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siam.tex
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siam.tex
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\documentclass[handout]{beamer}
%\documentclass[presentation]{beamer}
\usepackage[utf8]{inputenc}
\usepackage{amsmath, pdfpages, pdflscape, lscape, color, listings, hyperref, amssymb, graphicx, textcomp,varioref, afterpage, subcaption, float, bm, tikz, colortbl}
\global
\newcommand{\Fig}[1]{Figure \ref{#1}}
\newcommand{\fig}[1]{figure \ref{#1}}
\newcommand{\tab}[1]{table \ref{#1}}
\newcommand{\eq}[1]{equation \ref{#1}}
\newcommand{\Eq}[1]{Equation \ref{#1}}
\newcommand{\alg}[1]{algorithm \ref{#1}}
\newcommand{\Alg}[1]{Algorithm \ref{#1}}
\newcommand{\chp}[1]{chapter \ref{#1}}
\newcommand{\Chp}[1]{Chapter \ref{#1}}
\newcommand{\e}[1]{\cdot 10^{#1}}
\newcommand{\h}{\hbar}
\newcommand{\der}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\dder}[2]{\frac{\partial^2 #1}{\partial #2^2}}
\newcommand{\p}{\boldsymbol{P}}
\newcommand{\q}{\boldsymbol{q}}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert_{\!Q}}
\newcommand{\inner}[1]{\left\langle#1\right\rangle_{\!Q}}
\newcommand{\coef}[2]{\frac{\inner{#1,#2}}{\norm{#2}^2}}
\newcommand{\gooditem}[1]{\setbeamercolor{item}{fg=green}\item #1}
\newcommand{\baditem}[1]{\setbeamercolor{item}{fg=red}\item #1}
\DeclareMathOperator*{\argmin}{argmin}
\DeclareMathOperator*{\argmax}{argmax}
\newcommand{\E}[1]{\mbox{E}\!\left(#1\right)}
\newcommand{\Var}[1]{\mbox{Var}\!\left(#1\right)}
\newcommand{\Cov}[1]{\mbox{Cov}\!\left(#1\right)}
\setbeamertemplate{blocks}[rounded][shadow=false]
\usefonttheme[onlysmall]{structurebold}
% Use a bold face title font
\setbeamerfont{title}{series=\bfseries}
\setbeamerfont{frametitle}{series=\bfseries}
% This is not created yet, but should be...
%\usecolortheme{simula}
% Suppress navigation symbols
\setbeamertemplate{navigation symbols}{}
\definecolor{listingsstringcolor}{rgb}{0,0.5,0}
\definecolor{listingskeywordcolor}{rgb}{0,0,1}
\definecolor{listingsbasiccolor}{rgb}{0.0,0.0,0.0}
\definecolor{myred}{RGB}{190,0,0}
\newcommand{\listingsfont}{\sffamily}
\lstset{
language=python,
commentstyle=\color{red},
basicstyle=\color{listingsbasiccolor},
keywordstyle=\color{listingskeywordcolor},
stringstyle=\color{listingsstringcolor},
tabsize=4, % sets default tabsize to 2 spaces
morekeywords={as, assert, with, yield},
escapeinside={||},
basicstyle=\ttfamily\footnotesize,
columns=fixed
}
\newenvironment{chaospy}[1]
{\color{gray!30!black}
\usebackgroundtemplate{
\begin{tikzpicture}[remember picture, overlay]
\node[anchor = center, opacity=.15] (image) at (current page.center) {\includegraphics[scale=0.25]{chaospy_logo.jpg}};
\end{tikzpicture}}
\begin{frame}[fragile, environment=chaospy]
\frametitle{{#1}}}
{\end{frame}}
\definecolor{keywords}{RGB}{255,0,90}
\definecolor{comments}{RGB}{0,0,113}
\definecolor{red}{RGB}{160,0,0}
\definecolor{green}{RGB}{0,150,0}
\setbeamercolor{title}{fg=myred}
\setbeamercolor{frametitle}{fg=black!65!white,bg=white}
\setbeamercolor{normal text}{fg=black!75!white,bg=white}
