Commit 79c1540b authored by niklas.baumgarten's avatar niklas.baumgarten
Browse files

worked on slides

parent 976a60fc
\documentclass[18pt]{beamer}
\usepackage{defs}
\usepackage{templates/beamerthemekit}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}{Outline}
\tableofcontents
\end{frame}
\section*{Elliptic model problem}
\input{src/model_problem}
\input{src/example_fields}
\section*{Monte Carlo methods}
\input{src/monte_carlo_methods}
\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
\begin{frame}{Monte Carlo Methods I}
\begin{frame}{Multilevel Monte Carlo Method I}
\begin{itemize}
\item Goal: Estimate $\EE[\goal]$, where $\goal$ is some functional of the random
\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
convergence rate $\alpha$, thus
\begin{equation*}
\vert \mathbb{E}[\goal_h - \goal] \vert \lesssim h^\alpha,
\quad \vert \mathbb{E}[\goal_h - \goal] \vert \lesssim N^{-\alpha},
\quad N = \dim(V_h)
\end{equation*}
where , the cost to solve one realization is
\begin{equation*}
\mathcal{C}(\goal_h) \lesssim h^{-\gamma}, \quad \mathcal{C}(\goal_h) \lesssim
N^{\gamma}
\end{equation*}
\item Assume: $u_h(\omega, x)$ 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
N = \dim(V_h)
\end{equation}
and that the cost for one sample can be bounded with $\gamma > 0$ by
\begin{equation}
\label{eq:gamma-assumption}
\cost(\goal_h(\omega_m)) \lesssim h^{-\gamma}, \quad
\cost(\goal_h) \lesssim
N^{\gamma / d}
\end{equation}
and the variance of the difference $\goal_{h_l} - \goal_{h_{l-1}}$ decays with
\begin{equation*}
\norm{\mathbb{V}[\goal_{h_l} - \goal_{h_{l-1}}]} \lesssim h^\beta,
\quad \vert \mathbb{V}[\goal_{h_l} - \goal_{h_{l-1}}] \| \lesssim N^{-\beta}.
\end{equation*}
\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}
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}{Monte Carlo Methods II}
\begin{frame}{Multilevel Monte Carlo Method I}
\begin{itemize}
\item The standard Monte Carlo estimator for the approximated functional is
\item The Monte Carlo estimator for the approximated functional is
\begin{equation*}
\widehat{Q}_{h,M}^{MC} = \frac{1}{M} \sum_{i=1}^M Q_h(\omega_i).
\widehat{\goal}_{h,M}^{MC} = \frac{1}{M} \sum_{m=1}^M \goal_h(\omega_m)
\end{equation*}
\item The RMSE is then given by
\item The root mean square error (RMSE) is then given by
\begin{equation*}
e(\widehat{Q}^{MC}_{h,M})^2 = \mathbb{E} \left[ (\widehat{Q}^{MC}_{h,M} - \mathbb{E}[Q])^2 \right] = \underbrace{M^{-1} \mathbb{V}[Q_h]}_{\text{estimator error}} + \underbrace{\left( \mathbb{E}[Q_h - Q] \right)^2}_{\text{FEM error}}.
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 cost of ($\mathcal{C}_\epsilon(\widehat{Q}_h)$ is the cost to achieve $e(\widehat{Q}_h) < \epsilon$)
\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*}
\mathcal{C}(\widehat{Q}^{MC}_{h,M}) \lesssim M \cdot N^\gamma, \quad \mathcal{C}_{\epsilon}(\widehat{Q}^{MC}_{h,M}) \lesssim \epsilon^{-2 -\frac{\gamma}{\alpha}}.
\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*}
\end{itemize}
\end{frame}
......@@ -63,20 +72,20 @@
\begin{itemize}
\item The RMSE is then given by
\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} \mathbb{V}[Y_l]}_{\text{estimator error}} + \underbrace{\left( \mathbb{E}[Q_h - Q] \right)^2}_{\text{FEM error}}.
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}}.
\end{equation*}
\item This leads leads to a better computational cost since:
\begin{itemize}
\item Assume $Q_h \rightarrow Q$, then $\mathbb{V}[\left( Q_{h_l}(\omega_i) - Q_{h_{l-1}}(\omega_i) \right)] \rightarrow 0$.
\item Assume $Q_h \rightarrow Q$, then $\VV[\left( Q_{h_l}(\omega_i) - Q_{h_{l-1}}(\omega_i) \right)] \rightarrow 0$.
\item The $Q_{h_0}(\omega_i)$ are not getting more expensive for more accuracy.
\item The optimal choice for the sequence $M_l$ is given by
\begin{equation*}
M_l = \left\lceil 2 \epsilon^{-2} \sqrt{\frac{\mathbb{V}[Y_l]}{\mathcal{C}_l}} \left( \sum_{l=0}^L \sqrt{\mathbb{V}[Y_l] \mathcal{C}_l} \right) \right\rceil.
M_l = \left\lceil 2 \epsilon^{-2} \sqrt{\frac{\VV[Y_l]}{\cost_l}} \left( \sum_{l=0}^L \sqrt{\VV[Y_l] \cost_l} \right) \right\rceil.
\end{equation*}
\end{itemize}
\item This gives an overall cost of (given $\mathcal{C}_{\epsilon}$ is best case)
\item This gives an overall cost of (given $\cost_{\epsilon}$ is best case)
\begin{equation*}
\mathcal{C}(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L}) = \sum_{l=0}^L M_l \mathcal{C}_l, \quad \mathcal{C}_{\epsilon}(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L}) \lesssim \epsilon^{-2}.
\cost(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L}) = \sum_{l=0}^L M_l \cost_l, \quad \cost_{\epsilon}(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L}) \lesssim \epsilon^{-2}.
\end{equation*}
\end{itemize}
\end{frame}
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