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-
- \author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich)}
- \institute{UZH}
- \title[Specific \pdf~generation]{Specific \pdf~generation}
- \date{\fixme}
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- \author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich)}
- \institute{UZH}
- \title[Applications of MC methods]{Applications of MC methods}
- \date{\fixme}
-
-
- \begin{document}
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- {
- \setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
- \begin{frame}[c]%{\phantom{title page}}
- \begin{center}
- \begin{center}
- \begin{columns}
- \begin{column}{0.9\textwidth}
- \flushright\fontspec{Trebuchet MS}\bfseries \Huge {Applications of MC methods}
- \end{column}
- \begin{column}{0.2\textwidth}
- %\includegraphics[width=\textwidth]{SHiP-2}
- \end{column}
- \end{columns}
- \end{center}
- \quad
- \vspace{3em}
- \begin{columns}
- \begin{column}{0.44\textwidth}
- \flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}}
-
- \end{column}
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- \vspace{1em}
- % \footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
- \vspace{0.5em}
- \textcolor{normal text.fg!50!Comment}{Monte Carlo methods, \\ 26 May, 2016}
- \end{center}
- \end{frame}
- }
-
-
- \begin{frame}\frametitle{Optimization}
- \begin{minipage}{\textwidth}
- \begin{exampleblock}{Optimization problem:}
- We have a set $X \subset \mathbb{R}^m$ and a function $F : X \to \mathbb{R}$.\\
- Task:\\
- Find the optimum point:
- \begin{align*}
- x_{opt} \in X : \forall_{x \in X}~F(x) \geq F(x_{opt})
- \end{align*}
- \end{exampleblock}
- \ARROW This is completely different then normal function minimalization as we choose $x_{opt}$ from a set X. This makes a big differences for numerical computations.\\
- \ARROWR The MC algorithms for solving this problem:
- \begin{itemize}
- \item Hit and miss method - the simplest and the slowest.
- \item Sequence methods - MC interpretation of method of further approximations.
- \item Genetic methods, stat. optimization.
- \end{itemize}
-
-
- \end{minipage}
-
- \end{frame}
-
-
-
-
- \begin{frame}\frametitle{Optimization hit and miss}
- \begin{minipage}{\textwidth}
- \begin{footnotesize}
-
- \ARROWR The algorithm acts as follows:
- \begin{itemize}
- \item We generate $N$ points $x_1,...,x_N \in X$ from a constant \pdf~on $X$.
- \item We calculate the $F$ function value in the points $x_1,...,x_N$:
- \begin{align*}
- F_1=F(x_1),~~~F_2=F(x_2),~~~...,~~~F_N
- \end{align*}
- \item We calculate $F^{\ast}=\min \lbrace F_1,F_2,...,F_N \rbrace$.
- \item The solution is $x_j: F(x_j)=F^{\ast}$
- \end{itemize}
- \ARROW Precision:
- \begin{exampleblock}{}
- If $F^{\ast}=\min_{1\leq j \leq N} \lbrace F(x_j) \rbrace$ where $x_j=1,2,...N$ are random points from uniform \pdf~on $X$. Then with the probability $1-(1-\gamma)^N$ the volume of points $x$ for which $F(x)<F^{\ast}$ is smaller then $\epsilon$.
- \end{exampleblock}
- \ARROW We can say that with probability $1-(1-\gamma)^N$ the points $x_{opt}$ was localized with the probability $\gamma$. \\
- \ARROW the smaller the volume the better the accuracy.
-
-
- \end{footnotesize}
- \end{minipage}
-
- \end{frame}
-
-
- \begin{frame}\frametitle{Optimization hit and miss}
- \begin{minipage}{\textwidth}
- \begin{footnotesize}
-
- \ARROWR The algorithm acts as follows:
- \begin{itemize}
- \item We generate $N$ points $x_1,...,x_N \in X$ from a constant \pdf~on $X$.
- \item We calculate the $F$ function value in the points $x_1,...,x_N$:
- \begin{align*}
- F_1=F(x_1),~~~F_2=F(x_2),~~~...,~~~F_N
- \end{align*}
- \item We calculate $F^{\ast}=\min \lbrace F_1,F_2,...,F_N \rbrace$.
- \item The solution is $x_j: F(x_j)=F^{\ast}$
- \end{itemize}
- \ARROW Precision:
- \begin{exampleblock}{}
- If $F^{\ast}=\min_{1\leq j \leq N} \lbrace F(x_j) \rbrace$ where $x_j=1,2,...N$ are random points from uniform \pdf~on $X$. Then with the probability $1-(1-\gamma)^N$ the volume of points $x$ for which $F(x)<F^{\ast}$ is smaller then $\epsilon$.
