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Add more details about the number of Krylov methods #703

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Feb 15, 2023
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Add more details about the number of Krylov methods #703

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Add more details about the number of Krylov methods
amontoison Feb 5, 2023
d4582a9
Update example with Nvidia IC0 preconditioner
amontoison Feb 15, 2023
8af9f3a
Update the date
amontoison Feb 15, 2023
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20 changes: 12 additions & 8 deletions paper/paper.md
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Expand Up @@ -16,7 +16,7 @@ authors:
affiliations:
- name: GERAD and Department of Mathematics and Industrial Engineering, Polytechnique Montréal, QC, Canada.
index: 1
date: 29 January 2023
date: 15 February 2023
bibliography: paper.bib
header-includes: |
\usepackage{booktabs}
Expand Down Expand Up @@ -59,8 +59,10 @@ Krylov.jl aims to provide a unified interface for the largest collection of Kryl
- \textbf{Krylov processes}: \textsc{Arnoldi}, \textsc{Golub-Kahan}, \textsc{Hermitian Lanczos}, \textsc{Montoison-Orban}, \textsc{Non-Hermitian Lanczos}, \textsc{Saunders-Simon-Yip};
- \textbf{Krylov methods}: \textsc{Bicgstab}, \textsc{Bilq}, \textsc{Bilqr}, \textsc{Cg}, \textsc{Cg-lanczos}, \textsc{Cg-lanczos-shift}, \textsc{Cgls}, \textsc{Cgne}, \textsc{Cgs}, \textsc{Cr}, \textsc{Craig}, \textsc{Craigmr}, \textsc{Crls}, \textsc{Crmr}, \textsc{Diom}, \textsc{Dqgmres}, \textsc{Fgmres}, \textsc{Fom}, \textsc{Gmres}, \textsc{Gpmr}, \textsc{Lnlq}, \textsc{Lslq}, \textsc{Lsmr}, \textsc{Lsqr}, \textsc{Minres}, \textsc{Minres-qlp}, \textsc{Qmr}, \textsc{Symmlq}, \textsc{Tricg}, \textsc{Trilqr}, \textsc{Trimr}, \textsc{Usymlq}, \textsc{Usymqr}.

MATLAB [@MATLAB] and PETSc [@petsc] have nineteen and eighteen distinct Krylov methods, respectively.
Note that we only consider the number of Krylov methods that generate different iterates.
MATLAB [@MATLAB] and PETSc [@petsc] have eleven and eighteen distinct Krylov methods, respectively.
Note that we only consider the number of Krylov methods that generate different iterates without preconditioning.
Variants with preconditioning are not counted except it is a flexible one such as \textsc{Fgmres}.

Some processes and methods are not available elsewhere and are the product of our own research.
References for each process and method are available in the extensive [documentation](https://juliasmoothoptimizers.github.io/Krylov.jl/stable/).

Expand Down Expand Up @@ -185,19 +187,21 @@ b_gpu = CuVector(b_cpu)
# Incomplete Cholesky decomposition LLᴴ ≈ A with zero fill-in
P = ic02(A_gpu, 'O')

# Additional vector required for solving triangular systems
z = similar(CuVector{ComplexF64}, n)

# Solve Py = x
function ldiv_ic0!(y, P, x)
copyto!(y, x)
ldiv!(LowerTriangular(P), y) # Forward substitution with L
ldiv!(LowerTriangular(P)', y) # Backward substitution with Lᴴ
function ldiv_ic0!(P, x, y, z)
ldiv!(z, LowerTriangular(P), x) # Forward substitution with L
ldiv!(y, LowerTriangular(P)', z) # Backward substitution with Lᴴ
return y
end

# Linear operator that model the preconditioner P⁻¹
T = ComplexF64
symmetric = false
hermitian = true
P⁻¹ = LinearOperator(T, m, n, symmetric, hermitian, (y, x) -> ldiv_ic0!(y, P, x))
P⁻¹ = LinearOperator(T, m, n, symmetric, hermitian, (y, x) -> ldiv_ic0!(P, x, y, z))

# Solve a Hermitian positive definite system with an incomplete Cholesky factorization preconditioner
x, stats = cg(A_gpu, b_gpu, M=P⁻¹)
Expand Down