\setbeamercolor{structure}{fg=black,bg=white}
\graphicspath{{./figures/}}
\title{Chaospy: \\ A modular implementation of Polynomial Chaos expansions and Monte Carlo methods}
\author{Simen Tennøe\\ \vspace{5mm}\footnotesize Supervisors:\\ \vspace{0.5mm} Jonathan Feinberg, Hans Petter Langtangen, Gaute Einevoll and Geir Halnes}
\institute{University of Oslo, CINPLA}
\date{}
% \titlegraphic{\includegraphics[width=\textwidth]{chaospy_logo.jpg}}
\begin{document}
% \maketitle
% \begin{frame}
% \begin{tikzpicture}[remember picture, overlay]
% \node[anchor = center, opacity=.25] (image) at (current page.center) {\includegraphics[scale=0.25]{chaospy_logo.jpg}};
% \end{tikzpicture}
%
% \begin{center}
% \textbf{ \color{myred} \Large Chaospy: \\ \vspace{1mm} A modular implementation of polynomial\\ \vspace{1mm} chaos expansions and Monte Carlo methods}
% \end{center}
% \begin{center}
% \large \vspace{5mm} Simen Tennøe\\ \vspace{5mm}\footnotesize Supervisors:\\ \vspace{0.5mm} Jonathan Feinberg, Hans Petter Langtangen, Gaute Einevoll and Geir Halnes
% \end{center}
%
% \begin{center}
% \small University of Oslo, CINPLA
% \end{center}
% %\includegraphics[width=\textwidth]{chaospy_logo.jpg}
%
% \end{frame}
\begin{frame}
\begin{center}
\textbf{\color{myred}\Large Chaospy: \\ \vspace{1mm} A modular implementation of Polynomial\\ \vspace{1mm} Chaos expansions and Monte Carlo methods}
\end{center}
% \frametitle{\textbf{\color{myred}\Large Chaospy: \\ \vspace{1mm} A modular implementation of polynomial\\ \vspace{1mm} chaos expansions and Monte Carlo methods}}
\large \vspace{8mm} Simen Tennøe
\vspace{6mm}
\footnotesize Supervisors:
%
\vspace{1mm}
Jonathan Feinberg
Hans Petter Langtangen
Gaute Einevoll
Geir Halnes
\vspace{5mm}
\small University of Oslo, CINPLA
\begin{tikzpicture}[remember picture,overlay]
\node [xshift=0.2\paperwidth,yshift=-0.14\paperheight] at (current page.center)
{\includegraphics[width = 0.5\paperwidth ]{chaospy_logo.jpg}};
\end{tikzpicture}
\begin{tikzpicture}[remember picture,overlay]
\node [xshift=1.35cm,yshift=0.55cm] at (current page.south west)
{\includegraphics[width = 0.2\paperwidth ]{cinpla.png}};
\end{tikzpicture}
\end{frame}
\begin{frame}[fragile]{Chaospy is a Python toolbox for forward model UQ}
\begin{tikzpicture}[remember picture, overlay, font=\sffamily]
\node [align=left, xshift=-0.4\textwidth,yshift=0.15\textwidth] (image3) at (current page.center)
{\includegraphics[width = 0.2\textwidth]{chaospy_logo.jpg}};
\node[align=left] at (image3.east) {\hspace{5cm} \bf Properties of Chaospy};
\end{tikzpicture}
\pause
\begin{tikzpicture}[remember picture, overlay, font=\sffamily]
\node [align=left, xshift=-0.2\textwidth,yshift=-0.05\textwidth] (image1) at (current page.center)
{\includegraphics[width = 0.25\textwidth]{samples.png}};
\node[align=left] at (image1.east) {\hspace{4.5cm} \bf Monte Carlo methods};
\end{tikzpicture}
\pause
\begin{tikzpicture}[remember picture, overlay, font=\sffamily]
\node [align=left, xshift=0\textwidth,yshift=-0.25\textwidth] (image2) at (current page.center)
{\includegraphics[width = 0.25\textwidth]{pc.png}};
