f_2(x), \ldots, f_m(x))$.
 
Then
 
\[ \begin{align} f(x + \epsilon v) &= (f_1(x + \epsilon v), \ldots, f_m(x + \epsilon v)) \\ &= (f_1(x) + \epsilon[\nabla f_1(x) \cdot v], \dots, f_m(x) + \epsilon[\nabla f_m(x) \cdot v] \\ &= f(x) + [f'(x) \cdot v] \epsilon, \end{align} \]
 
To calculate the complete Jacobian, we calculate these directional derivatives in the $n$ different directions of the basis vectors, i.e. if
 
\[ d = d_0 + e_1 \epsilon_1 + \ldots + e_n \epsilon_n \]
 
for $e_i$ the $i$th basis vector, then
 
\[ f(d) = f(d_0) + Je_1 \epsilon_1 + \ldots + Je_n \epsilon_n \]
 
computes all columns of the Jacobian simultaneously.
 
Struct of Arrays Representation
 
Instead of thinking about a vector of dual numbers, thus we can instead think of dual numbers with vectors for the components. But if there are vectors for the components, then we can think of the grouping of dual components as a matrix. Thus define our multidimensional multi-partial dual number as:
 
\[ D_0 = [d_1,d_2,d_3,\ldots,d_n] \]
 
\[ \Sigma = \begin{bmatrix} d_{11} & d_{12} & \cdots & d_{1n} \\ d_{21} & d_{22} & & \vdots \\ \vdots & & \ddots & \vdots \\ d_{m1} & \cdots & \cdots & d_{mn} \end{bmatrix} \]
 
\[ \epsilon=[\epsilon_1,\epsilon_2,\ldots,\epsilon_m] \]
 
\[ D = D_0 + \Sigma \epsilon \]
 
where $D_0$ is a vector in $\mathbb{R}^n$, $\epsilon$ is a vector of dimensional signifiers and $\Sigma$ is a matrix in $\mathbb{R}^{n \times m}$ where $m$ is the number of concurrent differentiation dimensions. Each row of this is a dual number, but now we can use this to easily define higher dimensional primitives.
 
For example, let $f(x) = Ax$, matrix multiplication. Then, we can show with our dual number arithmetic that:
 
\[ f(D) = A*D_0 + A*\Sigma*\epsilon \]
 
is how one would compute the value of $f(D_0)$ and the derivative $f'(D_0)$ in all directions signified by the columns of $\Sigma$ simultaneously. Using multidimensional Taylor series expansions and doing the manipulations like before indeed implies that the arithmetic on this object should follow:
 
\[ f(D) = f(D_0) + f'(D_0)\Sigma \epsilon \]
 
where $f'$ is the total derivative or the Jacobian of $f$. This then allows our system to be highly efficient by allowing the definition of multidimensional functions, like linear algebra, to be primitives of multi-directional derivatives.
 
Higher derivatives
 
The above techniques can be extended to higher derivatives by adding more terms to the Taylor polynomial, e.g.
 
\[ f(a + \epsilon) = f(a) + \epsilon f'(a) + \frac{1}{2} \epsilon^2 f''(a) + o(\epsilon^2). \]
 
We treat this as a degree-2 (or degree-$n$, in general) polynomial and do polynomial arithmetic to calculate the new polynomials. The coefficients of powers of $\epsilon$ then give the higher-order derivatives.
 
For example, for a function $f: \mathbb{R}^n \to \mathbb{R}$ we have
 
\[ f(x + \epsilon v) = f(x) + \epsilon \left[ \sum_i (\partial_i f)(x) v_i \right] + \frac{1}{2}\epsilon^2 \left[ \sum_i \sum_j (\partial_{i,j} f) v_i v_j \right] \]
 
using Dual numbers with a single $\epsilon$ component. In this way we can compute coefficients of the (symmetric) Hessian matrix.
 
Application: solving nonlinear equations using the Newton method
 
As an application, we will see how to solve nonlinear equations of the form $f(x) = 0$ for functions $f: \mathbb{R}^n \to \mathbb{R}^n$.
 
Since in general we cannot do anything with nonlinearity, we try to reduce it (approximate it) with something linear. Furthermore, in general we know that it is not possible to solve nonlinear equations in closed form (even for polynomials of degree $\ge 5$), so we will need some kind of iterative method.
 
We start from an initial guess $x_0$. The idea of the Newton method is to follow the tangent line to the function $f$ at the point $x_0$ and find where it intersects the $x$-axis; this will give the next iterate $x_1$.
 
Algebraically, we want to solve $f(x_1) = 0$. Suppose that $x_1 = x_0 + \delta$ for some $\delta$ that is currently unknown and which we wish to calculate.
 
Assuming $\delta$ is small, we can expand:
 
\[ f(x_1) = f(x_0 + \delta) = f(x_0) + Df(x_0) \cdot \delta + \mathcal{O}(\| \delta \|^2). \]
 
Since we wish to solve
 
\[ f(x_0 + \delta) \simeq 0, \]
 
we put
 
\[ f(x_0) + Df(x_0) \cdot \delta = 0, \]
 
so that mathematically we have
 
\[ \delta = -[Df(x_0)]^{-1} \cdot f(x_0). \]
 
Computationally we prefer to solve the matrix equation
 
\[ J \delta = -f(x_0), \]
 
where $J := Df(x_0)$ is the Jacobian of the function; Julia uses the syntax \ ("backslash") for solving linear systems in an efficient way:
 
 using ForwardDiff, StaticArrays
 
 function newton_step(f, x0)
diff --git a/notes/16-From_Optimization_to_Probabilistic_Programming/index.html b/notes/16-From_Optimization_to_Probabilistic_Programming/index.html
index f60cebae..f059304c 100644
--- a/notes/16-From_Optimization_to_Probabilistic_Programming/index.html
+++ b/notes/16-From_Optimization_to_Probabilistic_Programming/index.html
@@ -24,7 +24,7 @@
 prob1 = ODEProblem(lotka_volterra,u0,tspan,_θ)
 sol = solve(prob1,Tsit5())
 plot(sol)
-
  
and from which we can get an ensemble of solutions:
 
+
  
and from which we can get an ensemble of solutions:
 
 prob_func = function (prob,i,repeat)
   remake(prob,p=rand.(θ))
 end
@@ -34,21 +34,21 @@
 
 using DiffEqBase.EnsembleAnalysis
 plot(EnsembleSummary(sol))
-
  
From just a few variables having probabilities, every variable has an induced probability: there is a probability distribution on the integrator states, the output at time t_i, etc.
 
