diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.log b/Lectures_my/MC_2016/Lecture1/mchrzasz.log index 4724d9e..1d7b721 100644 --- a/Lectures_my/MC_2016/Lecture1/mchrzasz.log +++ b/Lectures_my/MC_2016/Lecture1/mchrzasz.log @@ -1,4 +1,4 @@ -This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1) 3 MAR 2016 21:43 +This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1) 12 MAR 2016 13:53 entering extended mode restricted \write18 enabled. %&-line parsing enabled. diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf b/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf index e92de7e..5bde4e9 100644 --- a/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf +++ b/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf Binary files differ diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.tex b/Lectures_my/MC_2016/Lecture1/mchrzasz.tex index 6f060ef..f3e6ba1 100644 --- a/Lectures_my/MC_2016/Lecture1/mchrzasz.tex +++ b/Lectures_my/MC_2016/Lecture1/mchrzasz.tex @@ -519,7 +519,7 @@ \ARROW For 2 random variables we define: \begin{align} -E(cx+y)=cE(x)+E(b) \nonumber \\ +E(cx+y)=cE(x)+E(y) \nonumber \\ V(cx+y)=c^2V(x)+V(y)+2c Cov(x,y), \label{p1} \end{align} where $Cov(x,y)=E([x-E(x)][y-E(y)])$ is called the covariance. diff --git a/Lectures_my/MC_2016/Lecture3/mchrzasz.aux b/Lectures_my/MC_2016/Lecture3/mchrzasz.aux index 2180bb6..96ffb0f 100644 --- a/Lectures_my/MC_2016/Lecture3/mchrzasz.aux +++ b/Lectures_my/MC_2016/Lecture3/mchrzasz.aux @@ -153,14 +153,8 @@ \@writefile{nav}{\headcommand {\beamer@framepages {21}{21}}} \pgfsyspdfmark {pgfid69}{0}{0} \pgfsyspdfmark {pgfid70}{0}{0} -\HyPL@Entry{21<>} -\pgfsyspdfmark {pgfid71}{23867907}{17900937} -\@writefile{nav}{\headcommand {\slideentry {0}{0}{22}{22/22}{}{0}}} -\@writefile{nav}{\headcommand {\beamer@framepages {22}{22}}} -\pgfsyspdfmark {pgfid72}{0}{0} -\pgfsyspdfmark {pgfid73}{0}{0} -\@writefile{nav}{\headcommand {\beamer@partpages {1}{22}}} -\@writefile{nav}{\headcommand {\beamer@subsectionpages {1}{22}}} -\@writefile{nav}{\headcommand {\beamer@sectionpages {1}{22}}} -\@writefile{nav}{\headcommand {\beamer@documentpages {22}}} -\@writefile{nav}{\headcommand {\def \inserttotalframenumber {21}}} +\@writefile{nav}{\headcommand {\beamer@partpages {1}{21}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {1}{21}}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {1}{21}}} +\@writefile{nav}{\headcommand {\beamer@documentpages {21}}} +\@writefile{nav}{\headcommand {\def \inserttotalframenumber {20}}} diff --git a/Lectures_my/MC_2016/Lecture3/mchrzasz.log b/Lectures_my/MC_2016/Lecture3/mchrzasz.log index 7907753..657c9b2 100644 --- a/Lectures_my/MC_2016/Lecture3/mchrzasz.log +++ b/Lectures_my/MC_2016/Lecture3/mchrzasz.log @@ -1,4 +1,4 @@ -This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1) 7 MAR 2016 23:17 +This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1) 12 MAR 2016 13:47 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -3091,8 +3091,6 @@ [16 ] -File: images/FOAM2.png Graphic file (type QTm) - File: images/BG_lower.png Graphic file (type QTm) Overfull \vbox (19.18185pt too high) has occurred while \output is active [] @@ -3135,35 +3133,14 @@ [18 ] -File: images/BG_lower.png Graphic file (type QTm) - -Overfull \vbox (19.18185pt too high) has occurred while \output is active [] - -................................................. -. fontspec info: "no-scripts" -. -. 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For evaluation of an integral in a given hipper-surface normally one uses numerical or \mc crude methods. In general it is not a easy task! +\item Recursive algorithm that searches for hipper-surfaces in which the function is approximately close. For evaluation of an integral in a given hipper-surface normally one uses numerical or \mc~crude methods. In general it is not an easy task! \item Often the function is approximated by a given set of elementary functions. \end{itemize} \item Calculation phase \begin{itemize} \item The integral is calculated using mostly using Stratified Sampling and Importance Sampling, depending on exploration phase. -\item If a \mc program has capability to generated distributions accordingly to the function of which we want to calculate the integral, it's in this place where it happens. +\item If a \mc~program has capability to generated