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\section*{Elliptic model problem}
\section*{Monte Carlo methods}
\begin{frame}{Multilevel Monte Carlo Method I}
\item Goal: Estimate the expectation $\EE[\goal(\omega)]$, where $\goal$ is some
functional of the random
solution $u(\omega, x)$.
\item Assume: $u_h(\omeg, x)$ is the corresponding FEM solution with the
convergence rate $\alpha > 0$, i.e.
\item \underline{Problem:} Let $u(\omega)$ be a random
PDE solution on $\Omega$ and let $\goal(\omega)$ be some functional of $u(\omega)$.
Estimate $\EE[\goal(\omega)]$.
\item \underline{Assumptions:} Let $u_h(\omega, x) \in V_h$ be the
corresponding FEM solution with convergence rate $\alpha > 0$, i.e.
\abs{\EE[\goal_h - \goal]} \lesssim h^\alpha, \quad
\abs{\EE[\goal_h - \goal]} \lesssim N^{-\alpha / d}, \quad
N = \dim(V_h)
N = \dim(V_h),
and that the cost for one sample can be bounded with $\gamma > 0$ by
the cost for one sample can be bounded with $\gamma > 0$ by
\cost(\goal_h(\omega_m)) \lesssim h^{-\gamma}, \quad
\cost(\goal_h) \lesssim
N^{\gamma / d}
\cost(\goal_h(\omega_m)) \lesssim N^{\gamma / d}, \quad
\omega_m \in \Omega
and the variance of the difference $\goal_l - \goal_{l-1}$ decays with
and the variance of $\goal_l - \goal_{l-1}$ decays with $\beta > 0$
\abs{\VV[\goal_l - \goal_{l-1}]} \lesssim h^\beta, \quad
......@@ -27,20 +28,35 @@
\begin{frame}{Examples I}
\item \underline{Elliptic Model Problem:} Let $D \subset \RR^d$. Search for $u
\in V$, such that
- \div(\kappa(\omega,x) \nabla u(\omega,x)) = f(\omega,x)
with Neumann and Dirichlet boundary conditions.
\item Simple 1D Problem already with results?
\item Vielleicht gemittelte Lösung mit unterschiedlichen epsilon
\begin{frame}{Multilevel Monte Carlo Method II}
\item Monte Carlo (MC) estimator:
\item \underline{MC Estimator:} Draw $\omega_m \in \Omega$ and compute
\widehat{\goal}_{h,M}^{MC} = \frac{1}{M} \sum_{m=1}^M \goal_h(\omega_m)
\item Root mean square error (RMSE):
\item \underline{RMSE (Root mean square 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[\goal_h - \goal] \right)^2}_{\text{FEM error}}
\item Total cost:
\item \underline{Total cost:}
\cost(\widehat{\goal}^{MC}_{h,M}) \lesssim M \cdot N^\gamma, \quad
\cost_{\epsilon}(\widehat{\goal}^{MC}_{h,M}) \lesssim
......@@ -51,9 +67,15 @@
\begin{frame}{Examples II}
\item 2D Problem etwas irregulär
\begin{frame}{Multilevel Monte Carlo Methods III}
\item Main idea: Sample from several approximation levels
\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 Set $\goal_l - \goal_{l-1} \defeq \dgoal_l$ and $\goal_0 \defeq \dgoal_0$:
......@@ -79,7 +101,7 @@
\begin{frame}{Multilevel Monte Carlo Methods II}
\item Root mean square error (RMSE):
\item \underline{RMSE (Root mean square error):}
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}} +
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