Bayesian Estimation with Point Estimates: Bayes' Rule, Maximum Likelihood, and MAP
 
Recall from our previous studies that the difficult part of modeling is not necessarily the forward modeling approach, rather it's the incorporation of data or the estimation problem that is difficult. When your variables are now random distributions, how do you "fit" them?
 
The answer comes from Bayes' rule, which is the following. Assume you had a prior distribution $p(\theta)$ for the probability that $X$ is a given value $\theta$. Then the posterior probability distribution, $p(\theta|D)$, or the distribution which is updated to include data, is given by:
 
\[ p(\theta|D) = \frac{p(D|\theta)p(\theta)}{\int_\Omega p(D|\theta)p(\theta)d\theta} \]
 
The scaling factor on the denominator is simply a constant to make the distribution integrate 1 (so that the resulting function is a probability distribution!). The numerator is simply the prior distribution multiplied by the likelihood of seeing the data given the value of the random variable. The prior distribution must be given but notice that the likelihood has another name: the likelihood is the model.
 
The reason why it's the same thing is because the model is what tells you the expected outcomes given a value of the random variable, and your data is on an expected outcome! However, the likelihood encodes a little bit more information in that it again is a distribution and not a point estimate. We need to make a choice for our measurement distribution on our model's results.
 
Quick Question: Why is this referred to as measurement noise? Why is it not process noise?
 
A common choice for the measurement distribution is the Normal distribution. This comes from the Central Limit Theorem (CLT) which essentially states that, given enough interacting mechanisms, the average values of things "tend to become normally distributed". The true statement of the CLT is much more complex, but that is a decent working definition for practical use. The normal distribution is defined by two parameters, $\mu$ and $\sigma$, and is given by the following function:
 
\[ f(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2\sigma^2}\right) \]
 
This is a bell curve centered at $\mu$ with a variance of $\sigma$. Our best guess for the output, i.e. the model's prediction, should be the average measurement, meaning that $\mu$ is the result from the simulator. $\sigma$ is a parameter for how much measurement error we expect (some intuition on $\sigma$ will come soon).
 
Let's return to thinking about the ODE example. In this case, we have $\theta$ as a vector of random variables. This means that $u(t;\theta)$ is a random variable for the ODE $u'= ...$'s solution at a given point in time $t$. If we have a measurement at a time $t_i$ and assume our measurement noise is normally distributed with some constant measurement noise $\sigma$, then the likelihood of our data would be $f(x_i;u(t_i;\theta),\sigma)$ at each data point $(t_i,x_i)$. From probability we know that seeing the composition of events is given by the multiplication of probabilities, so the probability of seeing the full dataset given observations $D = (t_i,x_i)$ along the timeseries is:
 
\[ p(D|\theta) = \prod_i f(x_i;u(t_i;\theta),\sigma) \]
 
This can be read as: solve the model with the given parameters, and the probability of having seen the measurement is thus given by a product of normal distribution calculations. Note that in many cases the product is not numerically stable (and grows exponentially), and so the likelihood is transformed to the log-likelihood. To get this expression, we take the log of both sides and notice that the product becomes a summation, and thus:
 
\[ \begin{align} \log p(D|\theta) &= \sum_i \log f(x_i;u(t_i;\theta),\sigma)\\ &= \frac{N}{\log(\sqrt{2\pi}\sigma)} + \frac{1}{2\sigma^2} \sum_i -(x_i - u(t_i; \theta))^2 \end{align} \]
 
Notice that maximizing this log-likelihood is equivalent to minimizing the L2 norm of the solution against the data!. Thus we can see a few things:
 
     
  1. Previous parameter estimation by minimizing a norm against data can be seen as maximum likelihood with some measurement distribution. L2 norm corresponds to assuming measurement noise is normally distributed and all of the measurements have the same error variance.
     
  2. By the same derivation, having different error variances with normally distributed errors is equivalent to doing weighted L2 estimation.
     
 
This reformulation (generalization?) to likelihoods of probability distributions is known as maximum likelihood estimation (MLE), but is equivalent to our previous forms of parameter estimation using point estimates against data. However, this calculation is ignoring Bayes' rule, and is thus not finding the parameters which have the highest probability. To do that, we need to go back to Bayes' rule which states that:
 
\[ \log p(\theta|D) = \log p(D|\theta) + \log p(\theta) - C \]
 
Thus, maximizing the log-likelihood is "almost" the same as finding the most probable parameters, except that we need to add weights given $\log p(\theta)$ from our prior distribution! If we assume our prior distribution is flat, like a uniform distribution, then we have a non-informative prior and the maximum posterior point matches that of the maximum likelihood estimation. However, this formulation allows us to get point estimates in a way that takes into account prior knowledge, and is call maximum a posteriori estimation (MAP).
 