\node[align=left] at (image2.east) {\hspace{4cm} \bf Polynomial Chaos};
\end{tikzpicture}
\end{frame}
\begin{frame}{What is new in Chaospy}
\vspace{-7mm}
\begin{columns}
\column{.5\textwidth}
\begin{center}
\textbf<1>{Chaospy is modular and therefore very flexible}
\end{center}
\column{.5\textwidth}
\begin{center}
\includegraphics[width=0.5\textwidth]{lego.jpg}
\end{center}
\end{columns}
\pause
\vspace{7mm}
\begin{columns}
\column{.5\textwidth}
\begin{center}
\includegraphics[width=0.5\textwidth]{dependent.png}
\end{center}
\column{.5\textwidth}
\begin{center}
\textbf<2>{Chaospy has support for dependent variables}
\end{center}
\end{columns}
\pause
\vspace{7mm}
\begin{columns}
\column{.5\textwidth}
\begin{center}
\textbf<3>{Chaospy has a large collection of methods and distributions}
\end{center}
\column{.5\textwidth}
\begin{center}
\pause \textbf<4>{It is easy to compare different methods on given a problem}
\end{center}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Comparing Chaospy with Turns and Dakota}
\begin{center}
\newcolumntype{g}{>{\columncolor{gray}}l}
\def\arraystretch{1.1}
\begin{tabular}{ll|lll}
Feature & & Dakota & Turns & Chaospy\\\hline
Distributions && 11 & 26 & \textbf<3>{\textcolor<3>{green}{64}}\\
Copulas && 1 & 7 & \textbf<2->{\textcolor<2->{red}6}\\
Sampling schemes & & 4 & 7.5 & \textbf<2->{\textcolor<2->{red}7}\\
Orthogonal polynomial schemes & & 4 & 3 & \textbf<3>{\textcolor<3>{green}5}\\
Numerical integration strategies & & 7 & 0 & \textbf<3>{\textcolor<3>{green}7}\\
Regression methods & & 5 & 4 & \textbf<3>{\textcolor<3>{green}8}\\
Analytical metrics & & 6 & 6 & \textbf<3>{\textcolor<3>{green}7}\\
\end{tabular}
\end{center}
\end{frame}
%--- Next Frame ---%
\begin{frame}{Chaospy has support for many different methods}
\begin{itemize}[<+->]
\setlength{\itemsep}{5pt}
\item Monte Carlo with variance reduction techniques
\item Intrusive and non-intrusive polynomial chaos
\begin{itemize}
\vspace{5pt}
\setlength{\itemsep}{5pt}
\item Pseudo-spectral method
\item Point collocation/regression
\end{itemize}
\end{itemize}
\end{frame}
\begin{chaospy}{All Chaospy needs is a Python wrapper around the forward model}
\begin{lstlisting}[language=python]
def solver(*node):
# node: tuple of the uncertain stochastic parameters
model.set_parameters(node)
|\pause|
model.run()
|\pause|
results = model.post_processing()
|\pause|
return results
\end{lstlisting}
\end{chaospy}
\begin{frame}
\frametitle{Chaospy is a completly generic software; for simplicity we use a very simple example problem}
\vspace{-10mm}
\begin{align*}
\frac{d u(x)}{dx} & =-au(x),\qquad u(0) = I.
\end{align*}
\vspace{-5 mm}
\begin{itemize}
\item[$u$] The quantity of interest.
\item[$x$] Spatial location.
\item[$a,I$] Parameters containing uncertainties.
\end{itemize}
%\vspace{-5 mm}
\pause
%Initially assume model parameters:
\begin{align*}
a &\sim \text{Uniform(0, 0.1)} & I& \sim \text{Uniform(8, 10)}
\end{align*}
\pause
\vspace{5 mm}
%Analytical solution
%\[u(x; a, I) = Ie^{-ax}.\]
%\pause
%\vspace{5mm}
We want to compute E(u) and Var(u).
