Commit b588c292 by niklas.baumgarten

### worked on slides

parent e73467ab
 ... ... @@ -40,9 +40,13 @@ % Statistical Operators \newcommand{\EE}{\mathbb{E}} \newcommand{\VV}{\mathbb{V}} \DeclareMathOperator{\Cov}{Cov} % Algebraic Operators \renewcommand{\dim}{\operatorname{dim}} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\spann}{span} % Integration \renewcommand{\d}{\, \mathrm{d}} ... ... @@ -62,10 +66,26 @@ % Other \newcommand{\abs}[1]{{\left|#1\right|}} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\mat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\mat}[1]{\begin{pmatrix} #1 \end{pmatrix}} \newcommand{\set}[1]{\{#1\}} \newcommand{\sprod}[1]{{\langle#1\rangle}} \newcommand{\defeq}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt \hbox{\scriptsize.}\hbox{\scriptsize.}}}=} \newcommand{\eqdef}{=\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt \hbox{\scriptsize.}\hbox{\scriptsize.}}}} \ No newline at end of file \hbox{\scriptsize.}\hbox{\scriptsize.}}}} \DeclareMathOperator{\Toe}{Toe} \DeclareMathOperator{\BToe}{BToe} \DeclareMathOperator{\ToeSym}{ToeSym} \DeclareMathOperator{\BToeSym}{BToeSym} \DeclareMathOperator{\Circ}{Circ} \DeclareMathOperator{\BCirc}{BCirc} \DeclareMathOperator{\DFT}{DFT} \DeclareMathOperator{\IDFT}{IDFT} \DeclareMathOperator{\FFT}{FFT}
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 ... ... @@ -13,38 +13,43 @@ \titlepage \end{frame} \input{src/abstract} % \input{src/abstract} \begin{frame}{Outline} \tableofcontents \end{frame} \section*{Monte Carlo methods} \input{src/mlmc} % \input{src/experimental_setup} \section{Introduction and Example I} \input{src/introduction.tex} \section*{Stochastic Linear Transport problem} \section{Assumptions} \input{src/assumptions.tex} \section*{Stochastic Convection-Diffusion-Reaction problem} \section{Monte Carlo Estimator and Example II} \input{src/mc-estimator.tex} \section*{Acoustic wave propagation} % \input{src/alternative_acoustic} \section{Multilevel Monte Carlo Estimator and Example III} \input{src/mlmc-estimator.tex} \section*{Outlook and Conclusion} % \input{src/outlook} \section{Example IV - WIP} \input{src/example-iv.tex} \begin{frame}{References} \section{Outlook and Conclusion} \input{src/outlook.tex} \section{References} \input{src/references.tex} % \begin{frame}{References} % \bibliographystyle{acm} % \tiny{\bibliography{lit}} \end{frame} % \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} \input{src/mlmc-algorithm.tex} \input{src/random-field-creation.tex} \input{src/transport-est-solution.tex} % \input{src/old/regularity} % \input{src/old/finite_element_error} \end{document} \ No newline at end of file
 \documentclass[18pt]{beamer} %---------------------------------------------------------------------------------------- %% SLIDE FORMAT %---------------------------------------------------------------------------------------- \usepackage{templates/beamerthemekit} % 4:3 % \usepackage{templates/beamerthemekitwide} % (16:9) %---------------------------------------------------------------------------------------- % PACKAGES %---------------------------------------------------------------------------------------- \usepackage{amsmath,amssymb,amsthm} \usepackage{stmaryrd} \usepackage[utf8]{inputenc} \usepackage{subfigure} \usepackage[english]{babel} \usepackage{latexsym} \usepackage{mathtools} \usepackage{grffile} \usepackage{tabto} \usepackage{algorithm} \usepackage{algorithmic} \usepackage{pythonhighlight} \usepackage{color} \usepackage{url} %---------------------------------------------------------------------------------------- % USER