Bayesian Estimation of Posterior Distributions with Monte Carlo
 
The previous discussion still solely focused on getting point estimates for the most probable parameters. However, what if we wanted to find the distributions of the parameters, i.e. the full $p(D|\theta)$? Outside of very few small models, this cannot be done analytically and is thus the basic problem of probabilistic programming. There are two general approaches:
 
     
  1. Sampling-based approaches. Sample parameters $\theta_i$ in such a manner that the array $[\theta_i]$ converges to an array sampled from the true distribution, and thus with enough samples one can capture the distribution numerically.
     
  2. Variational inference. Find some way to represent the probability distribution and push forward the distributions at every step of the program.
     
 
Recovering Distributions from Sampled Points
 
It's clear from above that if you have a distribution, like Normal(5,1), that you can sample from the distribution to get an array of values which follow the distribution. However, in order for the following sampling approaches to make sense, we need to see how to recover a distribution from discrete samples. So let's say you had a bunch of normally distributed points:
 
+
  
From just a few variables having probabilities, every variable has an induced probability: there is a probability distribution on the integrator states, the output at time t_i, etc.
 
Bayesian Estimation with Point Estimates: Bayes' Rule, Maximum Likelihood, and MAP
 
Recall from our previous studies that the difficult part of modeling is not necessarily the forward modeling approach, rather it's the incorporation of data or the estimation problem that is difficult. When your variables are now random distributions, how do you "fit" them?
 
The answer comes from Bayes' rule, which is the following. Assume you had a prior distribution $p(\theta)$ for the probability that $X$ is a given value $\theta$. Then the posterior probability distribution, $p(\theta|D)$, or the distribution which is updated to include data, is given by:
 
\[ p(\theta|D) = \frac{p(D|\theta)p(\theta)}{\int_\Omega p(D|\theta)p(\theta)d\theta} \]
 
The scaling factor on the denominator is simply a constant to make the distribution integrate 1 (so that the resulting function is a probability distribution!). The numerator is simply the prior distribution multiplied by the likelihood of seeing the data given the value of the random variable. The prior distribution must be given but notice that the likelihood has another name: the likelihood is the model.
 
The reason why it's the same thing is because the model is what tells you the expected outcomes given a value of the random variable, and your data is on an expected outcome! However, the likelihood encodes a little bit more information in that it again is a distribution and not a point estimate. We need to make a choice for our measurement distribution on our model's results.
 
Quick Question: Why is this referred to as measurement noise? Why is it not process noise?
 
A common choice for the measurement distribution is the Normal distribution. This comes from the Central Limit Theorem (CLT) which essentially states that, given enough interacting mechanisms, the average values of things "tend to become normally distributed". The true statement of the CLT is much more complex, but that is a decent working definition for practical use. The normal distribution is defined by two parameters, $\mu$ and $\sigma$, and is given by the following function:
 
\[ f(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2\sigma^2}\right) \]
 
This is a bell curve centered at $\mu$ with a variance of $\sigma$. Our best guess for the output, i.e. the model's prediction, should be the average measurement, meaning that $\mu$ is the result from the simulator. $\sigma$ is a parameter for how much measurement error we expect (some intuition on $\sigma$ will come soon).
 
Let's return to thinking about the ODE example. In this case, we have $\theta$ as a vector of random variables. This means that $u(t;\theta)$ is a random variable for the ODE $u'= ...$'s solution at a given point in time $t$. If we have a measurement at a time $t_i$ and assume our measurement noise is normally distributed with some constant measurement noise $\sigma$, then the likelihood of our data would be $f(x_i;u(t_i;\theta),\sigma)$ at each data point $(t_i,x_i)$. From probability we know that seeing the composition of events is given by the multiplication of probabilities, so the probability of seeing the full dataset given observations $D = (t_i,x_i)$ along the timeseries is:
 
\[ p(D|\theta) = \prod_i f(x_i;u(t_i;\theta),\sigma) \]
 
This can be read as: solve the model with the given parameters, and the probability of having seen the measurement is thus given by a product of normal distribution calculations. Note that in many cases the product is not numerically stable (and grows exponentially), and so the likelihood is transformed to the log-likelihood. To get this expression, we take the log of both sides and notice that the product becomes a summation, and thus:
 
\[ \begin{align} \log p(D|\theta) &= \sum_i \log f(x_i;u(t_i;\theta),\sigma)\\ &= \frac{N}{\log(\sqrt{2\pi}\sigma)} + \frac{1}{2\sigma^2} \sum_i -(x_i - u(t_i; \theta))^2 \end{align} \]
 
Notice that maximizing this log-likelihood is equivalent to minimizing the L2 norm of the solution against the data!. Thus we can see a few things:
 
     
  1. Previous parameter estimation by minimizing a norm against data can be seen as maximum likelihood with some measurement distribution. L2 norm corresponds to assuming measurement noise is normally distributed and all of the measurements have the same error variance.
     
  2. By the same derivation, having different error variances with normally distributed errors is equivalent to doing weighted L2 estimation.
     
 
This reformulation (generalization?) to likelihoods of probability distributions is known as maximum likelihood estimation (MLE), but is equivalent to our previous forms of parameter estimation using point estimates against data. However, this calculation is ignoring Bayes' rule, and is thus not finding the parameters which have the highest probability. To do that, we need to go back to Bayes' rule which states that:
 
\[ \log p(\theta|D) = \log p(D|\theta) + \log p(\theta) - C \]
 
Thus, maximizing the log-likelihood is "almost" the same as finding the most probable parameters, except that we need to add weights given $\log p(\theta)$ from our prior distribution! If we assume our prior distribution is flat, like a uniform distribution, then we have a non-informative prior and the maximum posterior point matches that of the maximum likelihood estimation. However, this formulation allows us to get point estimates in a way that takes into account prior knowledge, and is call maximum a posteriori estimation (MAP).
 