\end{frame}
\begin{frame}[fragile]
{Monte Carlo integration can be used for any model}
\begin{figure}
\includegraphics[width=\textwidth]{MC.png}
\end{figure}
\end{frame}
%
% \begin{chaospy}{Creating multivariate probability distributions in Chaospy}
% \begin{lstlisting}[language=python]
% dist_a = cp.Uniform(0, 0.1)
% dist_I = cp.Uniform(8, 10)
% |\pause|
% dist = cp.J(dist_a, dist_I)
% \end{lstlisting}
% \end{chaospy}
\begin{chaospy}{Monte Carlo with Chaospy}
\scriptsize
\begin{lstlisting}[language=python]
import chaospy as cp
import numpy as np
|\pause|
dist_a = cp.Uniform(0, 0.1)
dist_I = cp.Uniform(8, 10)
|\pause|
# Joint distribution
dist = cp.J(dist_a, dist_I)
|\pause|
samples = dist.sample(size=1000)
|\pause|
# solver returns u(x), where x is fixed
# samples_u: list of all u(x) for each set of a and I
samples_u = [solver(a, I) for a, I in samples]
|\pause|
E = np.mean(samples_u, 0)
Var = np.var(samples_u, 0)
\end{lstlisting}
\end{chaospy}
\begin{frame}
\frametitle{Convergence of Monte Carlo is slow}
\vspace{-4mm}
\begin{align*}
\varepsilon_E &= \int|\E{u} - \E{\hat{u}}|\,dx & \qquad
\varepsilon_{Var} &= \int|\Var{u} - \Var{\hat{u}}|\,dx
\end{align*}
\vspace{-5mm}
\pause
\begin{center}
\includegraphics[width=0.9\textwidth]{MC_convergence_2D.png}
\end{center}
\end{frame}
\begin{frame}[fragile]
\frametitle{Chaospy has several variance reduction techniques for sampling a distribution}
\vspace{-5mm}
\begin{columns}
\column{.5\textwidth}
\begin{center}
\includegraphics[width=0.65\textwidth]{samples_H.png}
Hammersley sampling:
\scriptsize
\verb;nodes = dist.sample(100, "M");
\normalsize
\includegraphics[width=0.65\textwidth]{samples_H.png}
Halton sampling
\scriptsize
\verb;nodes = dist.sample(100, "H");
\normalsize
\end{center}
\column{.5\textwidth}
\begin{center}
\includegraphics[width=0.65\textwidth]{samples_L.png}
Latin Hypercube sampling:
\scriptsize
\verb;nodes = dist.sample(100, "L");
\normalsize
\includegraphics[width=0.65\textwidth]{samples_S.png}
Sobol sampling
\scriptsize
\verb;nodes = dist.sample(100, "S");
\normalsize
\end{center}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{The different sampling schemes available in Chaospy compared to Turns and Dakota}
\begin{center}
\def\arraystretch{1.1}
\begin{tabular}{ll|lll}
&& Dakota & Turns & Chaospy\\
Quasi-Monte Carlo scheme && & & \\\hline
Faure sequence && No & Yes & \textbf{\textcolor{red}{No}}\\
Halton sequence && Yes& Yes & \textbf{\textcolor{green}{Yes}}\\
Hammersley sequence && Yes & Yes & \textbf{\textcolor{green}{Yes}}\\
Haselgrove sequence && No & Yes & \textbf{\textcolor{red}{No}}\\
Korobov latice && No & No& \textbf{\textcolor{green}{Yes}}\\
Niederreiter sequence && No& Yes& \textbf{\textcolor{red}{No}}\\
Sobol sequence && No & Yes & \textbf{\textcolor{green}{Yes}}\\\\
Other methods &&&&\\\hline
Antithetic variables && No& No& \textbf{\textcolor{green}{Yes}}\\
Importance sampling && Yes &Yes& \textbf{\textcolor{green}{Yes}}\\
Latin Hypercube sampling && Yes & Limited & \textbf{\textcolor{green}{Yes}}\\
\end{tabular}
\end{center}
\end{frame}
\begin{chaospy}{Quasi-Monte Carlo with Latin Hypercube sampling}
\scriptsize
\begin{lstlisting}[language=python]
import chaospy as cp
import numpy as np
dist_a = cp.Uniform(0, 0.1)
dist_I = cp.Uniform(8, 10)
dist = cp.J(dist_a, dist_I)
samples = dist.sample(size=1000|\only<2>{, \color{myred}{rule="L"}}|)
samples_u = [solver(a, I) for a, I in samples]
E = np.mean(samples_u, 0)
Var = np.var(samples_u, 0)
\end{lstlisting}
\end{chaospy}
\begin{frame}
\frametitle{Convergence of quasi-Monte Carlo is better than Monte Carlo, but still slow}
\vspace{-5mm}
\begin{center}
\includegraphics[width=1\textwidth]{qMC-MC_convergence_2D.png}
\end{center}
\end{frame}
%\begin{frame}
%\frametitle{All random variables can with aid of the Rosenblatt transformations be transformed to/from the uniform distribution}
%\begin{figure}
%\includegraphics[width = \textwidth]{distributions.png}
%\end{figure}
%\end{frame}
%\begin{frame}
%\frametitle{Chaospy is built around the Rosenblatt transformation which enables several useful properties}
%\begin{itemize}[<+->]
%\item Sampling of all probability distributions.