DEFINED SYMBOLES, ENVIROMENTS AND NUMBERING %---------------------------------------------------------------------------------------- \setbeamertemplate{itemize/enumerate body begin}{\small} \setbeamertemplate{itemize/enumerate subbody begin}{\footnotesize} \newcommand{\C}{\mathbb{C}} \newcommand{\K}{\mathbb{K}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\E}{\mathbb{E}} \newcommand{\PP}{\mathbb{P}} \newcommand{\V}{\mathbb{V}} \newcommand{\amin}{a_{\text{min}}} \newcommand{\amax}{a_{\text{max}}} \DeclareMathOperator{\essinf}{essinf} \DeclareMathOperator{\esssup}{esssup} \DeclareMathOperator{\divergence}{div} \DeclareMathOperator{\Cov}{Cov} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\spann}{span} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\Toe}{Toe} \DeclareMathOperator{\BToe}{BToe} \DeclareMathOperator{\ToeSym}{ToeSym} \DeclareMathOperator{\BToeSym}{BToeSym} \DeclareMathOperator{\Circ}{Circ} \DeclareMathOperator{\BCirc}{BCirc} \DeclareMathOperator{\MCE}{MCE} \DeclareMathOperator{\MOCE}{MOCE} \DeclareMathOperator{\DFT}{DFT} \DeclareMathOperator{\IDFT}{IDFT} \DeclareMathOperator{\FFT}{FFT} %---------------------------------------------------------------------------------------- % START PRESENTATION %---------------------------------------------------------------------------------------- \title{Multilevel Monte Carlo Applications} \subtitle{Review and motivation} \author{Niklas Baumgarten, Christian Wieners} \institute{Institute for Applied and Numerical Mathematics} \begin{document} %title page \begin{frame} \titlepage \end{frame} %table of contents \begin{frame}{Outline} \tableofcontents \end{frame} %----------------------------------------% \section{Elliptic model problem} \input{model_problem} \input{example_fields2} %----------------------------------------% %----------------------------------------% \section{Monte Carlo methods} \input{monte_carlo_methods} \input{experimental_setup} \input{numerical_results} %----------------------------------------% %----------------------------------------% \section{Acoustic wave propagation} \input{alternative_acoustic} \section{Outlook and Conclusion} \input{outlook} \begin{frame}{References} \bibliographystyle{acm} \tiny{\bibliography{lit}} \end{frame} \section*{Backup} \input{the_random_field_model} \input{circulant_embedding} \input{weak_formulation} \input{existence} \input{regularity} \input{finite_element_error} \input{mlmc_algorithm} \end{document}
 \begin{frame}{Circulant embedding} \begin{itemize} \item The circulant embedding approach is a tool to create efficiently and independently identical distributed samples. \item The covariance matrix corresponding to (\ref{covariance_function}) has block symmetric Toeplitz structure, possibly with symmetric blocks (BST(S)TB). \begin{figure} \centering \subfigure{\includegraphics[width=0.35\linewidth]{./pictures/Matrices/Covariance matrix.png}} \hspace{1cm} \subfigure{\includegraphics[width=0.35\linewidth]{./pictures/Matrices/Embedded matrix.png}} \end{figure} \item This BST(S)TB matrix can be described with its first block rows and its first block columns. \end{itemize} \end{frame} \begin{frame}{Circulant embedding} \begin{itemize} \item Goal: Try to decompose the covariance matrix in an efficient way for \begin{equation*} X = RZ, \quad \text{where } A = RR^T. \end{equation*} \item This is done by embedding the BST(S)TB matrix in an block circulant matrix with circulant blocks.