distributions accordingly to the function of which we want to calculate the integral, it's in this place where it happens. \end{itemize} \end{enumerate} \ARROW There are algorithms where the exploration phase is linked with calculation phase. For each of the optimisation phase the integral is calculated as well. The result will be weighted average of those integrals! \begin{alertblock}{~} -This method might be bias! if in the extrapolation phase the function picks up a function peaks to late the whole method will lead to systematically bias results. +This method might be bias! if in the extrapolation phase the algorithm picks up a function peaks to late the whole method will lead to systematically bias results. \end{alertblock} @@ -329,7 +329,7 @@ \ARROW The first algorithm of this kind \texttt{RIWIAD} was proposed by Sheppeya \& Lautrupa in $1970$s. It was used to calculate integrals in cube $(0,1)^n$. \\ \ARROW It worked as follows: \begin{itemize} - \item At the begging the hipper-cube is divided in equal size sub cubes. In each of them the integral is calculated. + \item At the begging the hipper-cube is divided in equal size sub cubes and in each of them the integral is calculated. \item Based on the calculated integrals programs moves the boundaries to make the hipper-cubes smaller in the places where the function is greater and smaller where the function is smaller. \item The process starts over and continues over and over again. At each step the integral estimator and it's standard deviation is calculated. Form those a weighted average is constructed and it's standard deviation is constructed and its standard deviation. @@ -357,11 +357,11 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}\frametitle{Friedmanns algorithm} \begin{small} - \ARROW In the $1970$s J.Friedmann has also developed an adaptive \mc integration algorithm.\\ + \ARROW In the $1970$s J.Friedmann has also developed an adaptive \mc~integration algorithm.\\ \ARROW The algorithm was as follows: \begin{itemize} \item A probe function is constructed using a combination of Cauchy functions (Briet-Wigner), in which the peaks correspond to the local maxima of the integration function. In order to do so one needs to study the eigen functions in the neighbourhood of each peak (nasty thing...). - \item The Briet-Wigner is it falls down to $0$ slower then a Gauss distribution. + \item The Briet-Wigner is chosen as it falls down to $0$ slower then a Gauss distribution. \item The integral and the standard deviation is calculated based on the weighted averaged based on the probe function. @@ -376,12 +376,12 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}\frametitle{\texttt{DIVIONNE2} algorithm} \begin{footnotesize} -\ARROW J.Friedmann (1977): adaptive algorithm for \mc~integration based on recursive division of the integration area(available in the CERBLIB package).\\ +\ARROW J.Friedmann (1977): adaptive algorithm for \mc~integration based on recursive division of the integration area (available in the CERBLIB package).\\ \ARROW The algorithm: \begin{itemize} \item Multidimensional division of the hipper-cube. We divide each of the initial sub cubes to minimalise the spread of the function. \item After this the integral is calculated using Stratified Sampling. -\item We can generate a events accordingly to this function with this method. +\item We can generate events accordingly to this function with this method. \end{itemize} \ARROW Disadvantages: @@ -400,7 +400,7 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}\frametitle{\texttt{VEGAS} algorithm} \begin{footnotesize} -\ARROW J. G. P. Lepage (1978): adaptive algorithm for \mc~integration based on iterative division of the integration area(similar to \texttt{RIWID}).\\ +\ARROW J. G. P. Lepage (1978): adaptive algorithm for \mc~integration based on iterative division of the integration area (similar to \texttt{RIWID}).