%\vspace{3mm}
%\item Quasi-Monte Carlo sampling is available for any probability distribution.
% \vspace{3mm}
% \item Polynomial Chaos expansions are available for any probability distribution, even dependent variables
%\end{itemize}
% \end{frame}
\begin{frame}
\frametitle{Mapping in probability space; the idea behind Polynomial Chaos (PC) theory is to approximate our forward model with a polynomial}
\begin{align*}
u(x; q) & &\approx && \hat u_M(x;q) && =
&& \sum_{n=0}^N && {\color<3>{red}c_n(x)}\quad && {\color<4>{red}P_n(q)}\\
&& && && && && {\color<3>{red}\text{Coefficient}} && {\color<4>{red}\text{Polynomial}}
\end{align*}
$\hat u_M(x; q)$ is the mapping from the uncertain variables $q$ to the response variable $u$, $x$ is a fixed variable.
% \hspace{10.5mm} $q$ are the uncertain variables
% \begin{align*}
% N = \frac{(M + D)!}{M!D!} - 1, \quad D\text{ is the number of dimensions}\\
% \end{align*}
\pause
\vspace{1cm}
Mean and variance are calculated from $\hat u_M(x;q)$.
\end{frame}
%\begin{frame}
% \frametitle{Want to have the Best Linear Unbiased Estimator (BLUE)}
%We have the variance criteria
%\[ \Var{|u - \hat u_M|}\]
%\pause
%and want to choose the Best Linear Unbiased Estimator (BLUE):
%\begin{align*}
% \E{\hat u_M} &= u\\
% \Var{|u - \hat u_M|} &\leq \Var{|u - \hat u^*|} & &\forall u^* \in P_M\\
% &&& \forall M \in |\mathbb{Z}^+|
%\end{align*}
% All polynomials $P_n$ and coefficients $c_n$ are free.
%\end{frame}
%\begin{frame}
% \frametitle{We need to choose Askey-Wilson polynomials}
%Choosing the Best Linear Unbiased Estimator leads to:
%\begin{align*}
%\E{P_nP_m} &= 0 & \forall n \neq m\\
%\end{align*}
%This is fulfilled by choosing Askey-Wilson polynomials, a family of orthogonal polynomials.
%\end{frame}
%
% \begin{frame}
% \frametitle{The computational essence of Polynomial Chaos}
% \[\hat u_M(x;q) = \sum_{n=0}^N c_n(x) P_n(q)\]
% \vspace{10mm}
%
% We need to find the orthogonal polynomials $P_n(q)$ and the coefficients $c_n(x)$.
%
% \end{frame}
\begin{chaospy}{$P_n$ are orthogonal polynomials and are generaly calculated through the three-term discretized Stiltjes recursion}
\begin{lstlisting}[language=python]
dist = cp.Normal()
P = cp.orth_ttr(3, dist)
|\pause|
print P
[1.0, q0, q0^2-1.0, q0^3-3.0q0]
\end{lstlisting}
\end{chaospy}
\begin{frame}
\frametitle{Methods for generating expansions of orthogonal polynomials}
\begin{center}
\def\arraystretch{1.1}
\begin{tabular}{ll|lll}
Orthogonalization Method && Dakota &Turns & Chaospy \\\hline
Askey–Wilson scheme && Yes &Yes& \textbf{\textcolor{green}{Yes}}\\
Bertran recursion && No &No &\textbf{\textcolor{green}{Yes}}\\
Cholesky decomposition && No &No& \textbf{\textcolor{green}{Yes}}\\
Discretized Stieltjes && Yes &No& \textbf{\textcolor{green}{Yes}}\\
Modified Chebyshev && Yes & Yes& \textbf{\textcolor{red}{No}}\\
Modified Gram–Schmidt && Yes & Yes & \textbf{\textcolor{green}{Yes}}\\
\end{tabular}
\end{center}
\end{frame}
%
% \begin{frame}
% This leads to
% \begin{align*}
% \E{P_nP_m} &= 0 & \forall n \neq m\\
% c_n &= \frac{\E{uP_n}}{\E{P_n^2}}
% \end{align*}
%
% Solution:
% \begin{align*}
% \E{P_nP_m} &= 0 & &\text{Fulfilled by choosing orthogonal polynomials} \\
% c_n &&& \text{Can be approximated by quadrature rules, }\\
% &&& \text{the probability distribution is used as a weight function}
% \end{align*}
% \end{frame}
%\begin{frame}
%\frametitle{$c_n$ is approximated by quadrature rules, called the pseudo-spectral method}
% Choosing the Best Linear Unbiased Estimator also leads to:
%
%\begin{align*}
%c_n &= \frac{\E{uP_n}}{\E{P_n^2}}
%\end{align*}
%$c_n$ is approximated by quadrature rules where the probability distribution is used as a weight function.
