Commit bfe7f43d by niklas.baumgarten

worked on slides

parent 4a071a53
Pipeline #106734 passed with stages
in 3 minutes and 3 seconds
No preview for this file type
 ... ... @@ -16,10 +16,6 @@ \tableofcontents \end{frame} \section*{Elliptic model problem} \input{src/elliptic_model_problem.tex} \input{src/example_fields} \section*{Monte Carlo methods} \input{src/mlmc} \input{src/experimental_setup} ... ...
 \begin{frame}{Multilevel Monte Carlo Method I} \begin{itemize} \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. \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) 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 \label{eq:gamma-assumption} \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 $\beta > 0$ \label{eq:beta-assumption} \abs{\VV[\goal_l - \goal_{l-1}]} \lesssim h^\beta, \quad ... ... @@ -27,20 +28,35 @@ \end{itemize} \end{frame} \begin{frame}{Examples I} \begin{itemize} \item \underline{Elliptic Model Problem:} Let $D \subset \RR^d$. Search for $u \in V$, such that \label{eq:model_problem} - \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 \end{itemize} \end{frame} \begin{frame}{Multilevel Monte Carlo Method II} \begin{itemize} \item Monte Carlo (MC) estimator: \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 Root mean square error (RMSE): \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 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 ... ... @@ -51,9 +67,15 @@ \end{itemize} \end{frame} \begin{frame}{Examples II} \begin{itemize} \item 2D Problem etwas irregulär \end{itemize} \end{frame} \begin{frame}{Multilevel Monte Carlo Methods III} \begin{itemize} \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 $\goal_0 \defeq \dgoal_0$: \begin{align*} ... ... @@ -79,7 +101,7 @@ \begin{frame}{Multilevel Monte Carlo Methods II} \begin{itemize} \item Root mean square error (RMSE): \item \underline{RMSE (Root mean square error):} \begin{equation*} 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}} + ... ...
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!