Commit 4a071a53 authored by niklas.baumgarten's avatar niklas.baumgarten
Browse files

worked on slides

parent 9382fe7f
......@@ -54,12 +54,17 @@
\newcommand{\da}{\, \mathrm{d} a}
% Naming convcention
\newcommand{\goal}{\, \mathcal{Q}}
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\def \Title {Introductory Tutorial on M++}
\def \Titelshort {Tutorial}
\def \Lecturer {Prof. Dr.~Christian Wieners}
\def \Ubungsleiter {M.~Sc.~Niklas~Baumgarten}
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\usepackage[colorlinks=true,
pdftitle={Introductory Tutorial on M++},
pdfauthor={Niklas Baumgarten},
pdfpagemode={UseOutlines},pdfstartview={Fit},
linkcolor=blue,urlcolor=blue,
pdfsubject={Introductory Tutorial on M++}]{hyperref}
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Institute for Applied and Numerical Mathematics}
\\
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{\Lecturer} ~\textbullet~ {\small \Ubungsleiter}}
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No preview for this file type
\documentclass[18pt]{beamer}
\usepackage{defs}
\usepackage{templates/beamerthemekit}
\title{Multilevel Monte Carlo Applications}
\subtitle{Review of implementation and results}
\author{Niklas Baumgarten, Christian Wieners}
\institute{Institute for Applied and Numerical Mathematics}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}{Outline}
\tableofcontents
\end{frame}
\section*{Elliptic model problem}
\input{src/elliptic_model_problem.tex}
\input{src/example_fields}
\section*{Monte Carlo methods}
\input{src/mlmc}
\input{src/experimental_setup}
\section*{Intro}
\input{src/numerical_results}
\section*{Stochastic Linear Transport problem}
\section*{Stochastic Convection-Diffusion-Reaction problem}
\section*{Acoustic wave propagation}
% \input{src/alternative_acoustic}
\section*{Outlook and Conclusion}
% \input{src/outlook}
\begin{frame}{References}
\bibliographystyle{acm}
\tiny{\bibliography{lit}}
\end{frame}
\section*{Backup}
% \input{src/the_random_field_model}
% \input{src/circulant_embedding}
% \input{src/weak_formulation}
% \input{src/existence}
% \input{src/regularity}
% \input{src/finite_element_error}
% \input{src/mlmc_algorithm}
\end{document}
\ No newline at end of file
......@@ -3,12 +3,12 @@
\item Goal: Estimate the expectation $\EE[\goal(\omega)]$, where $\goal$ is some
functional of the random
solution $u(\omega, x)$.
\item Assume: $u_h(\omega, x)$ is the corresponding FEM solution with the
\item Assume: $u_h(\omeg, x)$ is the corresponding FEM solution with the
convergence rate $\alpha > 0$, i.e.
\begin{equation}
\label{eq:alpha-assumption}
\abs{\EE[\goal_h - \goal]} \lesssim h^\alpha, \quad
\abs{\EE[\goal_h - \goal]} \lesssim N^{-\alpha}, \quad
\abs{\EE[\goal_h - \goal]} \lesssim N^{-\alpha / d}, \quad
N = \dim(V_h)
\end{equation}
and that the cost for one sample can be bounded with $\gamma > 0$ by
......@@ -18,61 +18,72 @@
\cost(\goal_h) \lesssim
N^{\gamma / d}
\end{equation}
and the variance of the difference $\goal_{h_l} - \goal_{h_{l-1}}$ decays with
and the variance of the difference $\goal_l - \goal_{l-1}$ decays with
\begin{equation}
\label{eq:beta-assumption}
\abs{\mathbb{V}[\goal_{h_l} - \goal_{h_{l-1}}]} \lesssim h^\beta, \quad
\abs{\mathbb{V}[\goal_{h_l} - \goal_{h_{l-1}}]} \lesssim N^{-\beta / d}
\abs{\VV[\goal_l - \goal_{l-1}]} \lesssim h^\beta, \quad
\abs{\VV[\goal_l - \goal_{l-1}]} \lesssim N^{-\beta / d}
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}{Multilevel Monte Carlo Method I}
\begin{frame}{Multilevel Monte Carlo Method II}
\begin{itemize}
\item The Monte Carlo estimator for the approximated functional is
\begin{equation*}
\item Monte Carlo (MC) estimator:
\begin{align*}
\widehat{\goal}_{h,M}^{MC} = \frac{1}{M} \sum_{m=1}^M \goal_h(\omega_m)
\end{equation*}
\item The root mean square error (RMSE) is then given by
\begin{equation*}
\end{align*}
\item Root mean square error (RMSE):
\begin{align*}
e(\widehat{\goal}^{MC}_{h,M})^2 =
\EE \left[ (\widehat{\goal}^{MC}_{h,M} - \EE[\goal])^2 \right] =
\underbrace{M^{-1} \VV[\goal_h]}_{\text{estimator error}} +
\underbrace{\left( \EE[Q_h - Q] \right)^2}_{\text{FEM error}}.