%\end{frame}
\begin{chaospy}{The pseudo-spectral method, used to calculate $c_n$, needs numerical integration, which demands generating quadrature nodes and weights}
\scriptsize
\begin{lstlisting}[language=python]
dist = cp.Normal()
nodes, weights = cp.generate_quadrature(2, dist, rule="G")
|\pause|
print nodes
[[-1.73205081 0. 1.73205081]]
print weights
[ 0.16666667 0.66666667 0.16666667]
\end{lstlisting}
\normalsize
\end{chaospy}
\begin{frame}
\frametitle{Numerical integration strategies implemented in the three software toolboxes}
\begin{center}
\newcolumntype{g}{>{\columncolor{gray}}l}
\def\arraystretch{1.1}
\begin{tabular}{ll|lll}
Node and weight generators&& Dakota &Turns& Chaospy\\\hline
Clenshaw-Curtis quadrature && Yes& No &\textbf{\textcolor{green}{Yes}}\\
Cubature rules && Yes &No& \textbf{\textcolor{red}{No}}\\
Gauss-Legendre quadrature && Yes &No& \textbf{\textcolor{green}{Yes}}\\
Gauss-Patterson quadrature && Yes &No& \textbf{\textcolor{green}{Yes}}\\
Genz-Keister quadrature && Yes &No& \textbf{\textcolor{green}{Yes}}\\
Leja quadrature && No& No& \textbf{\textcolor{green}{Yes}}\\
Monte Carlo integration && Yes& No &\textbf{\textcolor{green}{Yes}}\\
Optimal Gaussian quadrature && Yes& No& \textbf{\textcolor{green}{Yes}}\\
\end{tabular}
\end{center}
\end{frame}
\begin{chaospy}{One slide is enough for the full implementation with the pseudo-spectral method
in Chaospy}
\scriptsize
% def u(x, a, I):
% return I*np.exp(-a*x)
% |\pause|
\begin{lstlisting}[language=python]
dist_a = cp.Uniform(0, 0.1)
dist_I = cp.Uniform(8, 10)
dist = cp.J(dist_a,dist_I)
|\pause|
P = cp.orth_ttr(2, dist)
|\pause|
nodes, weights = cp.generate_quadrature(3, dist)
|\pause|
samples_u = [solver(*node) for node in nodes.T]
|\pause|
u_hat = cp.fit_quadrature(P, nodes, weights, samples_u
rule="Gaussian")
|\pause|
mean = cp.E(u_hat, dist)
var = cp.Var(u_hat, dist)
\end{lstlisting}
\end{chaospy}
\begin{frame}
\frametitle{Convergence of polynomial chaos is much faster than the Monte Carlo methods}
\vspace{-5.8mm}
\begin{figure}
\includegraphics[width=1\textwidth]{qMC-MC-PC_convergence_2D.png}
\end{figure}
\end{frame}
%\begin{frame}
% \frametitle{Chaospy can be used for vastly more complicated problems, for example bloodflow simulations}
%
%\vspace{-5.8mm}
%
% \begin{columns}
% \column{.5\textwidth}
% \begin{center}
% \includegraphics[width=.6\textwidth]{arterialHumanPicture.png}
% \end{center}
% \column{.5\textwidth}
% \begin{center}
% \includegraphics[width=.6\textwidth]{arterialTreeMascot3D.png}
% \end{center}
% \end{columns}
%
%\vspace{5mm}
%\tiny
%Eck V, Feinberg J, Langtangen H, Hellevik L. Stochastic sensitivity analysis for timing and amplitude of pressure waves in the arterial system. Int J Numer Meth Biomed Engng. 2015;31(4):n/a-n/a. doi:10.1002/cnm.2711.