% \item For circulant matrices it holds \begin{equation*} B = W_{N,N} \Lambda W^*_{N,N}, \end{equation*} where $W_{N,N}$ is the Fourier matrix for 2 dimensions and $\Lambda = \diag(\lambda)$ with $\lambda$ being the vector of eigenvalues. The eigenvalues can be computed with $\lambda = N W_{N,N} b^r$, where here $b^r$ is a $N \times N$ matrix containing the first block rows. \end{itemize} \end{frame} \begin{frame}{Circulant embedding - Algorithm} \begin{algorithm}[H] \caption{Circulant embedding.} \label{Circulant embedding algorithm} \begin{algorithmic}[1] {\footnotesize \STATE Compute first rows and first columns of covariance matrix: $a^r$ and $a^c$ \hfill $\mathcal{O}(N^2)$ \STATE Embed $a^r$ and $a^c$ to create a circulant matrix: $b^r \leftarrow \texttt{MOCE}(a^r, a^c)$ \STATE Compute eigenvalues: $\lambda \leftarrow \texttt{FFT}(b^r)$ \hfill $\mathcal{O}(\tilde{N}^2 \log \tilde{N})$ \IF{all $\lambda \geq 0$} \STATE Compute and save square root: $\sqrt{\Lambda}$ \WHILE{sample needed} \STATE Generate complex random matrix: $\underline{Z} \leftarrow \texttt{Rnd}(\text{seed})$ \STATE Compute complex field: $\underline{X} \leftarrow \texttt{FFT}(\sqrt{\Lambda} \odot \underline{Z})$ \hfill $\mathcal{O}(\tilde{N}^2 \log \tilde{N})$ \STATE Yield the two independent samples: $X_1 = \text{Re}\{\underline{X}[0:N, 0:N]\}$ and \\ \hspace{1cm} $X_2 = \text{Im}\{\underline{X}[0:N, 0:N]\}$ \STATE Add $c(x)$ and compute log-normal field if desired \ENDWHILE \ELSE \STATE Compute padding: $\overline{a}^r, \overline{a}^c \leftarrow \texttt{Pad}(a^r, a^c)$ \STATE Set $a^r, a^c = \overline{a}^r, \overline{a}^c$ and go back to line 2. \ENDIF } \end{algorithmic} \end{algorithm} \end{frame} \ No newline at end of file
 \begin{frame}{Existence} \begin{lemma}[Ellipticity] {\footnotesize For almost all $\omega \in \Omega$, the bilinear form $b_\omega(u,v)$ is bounded and coercive in $H_0^1(D)$ with respect to the norm $\vert \cdot \vert_{H^1(D)}$ with the constants $\amax(\omega)$ and $\amin(\omega)$, respectively. Moreover, there exists a unique solution $u(\omega, \cdot) \in H_0^1(D)$ to the variational problem (\ref{weak_formulation}) and \begin{equation*} \vert u(\omega, \cdot) \vert_{H^1(D)} \lesssim \frac{\Vert f \Vert_{H^{t-1}(D)}}{\amin(\omega)}. \end{equation*}} \end{lemma} \begin{itemize} \item Proof idea: Straight forward calculations with assumptions and Lax-Milgram. \end{itemize} \begin{theorem}[Existence] {\footnotesize The weak solution $u$ of (\ref{eq:model_problem}-\ref{boundary_conditions2}) is unique and belongs to $L^p(\Omega, H_0^1(D))$ for all $p < p_*$.} \end{theorem} \begin{itemize} \item Proof idea: Extend the Lemma above on Bochner spaces with Hölder's inequality and the assumptions. \end{itemize} \end{frame} \ No newline at end of file
 \begin{frame}{Introduction and Example I} \begin{itemize} \item \underline{Problem:} Let $u(\omega, x) \in V$ be a random PDE solution on $\Omega$ and $D$. Let $\goal(\omega)$ be some functional of $u(\omega, x)$. Estimate $\EE[\goal(\omega)]$ \item \underline{Example (1D Elliptic Model Problem):} Let $D \subset \RR^{d=1}$. Search for $u \in V$, such that \begin{align*} - (\kappa(\omega,x) u'(\omega,x))' = 0, \quad u'(0) = 1, \quad u(1) = 0 \end{align*} with $\kappa(\omega, x) = \log(g(\omega, x))$, where $g$ is a Gaussian field \end{itemize} \end{frame} \begin{frame}{Assumptions} \begin{itemize} \item \underline{FEM (Finite Element Method:)} Let $u_h(\omega, x) \in V_h$ be the corresponding FEM solution to $u(\omega, x)$ and $\goal_h(\omega)$ be the functional \item \underline{Assumptions:} The FEM method