\\ \ARROW Let's calculate: $\int_0^1 f(x)dx$. \begin{itemize} \item We generate M random points from $\mathcal{U}(0,1)$. We calculate from them the integral and standard deviation. @@ -447,7 +447,7 @@ where $I_k$ and $\sigma_k$ are the integral and error in the k interaction. \item After the procedure stop we calculate the final results: \begin{align*} -\hat{I}=\sigma_I^2\sum_k\dfrac{I_k}{\sigma_k^2}~~\sigma_I= \left[\sum_k \dfrac{1}{\sigma_k^2}\right]^{-\frac{1}{2}} +\hat{I}=\sigma_I^2\sum_k\dfrac{I_k}{\sigma_k^2}~~~~~~~\sigma_I= \left[\sum_k \dfrac{1}{\sigma_k^2}\right]^{-\frac{1}{2}} \end{align*} \end{itemize} @@ -466,7 +466,7 @@ where $\alpha \in [1,2]$ sets the convergence speed. \ARROW When function has narrow peaks the $I_k$ and $\sigma_k$ might be wrongly calculated in early stages of iteraction. To fix this we can: \begin{align*} -I=\left[ \sum_k \dfrac{I_k^2}{\sigma_k^2}\right]^{-1} \sum_k I_k \left( \dfrac{I_k^2}{\sigma_k^2}\right),~~~ \sigma_I=I\left[\sum_k\dfrac{I_k^2}{\sigma_k^2}\right]^{-0.5} +I=\left[ \sum_k \dfrac{I_k^2}{\sigma_k^2}\right]^{-1} \sum_k I_k \left( \dfrac{I_k^2}{\sigma_k^2}\right),~~~~~ \sigma_I=I\left[\sum_k\dfrac{I_k^2}{\sigma_k^2}\right]^{-0.5} \end{align*} \ARROW If the number of interactions is to large then you cannot trust the algorithm! @@ -486,7 +486,7 @@ \begin{align*} p_x(x)= \dfrac{\sqrt{\int_0^1 dy \frac{f^2(x,y)}{p_y(y)} }}{\int_0^1dx \sqrt{\int_0^1 dy \dfrac{f^2(x,y)}{p_y(y)}}} \end{align*} -\ARROW So our 1D algorithm can be used to each of the axis (for x axis): +\ARROW So our 1D algorithm can be used to each of the axis (ex. for x axis): \begin{align*} (f_i)^2 = \sum_{x \in [ x_{i-1},x_i )} \sum_y \dfrac{f^2(x,y)}{p_y(y)}~~\sim ~~ \dfrac{1}{\Delta x_i} \int_{x_{i-1}}^{x_i}dx \int_0^{1}dy \dfrac{f^2(x,y)}{p_y(y)} \end{align*} @@ -502,9 +502,9 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}\frametitle{\texttt{VEGAS} algorithm - an example} \begin{footnotesize} -\ARROW An example of usage; let's calculate: +\ARROW An example of usage: let's calculate: \begin{align*} -I_n = \left(\dfrac{1}{a\sqrt{\pi}}\right)^n \int_0^1 \exp \left[ \dfrac{(x_n-0.5)^2}{a^2} \right] d^n x +I_n = \left(\dfrac{1}{a\sqrt{\pi}}\right)^n \int_0^1 \exp \left[ \dfrac{(x_n-0.5)^2}{a^2} \right] d^n x =1 \end{align*} \ARROW For the $n=9$, $a=0.1$ and $\alpha=1$ \begin{tabular}{|c|c c|c c|c|} @@ -561,13 +561,13 @@ \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 is bounded towards not so large number of dimensions. $(\leq 5)$. +\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. +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. @@ -603,18 +603,6 @@ \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} @@ -624,6 +612,8 @@ \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{center} + \begin{tabular}{c|c|c} \hline Standard deviation & 1D & nD\\ \hline @@ -635,6 +625,7 @@ +\end{center} \end{footnotesize} \end{frame} @@ -649,7 +640,7 @@ \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. +\item The integration uncertainties are hard to evaluate. \end{itemize} diff --git a/Lectures_my/MC_2016/Lecture4/images/FOAM.png b/Lectures_my/MC_2016/Lecture4/images/FOAM.png new file mode 100644 index 0000000..fbb1ba8 --- /dev/null +++ b/Lectures_my/MC_2016/Lecture4/images/FOAM.png Binary files differ diff --git a/Lectures_my/MC_2016/Lecture4/images/FOAM2.png b/Lectures_my/MC_2016/Lecture4/images/FOAM2.png new file mode 100644 index 0000000..6419d34 --- /dev/null +++ b/Lectures_my/MC_2016/Lecture4/images/FOAM2.png Binary files differ diff --git a/Lectures_my/MC_2016/Lecture4/images/MCMC.png 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Package biblatex Info: Trying to load language 'english'... 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(/usr/share/texmf/tex/latex/tipa/t3cmr.fd File: t3cmr.fd 2001/12/31 TIPA font definitions ) -LaTeX Font Info: ... okay on input line 213. +LaTeX Font Info: ... okay on input line 215. *geometry* driver: auto-detecting *geometry* detected driver: xetex @@ -2487,7 +2487,7 @@ ABD: EveryShipout initializing macros \AtBeginShipoutBox=\box119 -Package hyperref Info: Link coloring OFF on input line 213. +Package hyperref Info: Link coloring OFF on input line 215. (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty Package: nameref 2012/10/27 v2.43 Cross-referencing by name of section @@ -2497,9 +2497,9 @@ ) \c@section@level=\count448 ) -LaTeX Info: Redefining \ref on input line 213. -LaTeX Info: Redefining \pageref on input line 213. -LaTeX Info: Redefining \nameref on input line 213. +LaTeX Info: Redefining \ref on input line 215. +LaTeX Info: Redefining \pageref on input line 215. +LaTeX Info: Redefining \nameref on input line 215. (./mchrzasz.out) (./mchrzasz.out) \@outlinefile=\write10 @@ -2530,7 +2530,7 @@ tionary/translator-theorem-dictionary-English.dict Dictionary: translator-theorem-dictionary, Language: English ) -LaTeX Info: Redefining \includegraphics on input line 213. +LaTeX Info: Redefining \includegraphics on input line 215. *** you should *not* be loading the inputenc package *** XeTeX expects the source to be in UTF8 encoding @@ -2627,8 +2627,8 @@ Package biblatex Info: Trying to load bibliographic data... Package biblatex Info: ... file 'mchrzasz.bbl' not found. No file mchrzasz.bbl. -Package biblatex Info: Reference section=0 on input line 213. -Package biblatex Info: Reference segment=0 on input line 213. +Package biblatex Info: Reference section=0 on input line 215. +Package biblatex Info: Reference segment=0 on input line 215. 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(/usr/share/texlive/texmf-dist/tex/latex/polski/ot4cmtt.fd @@ -2779,8 +2779,11 @@ [2 ] +File: images/million-random-digits-open.jpg Graphic file (type QTm) + File: images/BG_lower.png Graphic file (type QTm) - + + Overfull \vbox (19.18185pt too high) has occurred while \output is active [] ................................................. @@ -2800,6 +2803,8 @@ [3 ] +LaTeX Font Info: External font `plex10' loaded for size +(Font) <9> on input line 308. File: images/BG_lower.png Graphic file (type QTm) Overfull \vbox (19.18185pt too high) has occurred while \output is active [] @@ -2821,15 +2826,8 @@ [4 ] -Overfull \hbox (17.33817pt too wide) detected at line 356 -\OML/plm/m/it/10.95 x[] \OT4/cmr/m/n/10.95 = \OML/plm/m/it/10.95 a[] \OT4/cmr/m -/n/10.95 + [] \OML/plm/m/it/10.95 h[]a[] \OT4/cmr/m/n/10.95 + [] [] \OML/plm/m/ -it/10.95 h[]h[]a[] \OT4/cmr/m/n/10.95 + [] [] [] \OML/plm/m/it/10.95 h[]h[]h[]a -[] \OT4/cmr/m/n/10.95 + \OML/plm/m/it/10.95 ::: - [] - File: images/BG_lower.png Graphic file (type QTm) - + Overfull \vbox (19.18185pt too high) has occurred while \output is active [] ................................................. @@ -2849,6 +2847,8 @@ [5 ] +File: images/shit.png Graphic file (type QTm) + File: images/BG_lower.png Graphic file (type QTm) Overfull \vbox (19.18185pt too high) has occurred while \output is active [] @@ -2870,16 +2870,8 @@ [6 ] -File: images/mark.png Graphic file (type QTm) - -Overfull \hbox (4.38205pt too wide) detected at line 429 -[][][] $[]$[][][][][][][][] $[]$[][][][][][][][] $[]$[][][][][][] - [] - -LaTeX Font Info: External font `plex10' loaded for size -(Font) <14.4> on input line 429. File: images/BG_lower.png Graphic file (type QTm) - + Overfull \vbox (19.18185pt too high) has occurred while \output is active [] ................................................. @@ -2920,13 +2912,13 @@ [8 ] -File: images/mark2.png Graphic file (type QTm) - -Overfull \hbox (4.38205pt too wide) detected at line 506 -[][][] $[]$[][][][][][][][] $[]$[][][][][][][][] $[]$[][][][][][] - [] - +LaTeX Font Info: Try loading font information for EU1+lmtt on input line 451 +. + (/usr/share/texlive/texmf-dist/tex/latex/euenc/eu1lmtt.fd +File: eu1lmtt.fd 2009/10/30 v1.6 Font defs for Latin Modern +) File: images/BG_lower.png Graphic file (type QTm) + Overfull \vbox (19.18185pt too high) has occurred while \output is active [] @@ -2947,6 +2939,17 @@ [9 ] +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr9! +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr6! +Missing character: There is no − in font plr9! 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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, \\ 26 November, 2015} + \textcolor{normal text.fg!50!Comment}{Monte Carlo methods, \\ 17 March, 2016} \end{center} \end{frame} } +\begin{frame}\frametitle{Random and pseudorandom numbers} + \begin{exampleblock}{John von Neumann:} +''Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.'' + \end{exampleblock} - - -\begin{frame}\frametitle{Markov Chain MC} +$\color{PineGreen}\Rrightarrow$ Random number: a given value that is taken by a random variable $ \twoheadrightarrow$ by definition cannot be predicted.