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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} (CERN)}
\institute{UZH}
\title[Application of MC methods]{Application of MC methods}
\date{\fixme}
\begin{document}
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\begin{frame}[c]%{\phantom{title page}}
\begin{center}
\begin{center}
\begin{columns}
\begin{column}{0.9\textwidth}
\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Application of MC methods}
\end{column}
\begin{column}{0.2\textwidth}
%\includegraphics[width=\textwidth]{SHiP-2}
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\quad
\vspace{3em}
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\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}}
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\hspace{0.3\textwidth}\includegraphics[height=1.3cm]{cern}
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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}{Experimental Methods in Particle Physics, \\ 12 October, 2017}
\end{center}
\end{frame}
}
\begin{frame}\frametitle{Classical methods of variance reduction}
\begin{footnotesize}
\ARROW In Monte Carlo methods the statistical uncertainty is defined as:
\begin{align*}
\sigma = \dfrac{1}{\sqrt{N}}\sqrt{V(f)}
\end{align*}
\ARROW Obvious conclusion:
\begin{itemize}
\item To reduce the uncertainty one needs to increase $N$.\\
$\rightrightarrows$ Slow convergence. In order to reduce the error by factor of 10 one needs to simulate factor of 100 more points!
\end{itemize}
\ARROW How ever the other handle ($V(f)$) can be changed! $\longrightarrow$ Lot's of theoretical effort goes\
into reducing this factor.\\
\ARROW We will discuss {\color{Mahogany}{four}} classical methods of variance reduction:
\begin{enumerate}
\item Stratified sampling.
\item Importance sampling.
\item Control variates.
\item Antithetic variates.
\end{enumerate}
\end{footnotesize}
\end{frame}
\begin{frame}\frametitle{Disadvantages of classical variance reduction methods}
\begin{footnotesize}
\ARROW All aforementioned methods(beside the Stratified sampling) require knowledge of the integration function!\\
\ARROW If you use the method in the incorrect way, you can easily get the opposite effect than intendant. \\
\ARROW Successful application of then require non negligible effort before running the program.\\
\ARROW A natural solution would be that our program is ''smart'' enough that on his own, he will learn something about our function while he is trying to calculate the integral.\\
\ARROW Similar techniques were already created for numerical integration!\\
\ARROW Truly adaptive methods are nontrivial to code but are widely available in external packages as we will learn.\\
\ARROW Naming conventions:
\begin{itemize}
\item Integration \mc - software that is able to compute JUST! integrals.
\item Generator \mc - software that BESIDES! beeing able to perform the integration is also capable of performing a generation of points accordingly to the integration function.
\end{itemize}
\end{footnotesize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now the Foam algorithm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{\texttt{FOAM} algorithm }
\begin{footnotesize}
\ARROW S.Jadach (2000), \href{http://arxiv.org/abs/physics/9910004}{arXiv:physics/9910004, Comp. Phys. Commun. 152 (2003) 55}. Adaptive method with recursive division of the integration domain in cells. \\
\ARROW There are two algorithms in dividing the integration domain:
\begin{itemize}
\item Symplectic: Cells are sympleces(hiper-triangles). This method can be applied to not so large number of dimensions. $(\leq 5)$.
\item Qubic: Cells are hiper-cubes. This might be applied in higher number dimensions. $(\leq20)$.
\end{itemize}
\ARROW The algorithm:
\begin{itemize}
\item Exploration phase:\\
The integration domain (hipper-cube) is divided recursively into cells. In each step only one cell is split. The splitting is not event! The procedure is stop when the number of cells reach a certain number that is set by us. One constructs an approximation function and based on this the integral is calculated.
\item Generation/Calculation Phase:\\
We generate random points accordingly to the distribution of approximation function and the integral is calculated using the Importance sampling based on the approximation function.
\end{itemize}
\end{footnotesize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{\texttt{FOAM} algorithm }
\begin{footnotesize}
\begin{center}
\includegraphics[width=0.95\textwidth]{FOAM.png}
\end{center}
\end{footnotesize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{\texttt{FOAM} algorithm }
\begin{footnotesize}
\begin{center}
\includegraphics[width=0.75\textwidth]{FOAM2.png}\\
\end{center}
\end{footnotesize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Neumann-Ulam method}
\begin{itemize}
\item For example lets try to solve this equation system:
\end{itemize}
\begin{equation}
\overrightarrow{x} =
\left(\begin{array}{c}
1.5 \\
-1.0\\
0.7 \end{array} \right)
+
\left(\begin{array}{ccc}
0.2 & 0.3 & 0.1 \\
0.4 & 0.3 & 0.2 \\
0.3 & 0.1 & 0.1 \end{array} \right) \overrightarrow{x}
\nonumber
\end{equation}
\begin{itemize}
\item The solution is $\overrightarrow{x}_0 = (2.154303, 0.237389, 1.522255)$.
\end{itemize}
\begin{columns}
\column{0.1in}
\column{2.5in}
\begin{itemize}
\item The propability matrix $h_{ij}$ has the shape:
\end{itemize}
\begin{tabular}{|c|cccc|}
\hline
$i/j$ & 1 & 2 & 3 & 0 \\ \hline
1 & 0.2 & 0.3 & 0.1 & 0.4 \\
2 & 0.4 & 0.3 & 0.2 & 0.1 \\
3 & 0.3 & 0.1 & 0.1 & 0.5 \\ \hline
\end{tabular}
\column{2.5in}
\begin{itemize}
\item An example solution:
\end{itemize}
\includegraphics[width=0.95\textwidth]{images/mark.png}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Neumann-Ulam dual method}
\begin{itemize}
\item Let's try to solve the equation system:
\end{itemize}
\begin{equation}
\overrightarrow{x} =
\left(\begin{array}{c}
1.5 \\
-1.0\\
0.7 \end{array} \right)
+
\left(\begin{array}{ccc}
0.2 & 0.3 & 0.1 \\
0.4 & 0.3 & 0.2 \\
0.1 & 0.1 & 0.1 \end{array} \right) \overrightarrow{x}
\nonumber
\end{equation}
\begin{itemize}
\item The solution is: $\overrightarrow{x}_0 = (2.0, 0.0, 1.0)$.
\item Let's put the initial probability as constant:
\end{itemize}
\begin{equation}
q_1=q_2=q_3=\dfrac{1}{3} \nonumber
\end{equation}
\begin{columns}
\column{0.1in}
\column{2.5in}
\begin{itemize}
\item The propability matrix $h_{ij}$ has the shape:
\end{itemize}
\begin{tabular}{|c|cccc|}
\hline
$i/j$ & 1 & 2 & 3 & 4 \\ \hline
1 & 0.2 & 0.4 & 0.1 & 0.3 \\
2 & 0.3 & 0.3 & 0.1 & 0.3 \\
3 & 0.1 & 0.2 & 0.1 & 0.6 \\ \hline
\end{tabular}
\column{2.5in}
\begin{itemize}
\item An example solution:
\end{itemize}
\includegraphics[width=0.95\textwidth]{images/mark2.png}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{center}
\begin{huge}
Q \& A
\end{huge}
\end{center}
\end{frame}
\backupbegin
\begin{frame}\frametitle{Backup}
\end{frame}
\backupend
\end{document}
Marcin Chrzaszcz