is convergent with 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 / d}, \quad N = \dim(V_h) 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(\omega_m)) \lesssim N^{\gamma / d}, \quad \omega_m \in \Omega The variance of $\goal_l - \goal_{l-1}$ decays with $\beta > 0$ \label{eq:beta-assumption} \abs{\VV[\goal_l - \goal_{l-1}]} \lesssim h^\beta, \quad \abs{\VV[\goal_l - \goal_{l-1}]} \lesssim N^{-\beta / d} \end{itemize} \end{frame} \begin{frame}{Monte Carlo Estimator} \begin{itemize} \item \underline{MC Estimator:} Draw $\omega_m \in\Omega$ and compute \begin{align*} \widehat{\goal}_{h,M}^{MC} = \frac{1}{M} \sum_{m=1}^M \goal_h(\omega_m) \end{align*} \item \underline{RMSE (Root Mean Square Error):} \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[\goal_h - \goal] \right)^2}_{\text{FEM error}} \end{align*} \item \underline{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}{Example II} \begin{itemize} \item \underline{Example (2D Elliptic Model Problem):} Let $D \subset \RR^{d=2}$. Search for $u \in V$, such that \begin{align*} - \div(\kappa(\omega,x) \nabla u(\omega,x)) \overset{\text{on }D}{=} 0, \quad \nabla u(x) \cdot n \overset{\text{on }\Gamma_N}{=} -1, \quad u(x) \overset{\text{on }\Gamma_D}{=} 0 \end{align*} with $\kappa(\omega, x) = \log(g(\omega, x))$, where $g$ is a Gaussian field with the covariance function $C(x, y) = \sigma^2 \exp(- \norm{x- y}_2^s / \lambda^s)$ \end{itemize} \begin{figure} \label{fig:2d-elliptic-model-problem} \centering \subfigure{\includegraphics[width=0.99\linewidth]{img/perm_potential.png}} \end{figure} \end{frame} \begin{frame}{Multilevel Monte Carlo Estimator I} \begin{itemize} \item \underline{Main Idea:} Draw samples from several levels $l \in \{ 0, \dots L \}$ and balance $\cost_l$ with $M_l$ \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 \underline{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 Estimator II} \begin{itemize} \item \underline{RMSE (Root Mean Square Error):} \begin{align*} 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{align*} \item \underline{Total Cost:} \begin{align*} \cost(\widehat{\goal}^{MLMC}_{h,\{ M_l \}_{l=0}^L}) \lesssim \sum_{l=0}^L M_l \cost_l, \quad \cost_{\epsilon}(\widehat{\goal}^{MLMC}_{h,\{M_l \}_{l=0}^L}) \lesssim \epsilon^{-2-(\gamma - \beta)/\alpha} \end{align*} Here, we assumed $\beta < \gamma$ \end{itemize} \end{frame} \begin{frame}{Multilevel Monte Carlo Estimator III} \begin{algorithm}[H] \caption{Multilevel Monte Carlo Estimator} \begin{algorithmic}[1] \label{alg:mlmc} {\footnotesize \STATE Choose $h_{l=0}^\text{init}, h_{l=1}^\text{init}, \dots, h_{l=L}^\text{init}$ and $M_{l=0}^\text{init}, M_{l=1}^\text{init}, \dots, M_{l=L}^\text{init}$ \STATE Set $\{{\vartriangle} M_l = M_l^\text{init}\}_{l = 0}^{L}$ and $\{M_l = 0\}_{l = 0}^{L}$ \WHILE {${\vartriangle} M_l > 0$ on any level} \FOR {levels with ${\vartriangle} M_l > 0$} \STATE $\dgoal_l, \, \mathcal{C}_l \leftarrow \texttt{MonteCarlo} ({\vartriangle} M_l, l)$ \STATE Update $\mathcal{C}_l$, $|\mathbb{E}[\dgoal_l]|$ and $\mathbb{V}[\dgoal_l]$ \STATE Set $M_l \leftarrow M_l + {\vartriangle} M_l$, ${\vartriangle} M_l = 0$ \ENDFOR \STATE Estimate $\alpha$, $\beta$, $\gamma$ with~\eqref{eq:alpha-assumption}, ~\eqref{eq:gamma-assumption} and~\eqref{eq:beta-assumption} \STATE Estimate $\{M_l^\text{opt}\}_{l = 0}^{L} with$ %~\eqref{eq:optimal-Ml} %\label{equation} $M_l \approx 2 \varepsilon^{-2} \sqrt{\frac{\VV[\dgoal_l]}{\mathcal{C}_l}} \left( \sum_{l=0}^L \sqrt{\mathbb{V}[\dgoal_l] \cost_l} \right)$ % \STATE Update $\{{\vartriangle} M_l\}_{l = 0}^{L} = \{ M_l^\text{opt} - M_l\}_{l = 0}^{L}$ \STATE Test for weak convergence %~\eqref{eq:convergence-test} % \label{eq:convergence-test} $|\EE[{\goal}_L - {\goal}_{L-1}]| \lesssim (2^\alpha - 1) \frac{\varepsilon}{\sqrt{2}}$ % \IF {not converged} \STATE Set $L \leftarrow L + 1$ and update $\{{\vartriangle} M_l\}_{l = 0}^{L}$ \ENDIF \ENDWHILE} \end{algorithmic} \end{algorithm} \end{frame} \begin{frame}{Example III} \begin{itemize} \item \underline{Example (2D Darcy Transport):} (...) \end{itemize} \end{frame}