\\ +%$\color{PineGreen}\Rrightarrow$ Sequence of random numbers $\twoheadrightarrow$ +$\color{PineGreen}\Rrightarrow$ Sources of truly random numbers: \begin{itemize} -\item Consider a finite possible states: $S_1$, $S_2$, ... -\item And the time steps of time, labelled as $1$, $2$, ... -\item At time $t$ the state is denoted $X_t$. -\item The conditional probability is defined as: +\item Mechanical +\item Physical \end{itemize} -\begin{equation} -P(X_t=S_j \vert X_{t-1}=S_{j-1},..., X_{1}=S_{1}) \nonumber -\end{equation} +$\color{PineGreen}\Rrightarrow$ Disadvantages of physical generators: \begin{itemize} -\item The Markov chain is then if the probability depends only on previous step. -\end{itemize} -\begin{equation} -P(X_t=S_j \vert X_{t-1}=S_{j-1},..., X_{1}=S_{1}) = P(X_t=S_j \vert X_{t-1}=S_{j-1} )\nonumber -\end{equation} -\begin{itemize} -\item For this reason this reason MCMC is also knows as drunk sailor walk. -\item Very powerful method. Used to solve linear eq. systems, invert matrix, solve differential equations, etc. -\end{itemize} -\end{frame} - - - -\begin{frame}\frametitle{Linear Equations} -\begin{itemize} -\item Lets say we have a linear equation system: -\end{itemize} -\begin{equation} -\begin{array}{lcl} X & = & pY + (1-p) A \\ Y & = & qX + (1-q)B \end{array} \nonumber -\end{equation} -\begin{itemize} -\item We know $A,B,p,q$; $X$ and $Y$ are meant to be determined. -\item Algorithm: -\begin{enumerate} -\item We choose first element of the first equation with probability $p$ and second with probability $1-p$. -\item We we choose the second one, the outcome of this MCMC is $W=A$. -\item If we choose the first we go to second equation and choose the first element with probability $q$ and the second with $1-q$. -\item We we choose the second one, the outcome of this MCMC is $W=B$. -\item If we choose the first we go to the first equation back again. -\item We repeat the procedure. -\end{enumerate} -\item We can estimate the solution of this system: -\end{itemize} -\begin{equation} -\hat{X} = \dfrac{1}{N}\sum_{i=1} W_i{~}{~}{~}{~}{~} \hat{\sigma_X}=\dfrac{1}{\sqrt{N-1}}\sqrt{\dfrac{1}{N} \sum_{i=1}^N W_i^2-\hat{X}^2} \nonumber -\end{equation} - -\end{frame} - -\begin{frame}\frametitle{Neumann-Ulam method} -\begin{itemize} -\item Let's try apply the basic MCMC method to solve a simple linear equation system: -\end{itemize} -\begin{equation} -A \overrightarrow{x} = \overrightarrow{b} \nonumber -\end{equation} -\begin{itemize} -\item The above system can be (always, see linear algebra lecture) translated into system: -\end{itemize} -\begin{equation} -\overrightarrow{x} = \overrightarrow{a} + H \overrightarrow{x} \nonumber -\end{equation} -\begin{itemize} -\item For this method we assume that the norm of the matrix is: -\end{itemize} -\begin{equation} -\Vert H \Vert = \underset{1 \leq i \leq n}{max} \sum_{j=1}^n \vert h_{ij} \vert <1 \nonumber \end{equation} -\begin{itemize} -\item Which we can write in a form: -\end{itemize} -\begin{equation} -(1 -H)\overrightarrow{x}=\overrightarrow{a} \nonumber -\end{equation} -\end{frame} - - - - -\begin{frame}\frametitle{Neumann-Ulam method} -\begin{itemize} -\item The solution would be then: -\end{itemize} -\begin{equation} -\overrightarrow{x}_0=(1 -H)^{-1}\overrightarrow{a} \nonumber -\end{equation} -\begin{itemize} -\item We can Taylor expend this: -\end{itemize} -\begin{equation} -\overrightarrow{x}_0=(1 -H)^{-1}\overrightarrow{a} = \overrightarrow{a} + H \overrightarrow{a} + H^2 \overrightarrow{a} + H^3 \overrightarrow{a} +.... \nonumber -\end{equation} -\begin{itemize} -\item For the $i$-th component of the $\overrightarrow{x}$ vector: -\end{itemize} -\begin{equation} -x_0^i= a_i + \sum_{j=1}^n h_{ij} a_{j_1} + \sum_{j_1=1}^n \sum_{j_2=1}^n h_{ij_1} h_{ij_2} a_{j_2} + \sum_{j_1=1}^n \sum_{j_2=1}^n \sum_{j_3=1}^n h_{ij_1} h_{ij_2} h_{ij_3} a_{j_3} + ...\nonumber -\end{equation} -\begin{itemize} -\item One can construct probabilistic behaviour of a system that follows the path of equation above. -\end{itemize} - -\end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Neumann-Ulam method} -\begin{itemize} -\item To do so we add to our matrix an additional column of the matrix: -\end{itemize} -\begin{equation} -h_{i,0} = 1-\sum_{j=1}^n h_{ij} > 0 \nonumber -\end{equation} -\begin{itemize} -\item The system has states: $\lbrace 0,1,2...,n\rbrace$ -\item State at $t$ time is denoted as $i_t$. -\item We make a random walk accordingly to to the following rules: -\begin{itemize} -\item At the begging of the walk ($t=0$) we are at $i_0$. -\item In the $t$ moment we are in the $i_t$ position then in $t+1$ time stamp we move to state $i_{t+1}$ with the probability $h_{i_t i_{t+1}}$. -\item We stop walking if we are in state $0$. -\end{itemize} -\item The path $X(\gamma) = (i_0, i_1, i_2, ..., i_k, 0)$ is called trajectory. -\item It can be proven that $x_i^0 =E \lbrace X (\gamma) \vert i_0=j \rbrace$. -\end{itemize} - - - -\end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Neumann-Ulam method, $\rm \color{RubineRed}{Lecture3/Markov}$} -\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 & 4 \\ \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 The problem with Neumann-Ulam method is that you need to repeat it for each of the coordinates of the $\overrightarrow{x}_0$ vector. -\item The dual method calculates the whole $\overrightarrow{x}_0$ vector. -\item The algorithm: -\begin{itemize} -\item On the indexes: $\lbrace0,1,...,n\rbrace$ we set a probability distribution:\\ $q_1, q_2,..., q_n$, $q_i>0$ and $\sum_i=1^n q_i=1$. -\item The starting point we select from $q_i$ distribution. -\item If in $t$ time we are in $i_t$ state then with probability $p(i_{t+1} \vert i_t) = h_{i_{t+1},i_{t}}$ in $t+1$ we will be in state $i_1$. For $i_{t+1}=0$ we define the probability: $h_{0,i_{t}}=1-\sum_{j=1}^n h_{j,i_{t}}$. Here we also assume that $h_{j,i_{t}} > 0$. -\item NOTE: there the matrix is transposed compared to previous method: $H^{T}$. -\item Again we end our walk when we are at state $0$. -\item For the trajectory: $\gamma = (i_0, i_1,...,i_k, 0)$, we assign the vector: -\end{itemize} -\begin{equation} -\overrightarrow{Y}(\gamma) = \dfrac{a_{i_0}}{ q_{i_{0}} p(0 \vert i_k) } \widehat{e}_{i_{k}} \in \mathcal{R}^n \nonumber -\end{equation} -\item The solution will be : $\overrightarrow{x}^0 = \dfrac{1}{N} \sum \overrightarrow{Y}(\gamma)$ +\item To slow for typical applications, especially the mechanical ones! +\item Not stable; small changes in boundary conditions might lead to completely different results! \end{itemize} \end{frame} - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Neumann-Ulam dual method, $\rm \color{RubineRed}{Lecture3/Markov2}$} -\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}\frametitle{Look elsewhere effect, $\rm \color{RubineRed}{Lecture3/LEE}$} -\begin{itemize} -\item Look elsewhere effect addresses the following problem: -\begin{itemize} -\item Imagine you observed a $3\sigma$ deviation in one of the observable that you measured. -\item Before you get excited one needs to understand if given the fact that you had so many measurements this might happen! -\end{itemize} -\item Example: Let's say we have measured 50 observables. What is the probability to observed 1 that is $3\sigma$ away from theory prediction? -\item Let's simulate 50 Gaussian distribution centred at 0 and width of 1. We count how simulations where at least one of the 50 numbers have the absolute value $>3$. -\item More complicated example: what if you observed 3 in a row $2\sigma$ fluctuations among 50 measurements? -\item This kind of studies are the best solvable by MC simulations. -\end{itemize} - - -\end{frame} - - - - - - -\begin{frame}\frametitle{Travelling Salesman Problem} -\begin{itemize} -\item Salesman starting from his base has to visit $n-1$ other locations and return to base headquarters. The problem is to find the shortest way. -\item For large $n$ the problem can't be solver by brutal force as the complexity of the problem is $(n-1)!$ -\item There exist simplified numerical solutions assuming factorizations. Unfortunately even those require anonymous computing power. -\item Can MC help? YES :) -\item The minimum distance $l$ has to depend on 2 factors: $P$ the area of the city the Salesman is travelling and the density of places he wants to visit: $\dfrac{n}{P}$ -\item Form this we can assume: -\end{itemize} -\begin{equation} -l \sim P^a (\dfrac{n}{P})^b=P^{a-b}n^b. \nonumber -\end{equation} - -\end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - - - - -\begin{frame}\frametitle{Traveling Salesman Problem} -\begin{itemize} -\item From dimension analysis: -\end{itemize} -\begin{equation} -a-b=\dfrac{1}{2}. \nonumber -\end{equation} -\begin{itemize} -\item To get $l$ we need square root of area. -\item From this it's obvious: -\end{itemize} -\begin{equation} -l \sim P^a (\dfrac{n}{P})^b=P^{0.5}n^{a-0.5}. \nonumber -\end{equation} -\begin{itemize} -\item Now we can multiply the area by alpha factor