Bayesian Estimation of Posterior Distributions with Monte Carlo
 
The previous discussion still solely focused on getting point estimates for the most probable parameters. However, what if we wanted to find the distributions of the parameters, i.e. the full $p(D|\theta)$? Outside of very few small models, this cannot be done analytically and is thus the basic problem of probabilistic programming. There are two general approaches:
 
     
  1. Sampling-based approaches. Sample parameters $\theta_i$ in such a manner that the array $[\theta_i]$ converges to an array sampled from the true distribution, and thus with enough samples one can capture the distribution numerically.
     
  2. Variational inference. Find some way to represent the probability distribution and push forward the distributions at every step of the program.
     
 
Recovering Distributions from Sampled Points
 
It's clear from above that if you have a distribution, like Normal(5,1), that you can sample from the distribution to get an array of values which follow the distribution. However, in order for the following sampling approaches to make sense, we need to see how to recover a distribution from discrete samples. So let's say you had a bunch of normally distributed points:
 
 X = Normal(5,1)
 x = [rand(X) for i in 1:100]
 scatter(x,[1 for i in 1:100])
-
  
Notice that there are more points in the areas of higher probability. Thus the density of sampled points gives us an estimate for the probability of having points in a given area. We can then count the number of points in a bin and divide by the total number of points in order to get the probability of being in a specific region. This is depicted by a histogram:
 
+
  
Notice that there are more points in the areas of higher probability. Thus the density of sampled points gives us an estimate for the probability of having points in a given area. We can then count the number of points in a bin and divide by the total number of points in order to get the probability of being in a specific region. This is depicted by a histogram:
 
 histogram(x)
-
  
and we see this converges when we get more points:
 
+
  
and we see this converges when we get more points:
 
 histogram([rand(X) for i in 1:10000],normed=true)
 using StatsPlots
 plot!(X,lw=5)
-
  
A continuous form of this is the kernel density estimate, which is essentially a smoothed binning approach.
 
+
  
A continuous form of this is the kernel density estimate, which is essentially a smoothed binning approach.
 
 using KernelDensity
 plot(kde([rand(X) for i in 1:10000]),lw=5)
 plot!(X,lw=5)
-
  
Thus, for the sampling-based approaches, we simply need to arrive at an array which is sampled according to the distribution that we want to estimate, and from that array we can recover the distribution.
 
Sampling Distributions with the Metropolis-Hastings Algorithm
 
The Metropolis-Hastings algorithm is the simplest form of Markov Chain Monte Carlo (MCMC) which gives a way of sampling the $\theta$ distribution. To see how this algorithm works, let's understand the ratio between two points in the posterior probability. If we have $x_i$ and $x_j$, the ratio of the two probabilities would be given by:
 
\[ \frac{p(x_i|D)}{p(x_j|D)} = \frac{p(D|x_i)p(x_i)}{p(D|x_j)p(x_j)} \]
 
(notice that the integration constant cancels). This motivates the idea that all we have to do is ensure we only go to a point $x_j$ from $x_i$ with probability difference that matches that ratio, and over time if we do this between "all points" we will have the right number of "each point" in the distribution (quotes because it's continuous). With a bit more rigour we arrive at the following algorithm:
 
     
  1. Starting at $x_i$, take $x_{i+1}$ from a sampling algorithm $g(x_{i+1}|x_i)$.
     
  2. Calculate $A = \min\left(1,\frac{p(D|x_{i+1})p(x_{i+1})g(x_i|x_{i+1})}{p(D|x_i)p(x_i)g(x_{i+1}|x_i)}\right)$. Notice that if we require $g$ to be symmetric, then this simplifies to the probability ratio $A = \min\left(1,\frac{p(D|x_{i+1})p(x_{i+1})}{p(D|x_i)p(x_i)}\right)$
     
  3. Use a random number to accept the step with a probability $A$. Go back to step 1, incrementing $i$ if accepted, otherwise just repeat.
     
 
I.e, we just walk around the space biasing the acceptance of a step by the factor $\frac{p(x_i|D)}{p(x_j|D)}$ and sooner or later we will have spent the right amount of time in each area, giving the correct distribution.
 
(This can be rigorously proven, and those details are left out.)
 
The Cost of Bayesian Estimation
 
Let's take a quick moment to understand the high cost of Bayesian posterior estimations. While before we were getting point estimates, now we are trying to recover a full probability distribution, and each accept/reject probability calculation requires evaluating the likelihood at some point. Remember, the likelihood is generated by our simulator, and thus every evaluation here is an ODE solver call or a neural network forward pass! This means that to get good distributions, we are solving the ODE hundreds of thousands of times, i.e. even more than when doing parameter estimation! This is something to keep in mind.
 
However, notice that this process is trivially parallelizable. We can just have parallel chains on going, i.e. start 16 processes all doing Metropolis-Hastings, and in the end they are all sampling from the same distribution, so the final array can simply be the pooled results of each chain.
 
Hamiltonian Monte Carlo
 
Metropolis-Hastings is easy to motivate and implement. However, it does not do well in high dimensional spaces because it searches in all directions. For example, it's common for the sampling distribution $g$ to be a multivariable distribution (i.e. normal in all directions). However, high dimensional objects commonly sit on low dimensional manifolds (known as the manifold hypothesis). If that's the case, the most probable set of parameters is something that is low dimensional. For example, parameters may compensate for one another, and so $\theta_1^2 + \theta_2^2 + \theta_3^2 = 1$ might be the manifold on which all of the most probable choices for $\theta$ lie, in which case we need sample on the sphere instead of all of $\mathbb{R}^3$.
 
However, it's quick to see that this will give Metropolis-Hastings some trouble, since it will use a normal distribution around the current point, and thus even if we start on the sphere, it will have a high chance of trying a point not on the sphere in the next round! This can be depicted as:
 
 
Recall that every single rejection is still evaluating the likelihood (since it's calculating an acceptance probability, finding it near zero, rejecting and starting again), and every likelihood call is calling our simulator, and so this is sllllllooooooooooooooowwwwwwwww in high dimensions!
 
