worked on slides

tex/slides.tex

-\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

tex/src/monte_carlo_methods.tex

-\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}