\end{equation*}
\item This yields a total cost of ($\cost_\epsilon(\widehat{Q}_h)$ is the cost to
\item achieve $e(\widehat{Q}_h) < \epsilon$)
\begin{equation*}
\cost(\widehat{Q}^{MC}_{h,M}) \lesssim M \cdot N^\gamma, \quad \cost_{\epsilon}(\widehat{Q}^{MC}_{h,M}) \lesssim \epsilon^{-2 -\frac{\gamma}{\alpha}}.
\end{equation*}
\underbrace{\left( \EE[\goal_h - \goal] \right)^2}_{\text{FEM error}}
\end{align*}
\item Total cost:
\begin{align*}
\cost(\widehat{\goal}^{MC}_{h,M}) \lesssim M \cdot N^\gamma, \quad
\cost_{\epsilon}(\widehat{\goal}^{MC}_{h,M}) \lesssim
\epsilon^{-2-\frac{\gamma}{\alpha}}
\end{align*}
Here, $\cost_\epsilon(\widehat{\goal}_h)$ is the cost to achieve
$e(\widehat{\goal}_h) < \epsilon$
\end{itemize}
\end{frame}
\begin{frame}{Multilevel Monte Carlo Methods I}
\begin{frame}{Multilevel Monte Carlo Methods III}
\begin{itemize}
\item The multilevel Monte Carlo (MLMC) method finds a way to balance the cost of individual samples and the total amount (confer \cite{charrier2013finite, giles2008multilevel, teckentrup2013further}).
\item Main idea MLMC: Sample from several approximation $Q_{h_l}$.
\item With the linearity of the expectation operator it holds
\begin{equation*}
\mathbb{E}[Q_{h_L}] = \mathbb{E}[Q_{h_0}] + \sum_{l=1}^L \mathbb{E}[Q_{h_l} - Q_{h_{l-1}}] = \sum_{l=0}^L \mathbb{E}[Y_l].
\end{equation*}
\item Now estimate each $Y_l$ with the classical MC method, thus
\begin{equation*}
\widehat{Y}^{MC}_{h,M_l} = \frac{1}{M_l} \sum_{i=1}^{M_l} \left( Q_{h_l}(\omega_i) - Q_{h_{l-1}}(\omega_i) \right), \quad
\widehat{Y}^{MC}_{h,M_0} = \frac{1}{M_0} \sum_{i=1}^{M_0} Q_{h_0}(\omega_i).
\end{equation*}
\item This gives the MLMC estimator
\begin{equation*}
\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L} = \sum_{l=0}^L \widehat{Y}^{MC}_{h,M_l} = \sum_{l=0}^L \frac{1}{M_l} \sum_{i=1}^{M_l} Y_l(\omega_{i}).
\end{equation*}
\item Main idea: Sample from several approximation levels
and balance cost per level $\cost_l$ with total sample amount per level $M_l$
\item Set $\goal_l - \goal_{l-1} \defeq \dgoal_l$ and $\goal_0 \defeq \dgoal_0$:
\begin{align*}
\EE[\goal_L] = \EE[\goal_0] + \sum_{l=1}^L \EE[\goal_l - \goal_{l-1}] =
\sum_{l=0}^L \EE[\dgoal_l]
\end{align*}
\item Estimate each $\dgoal_l$ with the MC method:
\begin{align*}
\widehat{\dgoal}^{MC}_{h,M_l} =
\frac{1}{M_l} \sum_{m=1}^{M_l} \left( \goal_l(\omega_m) -
\goal_{l-1} (\omega_m) \right), \quad
\widehat{\dgoal}^{MC}_{h,M_0} =
\frac{1}{M_0} \sum_{m=1}^{M_0} \goal_0 (\omega_m)
\end{align*}
\item MLMC estimator:
\begin{align*}
\widehat{\goal}^{MLMC}_{h,\{ M_l \}_{l=0}^L} =
\sum_{l=0}^L \widehat{\dgoal}^{MC}_{h,M_l} =
\sum_{l=0}^L \frac{1}{M_l} \sum_{m=1}^{M_l} \dgoal_l(\omega_{m})
\end{align*}
\end{itemize}
\end{frame}
\begin{frame}{Multilevel Monte Carlo Methods II}
\begin{itemize}
\item The RMSE is then given by
\item Root mean square error (RMSE):
\begin{equation*}
e(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L})^2 = \mathbb{E}\left[( \widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L} - \mathbb{E}[Q])^2 \right] = \underbrace{\sum_{l=0}^L \frac{1}{M_l} \VV[Y_l]}_{\text{estimator error}} + \underbrace{\left( \mathbb{E}[Q_h - Q] \right)^2}_{\text{FEM error}}.
e(\widehat{\goal}^{MLMC}_{h,\{ M_l \}_{l=0}^L})^2 =
\underbrace{\sum_{l=0}^L \frac{1}{M_l} \VV[\dgoal_l]}_{\text{estimator error}} +
\underbrace{\left( \EE[\goal_L - \goal] \right)^2}_{\text{FEM error}}.
\end{equation*}
\item This leads leads to a better computational cost since:
\begin{itemize}
......
......@@ -157,15 +157,7 @@ KIT -- University of the State of Baden-Wuerttemberg and\\National Laboratory of
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......@@ -178,7 +170,4 @@ KIT -- University of the State of Baden-Wuerttemberg and\\National Laboratory of
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