- \end{exampleblock}
- \ARROW We can say that with probability $1-(1-\gamma)^N$ the points $x_{opt}$ was localized with the probability $\gamma$. \\
- \ARROW the smaller the volume the better the accuracy.
-
-
- \end{footnotesize}
- \end{minipage}
-
- \end{frame}
-
-
-
- \begin{frame}\frametitle{Optimization hit and miss}
- \begin{minipage}{\textwidth}
- \begin{footnotesize}
- \begin{center}
- The smallest $N$ that obeys: $1-(1-\gamma)^N \geq 1 -\epsilon$:
- \includegraphics[width=0.7\textwidth]{images/table.png}
- \end{center}
- \ARROW Example: $\left[0,1\right]^m$ and $F:X \to \mathbb{R}$.\\
- How many points we need to generate to ahve $0.9$ probability to be have half of the range of each direction in each precision:
- \begin{itemize}
- \item For $m=1$: $\gamma=1/2~\rightarrowtail~N=4$.
- \item For $m=2$: $\gamma=1/4\rightarrowtail~N=9$.
- \item For $m=14$: $\gamma=2^{-14}\rightarrowtail~N>23~000$.
-
- \end{itemize}
- \ARROW Inefficient for multi-dimensions.
- \end{footnotesize}
- \end{minipage}
-
- \end{frame}
-
-
-
-
-
- \begin{frame}\frametitle{Optimization sequence}
- \begin{minipage}{\textwidth}
- \begin{footnotesize}
-
- \ARROW The algorithm:
- \begin{itemize}
- \item We choose the starting point $x_1 \in X$ from some \pdf~on $X$ set.
- \item After generating $x_1,x_2,...,x_n$ check if some conditions are meet.
- \begin{itemize}
- \item If YES then we stop and we put $x_n$ as solution.
- \item If NO then we generate $x_{n+1}$ form a \pdf~ that depends on already generated points.
- \end{itemize}
- \end{itemize}
-
- \ARROW The basic sequence algorithm:
- \begin{itemize}
- \item Choose $x_1$.
- \item After we have $x_1,x_2,...,x_n$ then we generate a temporary point $\xi_n$:
- \begin{align*}
- x_{n+1}=\begin{cases}
- x_n,~~~~~~~~~~{\rm if}~F(x_n+\xi_n) \geq F(x_n)-\epsilon\\
- x_n+\xi_n,~~{\rm if}~F(x_n+\xi_n) < F(x_n)-\epsilon \end{cases}
- \end{align*}
- \end{itemize}
-
-
-
- \end{footnotesize}
- \end{minipage}
-
- \end{frame}
-
-
-
-
- \begin{frame}\frametitle{Optimization sequence}
- \begin{minipage}{\textwidth}
- \begin{footnotesize}
-
- \ARROW From the above algorithm we will get a sequence:
- \begin{align*}
- F(x_1) \geq F(x_2) \geq F(x_3) \geq ...\geq F(x_n)\geq F(x_{n+1})...
- \end{align*}
- \ARROW If the function is bounded from the bottom the the above sequence is converging.\\
- \ARROWR How can we be sure it will converge to $x_{opt}$?
- \begin{center}
- \includegraphics[width=0.6\textwidth]{images/fig2.png}
- \end{center}
- \ARROW If we choose the correct the $P_n$ every sequence starting from $x^{\prime}$ will converge to $\overline{x}$, where $F$ has a local minimum.
-
-
-
-
- \end{footnotesize}
- \end{minipage}
-
- \end{frame}
-
-
- \begin{frame}\frametitle{Optimization sequence}
- \begin{minipage}{\textwidth}
- \begin{footnotesize}
- \ARROW There are two types of algorithms:
- \begin{itemize}
- \item If the algorithm can find the global minimum then we call it: global algorithm.
- \item If the algorithm can find only local minimum the we call it: local algorithm.
- \end{itemize}
- \ARROWR If in the sequence $\lbrace x_n \rbrace$ we find a point $x^{\prime}$ such that:
- \begin{align*}
- F(x_{opt}) < F(x^{\prime}) < F(x_{opt})+\epsilon
- \end{align*}
- then the above algorithm will converge only to $x^{\prime}$. \\
- \ARROW Of course we can change the $\epsilon$ such that we escape the $x^{\prime}$.
- \end{footnotesize}
- \end{minipage}
-
- \end{frame}
-
-
-
- \begin{frame}
- \begin{minipage}{\textwidth}
-
- \begin{center}
- \begin{Large}
- Q \& A
- \end{Large}
- \end{center}
-
- \end{minipage}
-
- \end{frame}
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