Commit 70177437 authored by niklas.baumgarten's avatar niklas.baumgarten
Browse files

worked on slides

parent cd7ac3d6
\begin{frame}{Introduction and Example I}
\begin{itemize}
\item \underline{Problem:} Let $u(\omega)$ be a random
PDE solution on $\Omega$ and let $\goal(\omega)$ be some functional of $u(\omega)$.
\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*}
......@@ -12,8 +12,6 @@
u(1) = 0
\end{align*}
with $\kappa(\omega, x) = \log(g(\omega, x))$, where $g$ is a Gaussian field
\item
\end{itemize}
\end{frame}
......@@ -56,10 +54,10 @@
\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{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:}
\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
......@@ -75,18 +73,26 @@
\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)) = 0, \quad
\nabla u(x) \cdot n = -1, \quad
u(x) = 0
- \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}
\centering
\subfigure{\includegraphics[width=0.99\linewidth]{img/perm_potential.png}}
\end{figure}
\end{frame}
\begin{frame}{Multilevel Monte Carlo Methods I}
\begin{frame}{Multilevel Monte Carlo Estimator I}
\begin{itemize}
\item \underline{Main idea:} Draw samples from several approximation levels
and balance cost per level $\cost_l$ with total sample amount per level $M_l$
\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}] =
......@@ -100,7 +106,7 @@
\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:}
\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} =
......@@ -109,83 +115,84 @@
\end{itemize}
\end{frame}
\begin{frame}{Multilevel Monte Carlo Methods II}
\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{\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:}
\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 $\beta < \gamma$.
\item Arguments (...)
Here, we assumed $\beta < \gamma$
\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{frame}{Multilevel Monte Carlo Estimator III}
\begin{algorithm}[H]
\caption{Multilevel Monte Carlo method}
\caption{Multilevel Monte Carlo Estimator}
\begin{algorithmic}[1]
\label{MLMC algorithm}
\label{alg:mlmc}
{\footnotesize
\STATE Set $l_0 = 3$, $L_0 = 5$ and the initial number of samples $M_0 = \{ 200, 100, 50 \}$
\STATE Set range of levels $\{l_0, \dots, L_0 \}$ and the number of needed samples $\{ \Delta M_l = M_0 \}_{l = 0}^{L}$
\WHILE {$ \Delta M_l > 0$ on any level}
\FOR {levels with needed samples}
\STATE Retrieve functionals and cost: $Y_l, \, \mathcal{C}_l \leftarrow \texttt{SubroutineEstimator}(\Delta M_l, l)$
\STATE Update statistics: $\mathcal{C}_l$, $|\E[Y_l]|$, $\V[Y_l]$ and set: $M_l = \Delta M_l$, $\Delta M_l = 0$
\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 exponents $\alpha$, $\beta$, $\gamma$ with the assumptions of the previous Theorem
\STATE Estimate optimal $M_l$, $l = 0, \dots, L$ with $M_l = \left\lceil 2 \epsilon^{-2} \sqrt{\frac{\V[Y_l]}{\mathcal{C}_l}} \left( \sum_{l=0}^L \sqrt{\V[Y_l] \mathcal{C}_l} \right) \right\rceil$
\STATE Test for weak convergence with $|\E[Q_{h_L} - Q_{h_{L-1}}]| < (2^\alpha - 1) \frac{\epsilon}{\sqrt{2}}$
\STATE If not converged, increase range of levels by one level and initialize new $M_L$
\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}
\begin{frame}{Example III}
\begin{itemize}
\item \underline{Example (2D Darcy Transport):} (...)
\end{itemize}
\end{frame}
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