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\begin{frame}{Assumptions}
\begin{itemize}
\item \underline{FEM (Finite Element Method):} Let $u_l(\omega, \cdot) \in V_l$
be the FEM solution to $u(\omega, \cdot)$ on level $l$. \\
Let $\goal_l(\omega)$ be the corresponding functional
\vspace{0.2cm}
\item \underline{Assumptions:} The FEM method is convergent with convergence
rate $\alpha > 0$, i.e.
\begin{equation}
\label{eq:alpha-assumption}
\abs{\EE[\goal_l - \goal]} \lesssim h_l^\alpha, \quad
\abs{\EE[\goal_l - \goal]} \lesssim N^{-\alpha / d}, \quad
N = \dim(V_l)
\end{equation}
The cost for one sample can be bounded with $\gamma > 0$ by
\begin{equation}
\label{eq:gamma-assumption}
\cost(\goal_l(\omega_m)) \lesssim h_l^{-\gamma}, \quad
\cost(\goal_l(\omega_m)) \lesssim N^{\gamma / d}, \quad
\omega_m \in \Omega
\end{equation}
The variance of $\goal_l - \goal_{l-1}$ decays with $\beta > 0$
\begin{equation}
\label{eq:beta-assumption}
\abs{\VV[\goal_l - \goal_{l-1}]} \lesssim h_l^\beta, \quad
\abs{\VV[\goal_l - \goal_{l-1}]} \lesssim N^{-\beta / d}
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}{Example IV}
\begin{itemize}
\item \underline{Example (2D Convection-Diffusion-Reaction):}
Let $\kappa$ be a log-normal field and
$q \colon \Omega \times \overline{D} \subset \RR^{d=2} \longrightarrow
\mathbb{R}^{d=2}$ be a random Darcy Flux, computed via the elliptic model problem.
\vspace{0.2cm}
Search for $c \colon \Omega \times [0, T] \times \overline{D} \longrightarrow
\mathbb{R}$, such that
\begin{align*}
\begin{cases}
\partial_t c(\omega) +
\div \big(-\kappa(\omega) \nabla c(\omega) + c(\omega) q(\omega) \big) &=
r(c), \, \, \, \text{on} \quad (0,T) \times D, \\
c(\omega) &= c_0, \quad \, \, \text{on} \quad t=0 \times D, \\
% \rho(\omega) &= \rho_{\text{in}},
% \quad \text{on} \quad [0, T] \times \Gamma_{\text{in}}
% = \{ x \in \partial D \colon q \cdot n < 0 \}
\end{cases},
\end{align*}
with some Dirichlet-, Neumann and mixed boundary conditions.
\item Solved with implicit euler rule and a Petrov-Galerkin approach
\item Stabilize with artificial diffusion
\end{itemize}
\end{frame}
\begin{frame}{Introduction and Example I}
\begin{itemize}
\item \underline{Problem:} Let $u(\omega, \cdot) \in V$ be a random
PDE solution on $\Omega$ and $D$. \\
Let $\goal(\omega) \in V'$ be some functional. \\
Estimate $\EE[\goal(\omega)]$
\vspace{0.2cm}
\item \underline{Example (1D Elliptic Model Problem):} Let $D \subset \RR^{d=1}$. \\
Search for $u \in V$, such that
\begin{align*}
\begin{cases}
- (\kappa(\omega,x) u'(\omega,x))' &= 0, \quad \text {on} \quad D \\
u'(0) &= 1, \quad \text{on} \quad \Gamma_N \\
u(1) &= 0, \quad \text{on} \quad \Gamma_D
\end{cases}
\end{align*}
with $\kappa(\omega, x) = \log(g(\omega, x))$, where $g$ is a Gaussian field
\end{itemize}
\end{frame}
\begin{frame}{Example I - Illustration}
\begin{figure}
\label{fig:elliptic-1d-l:7-m:0-3}
\centering
\subfigure{\includegraphics[width=0.99\linewidth]{img/elliptic-1d-l:7-m:0-3.png}}
\end{figure}
\begin{itemize}
\item \underline{Idea:} Draw samples of $\Omega$, compute the FEM solution,
and average over all solutions
\item \underline{Question:} When does this approach work?
