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e|Kr_9?*HdXQWKX>1QZ_&I59Of3MC~)PeuxfQ&LC( diff --git a/semester3/numcs/parts/02_quadrature/05_monte-carlo.tex b/semester3/numcs/parts/02_quadrature/05_monte-carlo.tex index 0c7d373..47055b8 100644 --- a/semester3/numcs/parts/02_quadrature/05_monte-carlo.tex +++ b/semester3/numcs/parts/02_quadrature/05_monte-carlo.tex @@ -1,2 +1,18 @@ \newsection +\newcommand{\tsigma}{\tilde{\sigma}} \subsection{Monte-Carlo Quadratur} +Bei der Monte-Carlo Quadratur wird, wie bei anderen Monte-Carlo-Algorithmen der Zufall genutzt + +\inlineremark Die Konvergenz ist sehr langsam ($\sqrt{N}$), aber nicht abhängig von der Dimension oder Glattheit. +Zudem kann das Ergebnis falsch sein, da es probabilistisch ist. + +Jede Monte-Carlo-Methode benötigt folgendes mit $X = [I_N - \tsigma_N, I_N + \tsigma_N]$: +\rmvspace +\begin{multicols}{2} + \begin{itemize}[noitemsep] + \item ein Gebiet für das ``Experiment'', hier $[0, 1]^d$ + \item gute Zufallszahlen + \item gute deterministische Berechnungen, hier $\tsigma_N$ und $I_N$ + \item Darstellung des Ergebnis, hier $\Pr[I \in X] = 0.683$ + \end{itemize} +\end{multicols}