What we need to do instead is ensure that we walk along the path of high probability. What we want to do is thus build a vector field that matches our high probability regions
 
 
and follow said vector field (following a vector field is solving what kind of equation?). The first idea one might have is to use the gradient. However, while this idea has the right intentions, the issue is that the gradient of the probability will average out all of the possible probabilities, and will thus flow towards the mode of the distribution:
 
 
To overcome this issue, we look to physical systems and see that a satellite orbiting a planet always nicely stays on some manifold instead of following the gradient:
 
 
The reason why it does is because it has momentum. Recall from basic physics that one way to describe a physical system is through Hamiltonian mechanics, where $H(x,p)$ is the energy associated with the state $(x,p)$ (normally $x$ is location and $p$ is momentum). Due to conservation of energy, the solution of the dynamical equations leads to $H(x,p)$ being constant, and thus the dynamics follow the level sets of $H$. From the Hamiltonian the dynamics of the system are:
 
\[ \begin{align} \frac{dx}{dt} &= \frac{dH}{dp}\\ &= -\frac{dH}{dx} \end{align} \]
 
Here we want our Hamiltonian to be our posterior probability, so that way we stay on the manifold of high probability. This means:
 
\[ H(x,p) = - \log \pi(x,p) \]
 
where $\pi(x,p) = \pi(p|x)\pi(x)$ (where I am now using $pi$ for probability since $p$ is momentum!). So to lift from a probability over parameters to one that includes momentum, we simply need to choose a conditional distribution $\pi(p|x)$. This would mean that
 
\[ \begin{align} H(x,p) &= -log \pi(p|x) - \log \pi(x)\\ &= K(p,x) + V(x) \end{align} \]
 
where $K$ is the kinetic energy and $V$ is the potential. Thus the potential energy is directly given by the posterior calculation, and the kinetic energy is thus a choice that is used to build the correct Hamiltonian. Hamiltonian Monte Carlo methods then dig into good ways to choose the kinetic energy function. This is done at the start (along with the choice of ODE solver time step) in such a way that it maximizes acceptance probabilities.
 
Connections to Differentiable Programming
 
\[ -\frac{dH}{dx} \]
 
requires calculating the gradient of the likelihood function with respect to the parameters, so we are once again using the gradient of our simulator! This means that all of our previous discussion on automatic differentiation and differentiable programming applies to the Hamiltonian Monte Carlo context.
 
There's another thread to follow that transformations of probability distributions are pushforwards of the Jacobian transformations (given the transformation of an integral formula), and this is used when doing variational inference.
 
Symplectic and Geometric Integration
 
One way to integrate the system of ODEs which result from the Hamiltonian system is to convert it to a system of first order ODEs and solve it directly. However, this loses information and can result in drift. This is demonstrated by looking at the long time solution of the pendulum:
 
+
  
Thus, for the sampling-based approaches, we simply need to arrive at an array which is sampled according to the distribution that we want to estimate, and from that array we can recover the distribution.
 
Sampling Distributions with the Metropolis-Hastings Algorithm
 
The Metropolis-Hastings algorithm is the simplest form of Markov Chain Monte Carlo (MCMC) which gives a way of sampling the $\theta$ distribution. To see how this algorithm works, let's understand the ratio between two points in the posterior probability. If we have $x_i$ and $x_j$, the ratio of the two probabilities would be given by:
 
\[ \frac{p(x_i|D)}{p(x_j|D)} = \frac{p(D|x_i)p(x_i)}{p(D|x_j)p(x_j)} \]
 
(notice that the integration constant cancels). This motivates the idea that all we have to do is ensure we only go to a point $x_j$ from $x_i$ with probability difference that matches that ratio, and over time if we do this between "all points" we will have the right number of "each point" in the distribution (quotes because it's continuous). With a bit more rigour we arrive at the following algorithm:
 
     
  1. Starting at $x_i$, take $x_{i+1}$ from a sampling algorithm $g(x_{i+1}|x_i)$.
     
  2. Calculate $A = \min\left(1,\frac{p(D|x_{i+1})p(x_{i+1})g(x_i|x_{i+1})}{p(D|x_i)p(x_i)g(x_{i+1}|x_i)}\right)$. Notice that if we require $g$ to be symmetric, then this simplifies to the probability ratio $A = \min\left(1,\frac{p(D|x_{i+1})p(x_{i+1})}{p(D|x_i)p(x_i)}\right)$
     
  3. Use a random number to accept the step with a probability $A$. Go back to step 1, incrementing $i$ if accepted, otherwise just repeat.
     
 
I.e, we just walk around the space biasing the acceptance of a step by the factor $\frac{p(x_i|D)}{p(x_j|D)}$ and sooner or later we will have spent the right amount of time in each area, giving the correct distribution.
 
(This can be rigorously proven, and those details are left out.)
 
The Cost of Bayesian Estimation
 
Let's take a quick moment to understand the high cost of Bayesian posterior estimations. While before we were getting point estimates, now we are trying to recover a full probability distribution, and each accept/reject probability calculation requires evaluating the likelihood at some point. Remember, the likelihood is generated by our simulator, and thus every evaluation here is an ODE solver call or a neural network forward pass! This means that to get good distributions, we are solving the ODE hundreds of thousands of times, i.e. even more than when doing parameter estimation! This is something to keep in mind.
 
However, notice that this process is trivially parallelizable. We can just have parallel chains on going, i.e. start 16 processes all doing Metropolis-Hastings, and in the end they are all sampling from the same distribution, so the final array can simply be the pooled results of each chain.
 