\end{itemize}
\end{frame}
\ No newline at end of file
\begin{frame}{Monte Carlo Estimator}
\begin{itemize}
\item \underline{MC Estimator:} Draw $\omega_m \in\Omega$ and compute
\begin{align*}
\widehat{\goal}_{l,M}^{MC} = \frac{1}{M} \sum_{m=1}^M \goal_l(\omega_m)
\end{align*}
\item \underline{RMSE (Root Mean Square Error):}
\begin{align*}
e(\widehat{\goal}^{MC}_{l,M})^2 =
\EE \left[ (\widehat{\goal}^{MC}_{l,M} - \EE[\goal])^2 \right] =
\underbrace{M^{-1} \VV[\goal_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}^{MC}_{l,M}) \lesssim M \cdot N^\gamma, \quad
\cost_{\epsilon}(\widehat{\goal}^{MC}_{l,M}) \lesssim
\epsilon^{-2-\frac{\gamma}{\alpha}}
\end{align*}
Here, $\cost_\epsilon(\widehat{\goal}_l)$ is the cost to achieve
$e(\widehat{\goal}_l) < \epsilon$
\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 $u(\omega, \cdot) \in V$, 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:elliptic-2d-l6-m:0-3}
\centering
\subfigure{\includegraphics[width=0.99\linewidth]{img/elliptic-2d-l6-m:0-3.png}}
{3 different input samples for $\kappa$}
\end{figure}
\end{frame}
\begin{frame}{Example II}
\begin{figure}
\label{fig:elliptic-2d-convergence}
\centering
\subfigure{\includegraphics[width=0.99\linewidth]{img/elliptic-2d-convergence.png}}
\end{figure}
\begin{itemize}
\item \underline{Observation:}
\begin{enumerate}
\item Monte Carlo approach might be too expensive for $d > 1$
(large $N$)
\item Regularity of the problem depends on covariance
function
\end{enumerate}
\end{itemize}
\end{frame}
\begin{frame}{Multilevel Monte Carlo Algorithm}
\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}
$M_l \approx 2 \varepsilon^{-2}
\sqrt{\frac{\VV[\dgoal_l]}{\mathcal{C}_l}} \left( \sum_{l=0}^L
\sqrt{\mathbb{V}[\dgoal_l] \cost_l} \right)$
% \end{equation}
\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}
%\begin{equation} \label{eq:convergence-test}
$|\EE[{\goal}_L - {\goal}_{L-1}]| \lesssim
(2^\alpha - 1) \frac{\varepsilon}{\sqrt{2}}$
%\end{equation}
\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}
\ No newline at end of file
\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}_{l,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}_{l,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}_{L,\{ M_l \}_{l=0}^L} =
\sum_{l=0}^L \widehat{\dgoal}^{MC}_{l,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}_{L,\{ 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}_{L,\{ M_l \}_{l=0}^L}) \lesssim
\sum_{l=0}^L M_l \cost_l, \quad
\cost_{\epsilon}(\widehat{\goal}^{MLMC}_{L,\{M_l \}_{l=0}^L}) \lesssim
\epsilon^{-2-(\gamma - \beta)/\alpha}
\end{align*}
Here, we assumed $\beta < \gamma$
\item Find $L$ with Richardson extrapolation and
$M_l \sim \sqrt{\frac{\VV[\dgoal_l]}{\mathcal{C}_l}}$ on the fly
\end{itemize}
\end{frame}
\begin{frame}{Example III}
\begin{itemize}
\item \underline{Example (2D Darcy Transport):}
Let $q \colon \Omega \times \overline{D} \subset \RR^{d=2} \longrightarrow
\mathbb{R}^{d=2}$ be a random Darcy Flux, computed via the elliptic model problem.
\vspace{0.2cm}
Search for $\rho \colon \Omega \times [0, T] \times \overline{D} \longrightarrow
\mathbb{R}$, such that
\begin{align*}
\begin{cases}
\partial_t \rho(\omega) + \div(\rho(\omega) q(\omega)) &= 0,
\quad \, \, \, \text{on} \quad (0,T) \times D, \\
\rho(\omega) &= \rho_0,
\quad \, \text{on} \quad t=0 \times D \\
\rho(\omega) &= \rho_{\text{in}},
\quad \text{on} \quad [0, T] \times \Gamma_{\text{in}}
% = \{ x \in \partial D \colon q \cdot n < 0 \}
\end{cases}
\end{align*}
\item Solved with implicit midpoint rule and dG-elements of second order
\end{itemize}
\end{frame}
\begin{frame}{Example III}
\begin{figure}
\label{fig:transport-l:5-m:0.png}
\centering
\subfigure{\includegraphics[width=0.99\linewidth]{img/transport-l:6-m:0.png}}
\end{figure}
\vspace{-1cm}
\begin{figure}
\label{fig:transport-lc:5-m:0.png}
\centering
\subfigure{\includegraphics[width=0.99\linewidth]{img/transport-lc:6-m:0.png}}
\end{figure}
\end{frame}
\begin{frame}{Example III}
\vspace{-0.3cm}
\begin{figure}
\label{fig:transport-convergence.png}
\centering
\subfigure{\includegraphics[width=0.95\linewidth]{img/transport-convergence.png}}
\end{figure}
\begin{itemize}
\item Values for Darcy flux: $\sigma=1.0$, $\lambda=0.1$ and $s=1.9$.
\item Goal functional: $\goal(\omega) = \int_D \rho(\omega, t = 1, x) \dx$
\item Initialization: $l_{\text{init}} = [4, 5, 6, 7]$
and $M_l^{\text{init}} = [16, 8, 4, 2]$.
\item Initial value: $\int_D \rho_0 \dx= 1$
\end{itemize}
\end{frame}
\begin{frame}{References}
\begin{enumerate}
\footnotesize{
\item Baumgarten, N. and Wieners, C. The parallel finite element system M++
with integrated multilevel preconditioning and multilevel Monte Carlo methods.
Computers \& Mathematics with Applications. 2020
\item Giles, M.B., 2015. Multilevel Monte Carlo methods.
Acta Numerica 24, 259–328.
\item Dietrich, C.R., Newsam, G.N., 1997. Fast and exact simulation of stationary
Gaussian processes through circulant embedding of the covariance matrix.
SIAM Journal on Scientific Computing 18, 1088–1107.
\item Charrier, J., Scheichl, R., Teckentrup, A.L., 2013.
Finite element error analysis
of elliptic PDEs with random coefficients and its application to multilevel
Monte Carlo methods. SIAM Journal on Numerical Analysis 51, 322–352.
\item Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L., 2011.
Multilevel Monte
Carlo methods and applications to elliptic PDEs with random coefficients.
Computing and Visualization in Science 14, 3.}
\end{enumerate}
\end{frame}
\begin{frame}{Example III - Estimated Solution}
\vspace{-1cm}
\begin{figure}
\label{fig:estimated-solution.png}
\centering
\subfigure{\includegraphics[width=0.99\linewidth]{img/transport-est-solution.png}}
\end{figure}
\end{frame}
\ No newline at end of file
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