that keeps the density constant then: -\end{itemize} -\begin{equation} -l \sim \alpha^0.5 \alpha6{a-0.5} = \alpha^a \nonumber -\end{equation} -\begin{itemize} -\item In this case the distance between the clients will not change, but the number of clients will increase by $\alpha$ so: -\end{itemize} -\begin{equation} -l \sim \alpha \nonumber -\end{equation} -\begin{itemize} -\item In the end we get: $a=1$ -\end{itemize} -\end{frame} - - - -\begin{frame}\frametitle{Traveling Salesman Problem} -\begin{itemize} -\item In total: -\end{itemize} -\begin{equation} -l \sim k (nP)^{0.5}\nonumber -\end{equation} -\begin{itemize} -\item Of course the k depends on the shape of the area and locations of client. However for large $n$ the k starts loosing the dependency. It's an asymptotically free estimator. -\item To use the above formula we need to somehow calculate k. -\item How to estimate this? Well make a TOY MC: take a square put uniformly $n$ points. Then we can calculate $l$. Then it's trivial: -\end{itemize} -\begin{equation} -k= l(nP)^{-0.5} \nonumber -\end{equation} -\end{frame} - - -\begin{frame}\frametitle{Traveling Salesman Problem} -\begin{itemize} -\item This kind of MC experiment might require large CPU power and time. The adventage is that once we solve the problem we can use the obtained k for other cases (it's universal constant!). -\item It turns out that: -\end{itemize} -\begin{equation} -k \sim \dfrac{3}{4} \nonumber -\end{equation} -\begin{itemize} -\item Ok, but in this case we can calculate $l$ but not the actual shortest way! Why the hell we did this exercise?! -\item Turns out that for most of the problems we are looking for the solution that is close to smallest $l$ not the exact minimum. -\end{itemize} -\end{frame} - -\begin{frame}\frametitle{War Games} - -\begin{itemize} -\item S. Andersoon 1966 simulated for Swedish government how would a tank battle look like. -\item Each of the sides has 15 tanks. that they allocate on the battle field. -\item The battle is done in time steps. -\item Each tank has 5 states: -\begin{itemize} -\item OK -\item Tank can only shoot -\item Tank can only move -\item Tank is destroyed -\item Temporary states -\end{itemize} -\item This models made possible to check different fighting strategies. - -\end{itemize} -\end{frame} - - - -\begin{frame} - +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Random numbers - history remark} +$\color{PineGreen}\Rrightarrow$ In the past there were books with random numbers: \begin{center} -\begin{Huge} - Q \& A -\end{Huge} +\includegraphics[width=0.55\textwidth]{images/million-random-digits-open.jpg} + \end{center} +$\color{PineGreen}\Rrightarrow$ It's obvious that they didn't become very popular ;)\\ +$\color{PineGreen}\Rrightarrow$ This methods are comming back!\\ +$\color{PineGreen}\twoheadrightarrow$ Storage device are getting more cheap and bigger (CD, DVD).\\ +$\color{PineGreen}\twoheadrightarrow$ 1995: G. Marsaglia, $650\rm MB$ of random numbers, ''White and Black Noise''. + + +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Pseudorandom numbers} +\begin{footnotesize} +\begin{alertblock}{} +Commercially available physical generators of random numbers are usually based on electronic noise. This kind of generators do not pass simple statistical tests! Before you use them check they statistical properties. +\end{alertblock} +\ARROW Pseudorandom numbers- numbers generated accordingly to strict mathematical formula.\\ +\ARROW Strictly speaking they are non random numbers, how ever they have all the statistical properties of random numbers.\\ +\ARROW How ever modern generators are so good that no one can distinguish the pseudo random numbers generated by then from true random numbers.\\ +\ARROW Mathematical methods of producing pseudorandom numbers: +\begin{itemize} +\item Good statistical properties of generated numbers. +\item Easy to use and fast! +\item Reproducible! +\end{itemize} +\ARROW Because of those properties the truelly random numbers are not used in practice any more! + + +\end{footnotesize} +\end{frame} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Middle square generator; von Neumann} +$\color{PineGreen}\Rrightarrow$ The first mathematical generator (middle square) was proposed by von Neumann (1964).