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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 is bounded towards 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} -\ARROW An example of usage; let's calculate: -\begin{align*} -I_n = (\dfrac{1}{a\sqrt{\pi}})^n \int_0^1 \exp \left[ \dfrac{(x_n-0.5)^2}{a^2} \right] d^n x -\end{align*} -\ARROW For the $n=9$, $a=0.1$ and $\alpha=1$. -{~}\\ + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{\texttt{FOAM} algorithm } +\begin{footnotesize} \begin{center} - -\begin{tabular}{|c|c|c|} -\hline -Number of points on axis & Integral value & Number of calculations \\ \hline \hline -$5$ & $71.364$ & $2 \cdot 10^6$ \\ -$6$ & $0.017$ & $10^7$ \\ -$10$ & $0.774$ & $10^9$ \\ -$15$ & $1.002$ & $3.8\cdot 10^9$ \\ \hline -\end{tabular} - +\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} +\ARROW E3.1 Using ROOT implementation of the FOAM algorithm calculate the integrals from exercise E2.3. + + \end{footnotesize} \end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{\texttt{FOAM} algorithm } +\begin{footnotesize} +\begin{center} +\includegraphics[width=0.75\textwidth]{FOAM2.png}\\ +\end{center} +\ARROW E3.1 Using ROOT implementation of the FOAM algorithm calculate the integrals from exercise E2.3. + + +\end{footnotesize} +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Monte Carlo vs numerical methods} +\begin{footnotesize} +\ARROW All numerical methods are based on evaluating the integral using linear combination of function: +\begin{align*} +I_Q = \sim_{i=1}^m \omega_i f(x_i) +\end{align*} +\ARROW Different methods have different weights $\omega_i$ and lattice point $x_i$.\\ +\ARROW Efficiency of Monte Carlo methods compared to the numerical ones: +\begin{tabular}{c|c|c} +\hline +Standard deviation & 1D & nD\\ \hline +Monte Carlo & $n^{-1/2}$ & $n^{-1/2}$ \\ +Trapezoidal Rule & $n^{-2}$ & $n^{-2/d}$\\ +Simpson Rule & $n^{-2}$ & $n^{-2/d}$\\ +m-point Gauss rule & $n^{-2m}$ & $n^{-2m/d}$\\ \hline +\end{tabular} +\end{footnotesize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Sum up} +\begin{footnotesize} +\ARROW In one dimension the Monte Carlo method is substantially slower then the numerical methods! Even the most simple ones.\\ +\ARROW In many dimensions the Monte Carlo methods rapidity gain the advantages! \\ +\ARROW For $d>4$ the \mc~method if faster then the Trapezoidal Rule.\\ +\ARROW For $d>8$ the \mc~method if faster then the Simpson Rule.\\ +\ARROW The disadvantages of the numerical methods: +\begin{itemize} +\item Hard to apply in multi dimensions. +\item Hard to apply in complex integration domains. +\item The integration uncertainty are hard to evaluate. +\end{itemize} +\end{footnotesize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Method of Moments} + +\begin{small} +$\color{PineGreen}\Rrightarrow$ Now real cool things!\\ +$\color{PineGreen}\Rrightarrow$ Let's consider we want to study a rare decay: $\PB^{\pm} \to \PK^{\pm} \Pmu \Pmu$. The decay is described by the following \ +PDF: +\begin{equation} +\dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} =\dfrac{3}{4}(1-F_H)(1-\cos^2 \theta_l)+F_H/2 + A_{FB}\cos \theta_l \nonumber +\end{equation} +$\color{PineGreen}\Rrightarrow$ PDF by construction is normalized: $\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} =1$ +\begin{columns} +\column{0.1in} +{~} +\column{2.2in} +\begin{itemize} +\item Normally we do a likelihood fit and we are done. +\item There is a second way! +\end{itemize} +\column{2.8in} +\includegraphics[width=0.95\textwidth]{images/Kmumu_LL.png} +\end{columns} + +\end{small} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Method of Moments} +\begin{footnotesize} +$\color{PineGreen}\Rrightarrow$ Let's calculate the integrals: +\begin{equation} +\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos \theta_l = \dfrac{2}{3}A_{FB} \nonumber +\end{equation} +\begin{equation} +\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos^2 \theta_l = \dfrac{1}{5} + \dfrac{2 F_H}{15} \nonumber +\end{equation} +$\color{PineGreen}\Rrightarrow$ So we can get our parameters that we searched for by doing a integration. So now what?\\ +$\color{PineGreen}\Rrightarrow$ Well nature is the best random number generator so let's take the data and treat and calculate the integral estimates: +\begin{equation} +\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos \theta_l = \dfrac{2}{3}A_{FB} = \dfrac{1}{N} \sum_{i=1}^N \cos \theta_{l,i} \nonumber +\end{equation} +\begin{equation} +\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos^2 \theta_l = \dfrac{1}{5} + \dfrac{2 F_H}{15}=\dfrac{1}{N} \sum_{i=1}^N cos^2 \theta_{l,i} \nonumber +\end{equation} +\ARROW E3.2 Calculate the $A_{FB}$ and $F_H$ using Method of moments. The events to be used to calculate them are here:\href{ http://www.physik.uzh.ch/lectures/mcm/lectures/mom.txt}{LINK} + + +\end{footnotesize} +\end{frame} + \backupbegin