Hamiltonian Monte Carlo
 
Metropolis-Hastings is easy to motivate and implement. However, it does not do well in high dimensional spaces because it searches in all directions. For example, it's common for the sampling distribution $g$ to be a multivariable distribution (i.e. normal in all directions). However, high dimensional objects commonly sit on low dimensional manifolds (known as the manifold hypothesis). If that's the case, the most probable set of parameters is something that is low dimensional. For example, parameters may compensate for one another, and so $\theta_1^2 + \theta_2^2 + \theta_3^2 = 1$ might be the manifold on which all of the most probable choices for $\theta$ lie, in which case we need sample on the sphere instead of all of $\mathbb{R}^3$.
 
However, it's quick to see that this will give Metropolis-Hastings some trouble, since it will use a normal distribution around the current point, and thus even if we start on the sphere, it will have a high chance of trying a point not on the sphere in the next round! This can be depicted as:
 
 
Recall that every single rejection is still evaluating the likelihood (since it's calculating an acceptance probability, finding it near zero, rejecting and starting again), and every likelihood call is calling our simulator, and so this is sllllllooooooooooooooowwwwwwwww in high dimensions!
 
What we need to do instead is ensure that we walk along the path of high probability. What we want to do is thus build a vector field that matches our high probability regions
 
 
and follow said vector field (following a vector field is solving what kind of equation?). The first idea one might have is to use the gradient. However, while this idea has the right intentions, the issue is that the gradient of the probability will average out all of the possible probabilities, and will thus flow towards the mode of the distribution:
 
 
To overcome this issue, we look to physical systems and see that a satellite orbiting a planet always nicely stays on some manifold instead of following the gradient:
 
 
The reason why it does is because it has momentum. Recall from basic physics that one way to describe a physical system is through Hamiltonian mechanics, where $H(x,p)$ is the energy associated with the state $(x,p)$ (normally $x$ is location and $p$ is momentum). Due to conservation of energy, the solution of the dynamical equations leads to $H(x,p)$ being constant, and thus the dynamics follow the level sets of $H$. From the Hamiltonian the dynamics of the system are:
 
\[ \begin{align} \frac{dx}{dt} &= \frac{dH}{dp}\\ &= -\frac{dH}{dx} \end{align} \]
 
Here we want our Hamiltonian to be our posterior probability, so that way we stay on the manifold of high probability. This means:
 
\[ H(x,p) = - \log \pi(x,p) \]
 
where $\pi(x,p) = \pi(p|x)\pi(x)$ (where I am now using $pi$ for probability since $p$ is momentum!). So to lift from a probability over parameters to one that includes momentum, we simply need to choose a conditional distribution $\pi(p|x)$. This would mean that
 
\[ \begin{align} H(x,p) &= -log \pi(p|x) - \log \pi(x)\\ &= K(p,x) + V(x) \end{align} \]
 
where $K$ is the kinetic energy and $V$ is the potential. Thus the potential energy is directly given by the posterior calculation, and the kinetic energy is thus a choice that is used to build the correct Hamiltonian. Hamiltonian Monte Carlo methods then dig into good ways to choose the kinetic energy function. This is done at the start (along with the choice of ODE solver time step) in such a way that it maximizes acceptance probabilities.
 
Connections to Differentiable Programming
 
\[ -\frac{dH}{dx} \]
 
requires calculating the gradient of the likelihood function with respect to the parameters, so we are once again using the gradient of our simulator! This means that all of our previous discussion on automatic differentiation and differentiable programming applies to the Hamiltonian Monte Carlo context.
 
There's another thread to follow that transformations of probability distributions are pushforwards of the Jacobian transformations (given the transformation of an integral formula), and this is used when doing variational inference.
 
Symplectic and Geometric Integration
 
One way to integrate the system of ODEs which result from the Hamiltonian system is to convert it to a system of first order ODEs and solve it directly. However, this loses information and can result in drift. This is demonstrated by looking at the long time solution of the pendulum:
 
 using ParameterizedFunctions
 u0 = [1.,0.]
 harmonic! = @ode_def HarmonicOscillator begin
diff --git a/notes/17-Global_Sensitivity_Analysis/index.html b/notes/17-Global_Sensitivity_Analysis/index.html
index 4bbd6875..c5f5d4e9 100644
--- a/notes/17-Global_Sensitivity_Analysis/index.html
+++ b/notes/17-Global_Sensitivity_Analysis/index.html
@@ -9,4 +9,4 @@
 using LatinHypercubeSampling
 p = LHCoptim(120,2,1000)
 scatter(p[1][:,1],p[1][:,2])
-
  
For a reference library with many different quasi-Monte Carlo samplers, check out QuasiMonteCarlo.jl.
 
Fourier Amplitude Sensitivity Sampling (FAST) and eFAST
 
The FAST method is a change to the Sobol method to allow for faster convergence. First transform the variables $x_i$ onto the space $[0,1]$. Then, instead of the linear decomposition, one decomposes into a Fourier basis:
 
\[ f(x_i,x_2,\ldots,x_n) = \sum_{m_1 = -\infty}^{\infty} \ldots \sum_{m_n = -\infty}^{\infty} C_{m_1m_2\ldots m_n}\exp\left(2\pi i (m_1 x_1 + \ldots + m_n x_n)\right) \]
 
where
 
\[ C_{m_1m_2\ldots m_n} = \int_0^1 \ldots \int_0^1 f(x_i,x_2,\ldots,x_n) \exp\left(-2\pi i (m_1 x_1 + \ldots + m_n x_n)\right) \]
 
The ANOVA like decomposition is thus
 
\[ f_0 = C_{0\ldots 0} \]
 
\[ f_j = \sum_{m_j \neq 0} C_{0\ldots 0 m_j 0 \ldots 0} \exp (2\pi i m_j x_j) \]
 
\[ f_{jk} = \sum_{m_j \neq 0} \sum_{m_k \neq 0} C_{0\ldots 0 m_j 0 \ldots m_k 0 \ldots 0} \exp \left(2\pi i (m_j x_j + m_k x_k)\right) \]
 
