Commit 79c1540b by niklas.baumgarten

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. \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) and that the cost for one sample can be bounded with $\gamma > 0$ by \label{eq:gamma-assumption} \cost(\goal_h(\omega_m)) \lesssim h^{-\gamma}, \quad \cost(\goal_h) \lesssim N^{\gamma / d} 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*} \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{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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