\\{~}\\ +$\color{PineGreen}\looparrowright$ Formula: $ +\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt, + boxrule=0pt,frame hidden]{ X_n= \lfloor X_{n-1}^2\cdot 10 ^{-m} \rfloor - \lfloor X_{n-1}^2\cdot 10^{-3m} \rfloor \cdot 10^{2m}} +$ + +$\color{PineGreen}\looparrowright$ where $X_0$ is a constant (seed), $\lfloor\cdot\rfloor$ is the cut-off of a number to integer.\\ +$\color{PineGreen}\Rrightarrow$ Example:\\ +{~}{~}Let's put $m=2$ and $X_0=2045$:\\ +\begin{columns} +\column{0.1\textwidth} +{~} +\column{0.4\textwidth} +$\color{PineGreen}\looparrowright$ $X_0^2=\underbrace{04}_{\rm rej}1820\underbrace{25}_{\rm rej}$ +\column{0.3\textwidth} + $\Rightarrow X_1=1820$ +\end{columns} +\begin{columns} +\column{0.1\textwidth} +{~} +\column{0.4\textwidth} +$\color{PineGreen}\looparrowright$ $X_1^2=\underbrace{03}_{\rm rej}3124\underbrace{00}_{\rm rej}$ +\column{0.3\textwidth} + $\Rightarrow X_1=3124$ +\end{columns} +$\color{PineGreen}\looparrowright$ Simple generator but unfortunately quite bad generator. Firstly the sequences are very short and strongly dependent on the $X_0$ number. + + + +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Middle square generator; von Neumann} +$\color{PineGreen}\Rrightarrow$ This was a first generator written and it's a good example how to not write generators.\\ +$\color{PineGreen}\Rrightarrow$ It's highly non stable! +\includegraphics[width=0.8\textwidth]{images/shit.png} + +\ARROW E 4.1 Write the von Neumann Middle square generator. + +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{General schematic } +\begin{small} +\ARROW Typical \mc~generator layout: +\begin{itemize} +\item We choose initial constants: $X_0$, $X_1$, ... $X_{k-1}$. +\item The $k$ number if calculated based on the previous ones: +\end{itemize} +\begin{align*} +X_k=f(X_0, ..., X_{k-1}), +\end{align*} +\ARROW Typically one generates $0/1$ which are then converted towards double precision numbers with $\mathcal{U}(0,1)$. \\ +\ARROW Generator period ($P, l$ integer numbers): $P$ is the period: +\begin{align*} +\exists _{l, P}: X_i=X_{i+j\cdot P}~~ \forall_{j \in \mathbb{I^+}}~ \forall_{i>l} +\end{align*} +\ARROW In post of the cases the period can be calculated analytically, although this is sometimes not trivial.\\ +\ARROW There is a recommendation about the period of a generator. For $N$ numbers we usually require: +\begin{align*} +N\ll P +\end{align*} +\ARROW In practice: $Nq$. Then: $b_n=b_{n-p}~{\rm xor}~b_{n-q}$ + +$\color{PineGreen}\Rrightarrow$ How to get numbers from bits (for example):\\ +$U_i = \sum_{j=1}^L 2^{-j} b_{is+j},~s{ +\includegraphics[width=0.95\textwidth]{images/gen1.png} +} +\only<2>{ +\includegraphics[width=0.95\textwidth]{images/gen2.png} +} +\only<3>{ +\includegraphics[width=0.95\textwidth]{images/gen3.png} +} +\only<4>{ +\includegraphics[width=0.95\textwidth]{images/gen4.png} +} +\only<5>{ +\includegraphics[width=0.95\textwidth]{images/gen5.png} +} +\only<6>{ +\includegraphics[width=0.95\textwidth]{images/gen6.png} +} +\end{small} +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Detector simulation} + +\begin{small} +$\color{PineGreen}\Rrightarrow$ Things do not get simpler on the detector side simulation.\\ +$\color{PineGreen}\Rrightarrow$ Lots of effects need to be taken into account: +\begin{columns} +\column{0.2in} +{~} +\column{2in} +$\color{PineGreen}\rightarrowtail$ Bremsstrahlung\\ +$\color{PineGreen}\rightarrowtail$ Interactions with different detector materials\\ +$\color{PineGreen}\rightarrowtail$ Particle identification\\ +$\color{PineGreen}\rightarrowtail$ Showers\\ +\column{3in} +\includegraphics[width=0.95\textwidth]{{images/lhcb2_h-640x408}.jpg} +\end{columns} +$\color{PineGreen}\Rrightarrow$ Example of generators:\\ +$\color{PineGreen}\rightarrowtail$ FLUKA\\ +$\color{PineGreen}\rightarrowtail$ Geant + +\end{small} +\end{frame} +\fi + +\begin{frame}\frametitle{Wrap up} +\begin{small} +$\color{PineGreen}\Rrightarrow$ Things to remember: +\begin{itemize} +\item Computer cannot produce random numbers, only pseudorandom numbers. +\item We use pseudorandon numbers as random numbers if they are statistically acting the same as random numbers. +\item Linear generators are not commonly used nowadays. +\item State of the art generators are the ones based on Kolomogorows theorem. +\end{itemize} +\end{small} +\end{frame} + + + \backupbegin \begin{frame}\frametitle{Backup}