The first order conditional variance is thus:
 
\[ V_j = \int_0^1 f_j^2 (x_j) dx_j = \sum_{m_j \neq 0} |C_{0\ldots 0 m_j 0 \ldots 0}|^2 \]
 
or
 
\[ V_j = 2\sum_{m_j = 1}^\infty \left(A_{m_j}^2 + B_{m_j}^2 \right) \]
 
where $C_{0\ldots 0 m_j 0 \ldots 0} = A_{m_j} + i B_{m_j}$. By Fourier series we know this to be:
 
\[ A_{m_j} = \int_0^1 \ldots \int_0^1 f(x)\cos(2\pi m_j x_j)dx \]
 
\[ B_{m_j} = \int_0^1 \ldots \int_0^1 f(x)\sin(2\pi m_j x_j)dx \]
 
Implementation via the Ergodic Theorem
 
Define
 
\[ X_j(s) = \frac{1}{2\pi} (\omega_j s \mod 2\pi) \]
 
By the ergodic theorem, if $\omega_j$ are irrational numbers, then the dynamical system will never repeat values and thus it will create a solution that is dense in the plane (Let's prove a bit later). As an animation:
 
 
(here, $\omega_1 = \pi$ and $\omega_2 = 7$)
 
This means that:
 
\[ A_{m_j} = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(x)\cos(m_j \omega_j s)ds \]
 
\[ B_{m_j} = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(x)\sin(m_j \omega_j s)ds \]
 
i.e. the multidimensional integral can be approximated by the integral over a single line.
 
One can satisfy this approximately to get a simpler form for the integral. Using $\omega_i$ as integers, the integral is periodic and so only integrating over $2\pi$ is required. This would mean that:
 
\[ A_{m_j} \approx \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\cos(m_j \omega_j s)ds \]
 
\[ B_{m_j} \approx \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\sin(m_j \omega_j s)ds \]
 
It's only approximate since the sequence cannot be dense. For example, with $\omega_1 = 11$ and $\omega_2 = 7$:
 
 
A higher period thus gives a better fill of the space and thus a better approximation, but may require a more points. However, this transformation makes the true integrals simple one dimensional quadratures which can be efficiently computed.
 
To get the total index from this method, one can calculate the total contribution of the complementary set, i.e. $V_{c_i} = \sum_{j \neq i} V_j$ and then
 
\[ S_{T_i} = 1 - S_{c_i} \]
 
Note that this then is a fast measure for the total contribution of variable $i$, including all higher-order nonlinear interactions, all from one-dimensional integrals! (This extension is called extended FAST or eFAST)
 
Proof of the Ergodic Theorem
 
Look at the map $x_{n+1} = x_n + \alpha (\text{mod} 1)$, where $\alpha$ is irrational. This is the irrational rotation map that corresponds to our problem. We wish to prove that in any interval $I$, there is a point of our orbit in this interval.
 
First let's prove a useful result: our points get arbitrarily close. Assume that for some finite $\epsilon$ that no two points are $\epsilon$ apart. This means that we at most have spacings of $\epsilon$ between the points, and thus we have at most $\frac{2\pi}{\epsilon}$ points (rounded up). This means our orbit is periodic. This means that there is a $p$ such that
 
\[ x_{n+p} = x_n \]
 
which means that $p \alpha = 1$ or $p = \frac{1}{\alpha}$ which is a contradiction since $\alpha$ is irrational.
 
Thus for every $\epsilon$ there are two points which are $\epsilon$ apart. Now take any arbitrary $I$. Let $\epsilon < d/2$ where $d$ is the length of the interval. We have just shown that there are two points $\epsilon$ apart, so there is a point that is $x_{n+m}$ and $x_{n+k}$ which are $<\epsilon$ apart. Assuming WLOG $m>k$, this means that $m-k$ rotations takes one from $x_{n+k}$ to $x_{n+m}$, and so $m-k$ rotations is a rotation by $\epsilon$. If we do $\frac{1}{\epsilon}$ rounded up rotations, we will then cover the space with intervals of length epsilon, each with one point of the orbit in it. Since $\epsilon < d/2$, one of those intervals is completely encapsulated in $I$, which means there is at least one point in our orbit that is in $I$.
 
Thus for every interval we have at least one point in our orbit that lies in it, proving that the rotation map with irrational $\alpha$ is dense. Note that during the proof we essentially showed as well that if $\alpha$ is rational, then the map is periodic based on the denominator of the map in its reduced form.
 
A Quick Note on Parallelism
 
Very quick note: all of these are hyper parallel since it does the same calculation per parameter or trajectory, and each calculation is long. For quasi-Monte Carlo, after generating "good enough" trajectories, one can evaluate the model at all points in parallel, and then simply do the GSA index measurement. For FAST, one can do each quadrature in parallel.
 
 
 Published from global_sensitivity.jmd using Weave.jl v0.10.12 on 2024-07-26. 
 
 
  
   
\ No newline at end of file
+
  
For a reference library with many different quasi-Monte Carlo samplers, check out QuasiMonteCarlo.jl.
 