%
%\end{frame}
\begin{chaospy}{Chaospy is an ideal tool for research in UQ for the statistics expert}
With a few lines of Python code it is easy to customize:
\begin{itemize}[<+->]
\setlength{\itemsep}{5pt}
\item distributions
\item polynomials
\item integration schemes
\item sampling schemes
\item statistical analysis of the result
\end{itemize}
\pause
\vspace{5mm}
Custom polynomials:
\scriptsize
\begin{lstlisting}[language=python]
q0, q1 = cp.variable(2)
|\pause|
phi = cp.Poly([1, q0, q1, q0**2 - 1, q0*q1])
|\pause|
print phi
[1, q0, q1, q0^2-1, q0q1]
\end{lstlisting}
\end{chaospy}
\begin{frame}
\frametitle{Chaospy handles Polynomial Chaos expansions with stochastically dependent variables}
Diffusion in layered media with uncertain boundary, $l$, and uncertain diffusion constants, $D_0$, $D_1$.
\begin{figure}
\includegraphics[height = 0.4\textheight]{layered_media.png}
\end{figure}
Uncertain $l$ slows down convergence, but introduction of auxiliary \emph{dependent} variables restores convergence.
% Standard polynomial chaos does not work very well, but we can transform our variables to get around the problem.
% The variables then become dependent.
%\pause
%Dependent variables break the orthogonality properties of the polynomials.
%\pause
%Chaospy uses Rosenblatt transformations to solve this problem.
\end{frame}
%\begin{chaospy}{Chaospy handles Polynomial Chaos expansions with stochastically dependent variables}
% \scriptsize
%\begin{lstlisting}[language=Python]
%
%C = [[1,0.5],[0.5,1]]
%mu = np.array([0, 0])
%dist_Q = cp.MvNormal(mu, C)|\pause|
%dist_R = cp.J(cp.Normal(), cp.Normal())
%
%P = cp.orth_ttr(M, dist_R)|\pause|
%nodes_R, weights_R = cp.generate_quadrature(M+1, dist_R)|\pause|
%nodes_Q = dist_Q.inv(dist_R.fwd(nodes_R))|\pause|
%weights_Q = weights_R*\
% dist_Q.pdf(nodes_Q)/dist_R.pdf(nodes_R)
%|\pause|
%samples_u = [solver(*node) for node in nodes_Q.T]|\pause|
%u_hat = cp.fit_quadrature(P, nodes_R, weights_Q, samples_u)
%\end{lstlisting}
%\end{chaospy}
%\begin{chaospy}{The polynomial approximation can be used as a surrogate model, which allows for computationally cheap statistical analysis}
% \scriptsize
% \begin{lstlisting}[language=python]
%u_hat = cp.fit_quadrature(P, nodes, weights, solves)
%|\pause|
%samples_q = dist.sample(10**6)
%samples_u = u_hat(*samples_q)
%|\pause|
%mean = np.mean(samples_u, 1)
%var = np.var(samples_u, 1)
%\end{lstlisting}
%\end{chaospy}
%
% \begin{frame}{Advantages and disadvantages of Chaospy}{}
% \onslide<1->\bf \color{green} {Advantages}
% \begin{itemize}
% \onslide<3->\gooditem The programming syntax resembles the mathematics
% \onslide<1->\gooditem Modular software design
% \onslide<5->\gooditem Supports both Monte Carlo and Polynomial Chaos
% \onslide<6->\gooditem Handles dependent variables
% \end{itemize}
% \vspace{8mm}
% \onslide<1->\bf\color{red} Disadvantages
% \begin{itemize}
% \onslide<2->\baditem No gui
% \onslide<4->\baditem The programming syntax lies close to the mathematical theory
% \onslide<7->\baditem Slow
% \end{itemize}
%
% \end{frame}
% \begin{frame}
% \frametitle{Some random variables are dependent}
% \begin{figure}
% \includegraphics[width = 0.9\textwidth]{dependent.png}
% \end{figure}
%
% \end{frame}
%
% \begin{frame}{Transformations manipulates probability distributions}
% \begin{figure}
% \includegraphics[width=\textwidth]{trans2.png}