 \begin{frame}{Multilevel Monte Carlo method} \begin{itemize} \item Main idea MLMC: Sample from several approximation $Q_h$ on different fine triangulations. \item With the linearity of the expectation operator it holds \begin{equation*} \E[Q_h] = \E[Q_{h_0}] + \sum_{l=1}^L \E[Q_{h_l} - Q_{h_{l-1}}] = \sum_{l=1}^L \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*} \end{itemize} \end{frame} \begin{frame}{Multilevel Monte Carlo method} \begin{itemize} \item The RMSE is then given by \begin{equation*} e(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L})^2 = \E\left[( \widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L} - \E[Q])^2 \right] = \underbrace{\sum_{l=0}^L \frac{1}{M_l} \V[Y_l]}_{\text{estimator error}} + \underbrace{\left( \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 $\V[\left( Q_{h_l}(\omega_i) - Q_{h_{l-1}}(\omega_i) \right)] \rightarrow 0$. \item Even with increasing required accuracy samples of $Q_{h_0}$ are not getting more expensive. \item The optimal choice for the sequence $M_l$ is given by \begin{equation*} M_l = \left\lceil 2 \epsilon^{-2} \sqrt{\frac{\V[Y_l]}{\mathcal{C}_l}} \left( \sum_{l=0}^L \sqrt{\V[Y_l] \mathcal{C}_l} \right) \right\rceil. \end{equation*} \end{itemize} \item This gives an overall cost of \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}(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L}) \lesssim \sum_{l=0}^L \sqrt{\V[Y_l] \mathcal{C}_l}. \end{equation*} \end{itemize} \end{frame} \begin{frame}{Multilevel Monte Carlo method} \begin{theorem}[Multilevel Monte Carlo method] \label{MLMC_theorem} {\footnotesize Suppose that there are positive constants $\alpha, \beta, \gamma, c_1, c_2, c_3 > 0$ such that $\alpha \geq \frac{1}{2} \min(\beta, \gamma)$ and \begin{enumerate} {\footnotesize \item $\left| \E[Q_{h_l} - Q] \right| \leq c_1 h_l^\alpha$ \item $\V[Q_{h_l} - Q_{h_{l-1}}] \leq c_2 h_l^\beta$ \item $\mathcal{C}_l \leq c_3 h_l^{- \gamma}$.} \end{enumerate} Then, for any $0 < \epsilon < \frac{1}{e}$, there exists an $L$ and a sequence $\{ M_l \}_{l=0}^L$, such that \begin{equation*} e(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L})^2 = \E\left[( \widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L} - \E[Q])^2 \right] < \epsilon^2 \end{equation*} and \begin{equation*} \mathcal{C}_\epsilon(\widehat{Q}^{MLMC}_{h,\{ M_l \}_{l=0}^L}) \lesssim \begin{cases} \epsilon^{-2}, &\text{if } \beta > \gamma \\ \epsilon^{-2}\log(\epsilon)^2, &\text{if } \beta = \gamma \\ \epsilon^{-2-(\gamma - \beta)/\alpha}, &\text{if } \beta < \gamma \end{cases}, \end{equation*} where the hidden constant depends on $c_1, c_2, c_3$.} \end{theorem} \end{frame} \begin{frame}{Multilevel Monte Carlo method - Algorithm} \begin{itemize} \item Challenge in using the MLMC method is showing that the assumptions hold \item The MLMC approach can be parallelized in an easy manner \end{itemize} \vspace{-0.4cm} \begin{algorithm}[H] \caption{Multilevel Monte Carlo method} \begin{algorithmic}[1] \label{MLMC algorithm} {\footnotesize