Fourier Amplitude Sensitivity Sampling (FAST) and eFAST
 
The FAST method is a change to the Sobol method to allow for faster convergence. First transform the variables $x_i$ onto the space $[0,1]$. Then, instead of the linear decomposition, one decomposes into a Fourier basis:
 
\[ f(x_i,x_2,\ldots,x_n) = \sum_{m_1 = -\infty}^{\infty} \ldots \sum_{m_n = -\infty}^{\infty} C_{m_1m_2\ldots m_n}\exp\left(2\pi i (m_1 x_1 + \ldots + m_n x_n)\right) \]
 
where
 
\[ C_{m_1m_2\ldots m_n} = \int_0^1 \ldots \int_0^1 f(x_i,x_2,\ldots,x_n) \exp\left(-2\pi i (m_1 x_1 + \ldots + m_n x_n)\right) \]
 
The ANOVA like decomposition is thus
 
\[ f_0 = C_{0\ldots 0} \]
 
\[ f_j = \sum_{m_j \neq 0} C_{0\ldots 0 m_j 0 \ldots 0} \exp (2\pi i m_j x_j) \]
 
\[ f_{jk} = \sum_{m_j \neq 0} \sum_{m_k \neq 0} C_{0\ldots 0 m_j 0 \ldots m_k 0 \ldots 0} \exp \left(2\pi i (m_j x_j + m_k x_k)\right) \]
 
The first order conditional variance is thus:
 
\[ V_j = \int_0^1 f_j^2 (x_j) dx_j = \sum_{m_j \neq 0} |C_{0\ldots 0 m_j 0 \ldots 0}|^2 \]
 
or
 
\[ V_j = 2\sum_{m_j = 1}^\infty \left(A_{m_j}^2 + B_{m_j}^2 \right) \]
 
where $C_{0\ldots 0 m_j 0 \ldots 0} = A_{m_j} + i B_{m_j}$. By Fourier series we know this to be:
 
\[ A_{m_j} = \int_0^1 \ldots \int_0^1 f(x)\cos(2\pi m_j x_j)dx \]
 
\[ B_{m_j} = \int_0^1 \ldots \int_0^1 f(x)\sin(2\pi m_j x_j)dx \]
 
Implementation via the Ergodic Theorem
 
Define
 
\[ X_j(s) = \frac{1}{2\pi} (\omega_j s \mod 2\pi) \]
 
By the ergodic theorem, if $\omega_j$ are irrational numbers, then the dynamical system will never repeat values and thus it will create a solution that is dense in the plane (Let's prove a bit later). As an animation:
 
 
(here, $\omega_1 = \pi$ and $\omega_2 = 7$)
 
This means that:
 
\[ A_{m_j} = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(x)\cos(m_j \omega_j s)ds \]
 
\[ B_{m_j} = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(x)\sin(m_j \omega_j s)ds \]
 
i.e. the multidimensional integral can be approximated by the integral over a single line.
 
One can satisfy this approximately to get a simpler form for the integral. Using $\omega_i$ as integers, the integral is periodic and so only integrating over $2\pi$ is required. This would mean that:
 
\[ A_{m_j} \approx \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\cos(m_j \omega_j s)ds \]
 
\[ B_{m_j} \approx \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\sin(m_j \omega_j s)ds \]
 
It's only approximate since the sequence cannot be dense. For example, with $\omega_1 = 11$ and $\omega_2 = 7$:
 
 
A higher period thus gives a better fill of the space and thus a better approximation, but may require a more points. However, this transformation makes the true integrals simple one dimensional quadratures which can be efficiently computed.
 
To get the total index from this method, one can calculate the total contribution of the complementary set, i.e. $V_{c_i} = \sum_{j \neq i} V_j$ and then
 
\[ S_{T_i} = 1 - S_{c_i} \]
 
Note that this then is a fast measure for the total contribution of variable $i$, including all higher-order nonlinear interactions, all from one-dimensional integrals! (This extension is called extended FAST or eFAST)
 
Proof of the Ergodic Theorem
 
Look at the map $x_{n+1} = x_n + \alpha (\text{mod} 1)$, where $\alpha$ is irrational. This is the irrational rotation map that corresponds to our problem. We wish to prove that in any interval $I$, there is a point of our orbit in this interval.
 
First let's prove a useful result: our points get arbitrarily close. Assume that for some finite $\epsilon$ that no two points are $\epsilon$ apart. This means that we at most have spacings of $\epsilon$ between the points, and thus we have at most $\frac{2\pi}{\epsilon}$ points (rounded up). This means our orbit is periodic. This means that there is a $p$ such that
 
\[ x_{n+p} = x_n \]
 
which means that $p \alpha = 1$ or $p = \frac{1}{\alpha}$ which is a contradiction since $\alpha$ is irrational.
 
Thus for every $\epsilon$ there are two points which are $\epsilon$ apart. Now take any arbitrary $I$. Let $\epsilon < d/2$ where $d$ is the length of the interval. We have just shown that there are two points $\epsilon$ apart, so there is a point that is $x_{n+m}$ and $x_{n+k}$ which are $<\epsilon$ apart. Assuming WLOG $m>k$, this means that $m-k$ rotations takes one from $x_{n+k}$ to $x_{n+m}$, and so $m-k$ rotations is a rotation by $\epsilon$. If we do $\frac{1}{\epsilon}$ rounded up rotations, we will then cover the space with intervals of length epsilon, each with one point of the orbit in it. Since $\epsilon < d/2$, one of those intervals is completely encapsulated in $I$, which means there is at least one point in our orbit that is in $I$.
 
Thus for every interval we have at least one point in our orbit that lies in it, proving that the rotation map with irrational $\alpha$ is dense. Note that during the proof we essentially showed as well that if $\alpha$ is rational, then the map is periodic based on the denominator of the map in its reduced form.
 
A Quick Note on Parallelism
 
Very quick note: all of these are hyper parallel since it does the same calculation per parameter or trajectory, and each calculation is long. For quasi-Monte Carlo, after generating "good enough" trajectories, one can evaluate the model at all points in parallel, and then simply do the GSA index measurement. For FAST, one can do each quadrature in parallel.
 
 
 Published from global_sensitivity.jmd using Weave.jl v0.10.12 on 2024-07-26. 
 
 
  
   
\ No newline at end of file