% \end{figure}
% \end{frame}
%
% \begin{frame}
% \frametitle{Dependent variables break the orthogonality property of Polynomal Chaos!}
% \scriptsize
% \[ P_{\mathfrak i} = P_{i_1}^{(1)}P_{i_2}^{(2)}\cdots P_{i_D}^{(D)} \qquad \mathfrak i = (i_1,i_2,...,i_D)\]
% \pause
% \begin{align*}
% \inner{ P_\mathfrak{i},P_\mathfrak{j}} &= E(P_\mathfrak{i},P_\mathfrak{j})\\
% \onslide<3-> {&=\E{P_{i_1}^{(1)}\cdots P_{i_D}^{(D)}P_{j_1}^{(1)}\cdots P_{j_D}^{(D)}}}\\
% \onslide<4-> {&=\E{P_{i_1}^{(1)}P_{j_1}^{(1)}}\cdots \E{P_{i_D}^{(D)}P_{j_D}^{(D)}}}\\
% \onslide<5-> {&=\dots}\\
% \onslide<5-> {&=\norm{P_{\mathfrak{i}}^{(1)}}\delta_{\mathfrak{i}\mathfrak{j}}}
% \end{align*}
% \onslide<6->
% \begin{alert}{But the problem is:}
% \[\E{uv} \neq \E{u}\E{v} \qquad \text{when $u$ and $v$ are
% stochastically dependent}\]
% \end{alert}
% \end{frame}
% \begin{chaospy}{Point collocation with Rosenblatt transformation}
% \scriptsize
% \begin{lstlisting}[language=Python]
% def u(x,a, I):
% return I*np.exp(-a*x)
% |\pause|
% dist_R = cp.J(cp.Normal(), cp.Normal())
% C = [[1, 0.5], [0.5, 1]]
% mu = [0, 0]
% dist_Q = cp.MvNormal(mu, C)
% |\pause|
% P = cp.orth_ttr(M, dist_R)|\pause|
% nodes_R = dist_R.sample(2*len(P), "M")|\pause|
% nodes_Q = dist_Q.inv(dist_R.fwd(nodes_R))
% |\pause|
% x = np.linspace(0, 1, 100)
% samples_u = [u(x, *node) for node in nodes_Q.T]|\pause|
% u_hat = cp.fit_regression(P, nodes_R, samples_u)
% \end{lstlisting}
% \end{chaospy}
\begin{frame}{Summary: Chaospy is a Python toolbox for forward model UQ with advanced Monte Carlo methods and Polynomial Chaos expansions}
\vspace{-5mm}
\begin{columns}
\column{.5\textwidth}
\begin{center}
\bf{Chaospy is modular, flexible, with syntax that resembles the mathematics}
\end{center}
\column{.5\textwidth}
\begin{center}
\includegraphics[width=0.6\textwidth]{lego.jpg}
\end{center}
\end{columns}
\vspace{10mm}
\begin{columns}
\column{.5\textwidth}
\begin{center}
\bf{A vast collection of methods, ideal for method comparisons}
\end{center}
\column{.5\textwidth}
\begin{center}
\includegraphics[width=0.6\textwidth]{bookshelf.jpg}
\end{center}
\end{columns}
%\begin{tikzpicture}[remember picture, overlay, font=\sffamily]
% \node [align=left, xshift=0.3\textwidth,yshift=0.1\textwidth] (image1) at (current page.center)
% {\includegraphics[width = 0.3\textwidth]{samples.png}};
% \node[align=left] at (image1.west) {\hspace{-7cm} \bf Advanced Monte Carlo methods};
% \end{tikzpicture}
%\begin{tikzpicture}[remember picture, overlay, font=\sffamily]
% \node [align=left, xshift=0.3\textwidth,yshift=-0.15\textwidth] (image2) at (current page.center)
% {\includegraphics[width = 0.25\textwidth]{pc.png}};
% \node[align=left] at (image2.west) {\hspace{-7.95cm} \bf Polynomial Chaos expansions};
% \end{tikzpicture}
% \begin{tikzpicture}[remember picture, overlay, font=\sffamily]
% \node [align=left, xshift=0.49\textwidth,yshift=-0.37\textwidth] (image2) at (current page.center)
% {\includegraphics[width = 0.35\textwidth]{cinpla.png}};
% \end{tikzpicture}
%
% \begin{tikzpicture}[remember picture, overlay, font=\sffamily]
% \node [align=left, xshift=-0.45\textwidth,yshift=-0.37\textwidth] (image2) at (current page.center)
% {\includegraphics[width = 0.18\textwidth]{chaospy_logo.jpg}};
% \end{tikzpicture}
% \pause
% \begin{tikzpicture}[remember picture, overlay, font=\sffamily]