From 99b479d56d5a71a4eefc42686af8cf407ccf0d3e Mon Sep 17 00:00:00 2001 From: hiro98 Date: Sun, 21 Jun 2020 21:25:59 +0200 Subject: [PATCH] add integration slides --- talk/slides.tex | 137 +++++++++++++++++++++++++++++++++++++++++------ talk/vortrag.org | 1 + 2 files changed, 123 insertions(+), 15 deletions(-) diff --git a/talk/slides.tex b/talk/slides.tex index 0687e3e..11ba7e8 100644 --- a/talk/slides.tex +++ b/talk/slides.tex @@ -15,10 +15,10 @@ labelformat=brace, position=top]{subcaption} \graphicspath{ {figs/} } \usetheme{Antibes} -\usepackage{eulerpx} +% \usepackage{eulerpx} \usepackage{ifdraft} -\usefonttheme[onlymath]{serif} +% \usefonttheme[onlymath]{serif} \setbeamertemplate{itemize items}[default] \setbeamertemplate{enumerate items}[default] \AtBeginSection[] @@ -209,18 +209,12 @@ labelformat=brace, position=top]{subcaption} \begin{block}{Task: calculate \(\abs{\mathcal{M}}^2\)} \begin{enumerate}[<+->] - \item translate diagrams to - matrix elements - \item use Casimir's trick to - average over spins - \item use completeness - relation to sum over - photon polarizations - \item use trace identities - to compute the absolute - square - \item simplify with - trigonometric identities + \item translate diagrams to matrix elements + \item use Casimir's trick to average over spins + \item use completeness relation to sum over photon + polarizations + \item use trace identities to compute the absolute square + \item simplify with trigonometric identities \end{enumerate} \end{block} \pause Here: Quark masses @@ -286,5 +280,118 @@ labelformat=brace, position=top]{subcaption} \end{minipage} \end{figure} \end{frame} + +\section{Monte Carlo Methods} + +\note[itemize]{ +\item Gradually bring in knowledge through distribution. +} +\begin{frame} + \begin{block}{Basic Ideas} + \begin{itemize} + \item<+-> Given some unknown function + \(f\colon \vb{x}\in\Omega\subset\mathbb{R}^n\mapsto\mathbb{R}\) \ldots + \item<+-> \ldots\ how do we answer questions about \(f\)? + \end{itemize} + \;\;\onslide<+->{\(\implies\) Sample it at random points.} + \end{block} + \pause + \begin{block}{Main Applications} + \begin{enumerate} + \item<+-> integrate \(f\) over some volume \(\Omega\) + \item<+-> treat \(f\) as distribution and take random samples + \end{enumerate} + \end{block} +\end{frame} + +\subsection{Integration} +\note[itemize]{ +\item omitting details (law of big numbers, central limit theorem) +\item at least three angles of attack +\item some sort of importance sampling, volume: stratified sampling +} +\begin{frame} + \begin{itemize} + \item<+-> we have: + \(f\colon \vb{x}\in\Omega\subset\mathbb{R}^n\mapsto\mathbb{R}\) + and \(\rho\colon \vb{x}\in\Omega\mapsto\mathbb{R}_{> 0}\) with + \(\int_{\Omega}\rho(\vb{x})\dd{\vb{x}} = 1\). + \item<+-> we seek: + \begin{equation} + \label{eq:baseintegral} + I = \int_\Omega f(\vb{x}) \dd{\vb{x}} + \onslide<+->{= \int_\Omega + \qty[\frac{f(\vb{x})}{\rho(\vb{x})}] \rho(\vb{x}) \dd{\vb{x}} = \EX{\frac{F}{\Rho}}} + \end{equation} + \item<+-> numeric approximation: + \begin{equation} + \label{eq:approxexp} + \EX{\frac{F}{\Rho}} \approx + \frac{1}{N}\sum_{i=1}^N\frac{f(\vb{x_i})}{\rho(\vb{x_i})} + \xrightarrow{N\rightarrow\infty} I + \end{equation} + \item<+-> error approximation: + \begin{align} + \sigma_I^2 &= \frac{\textcolor<+->{red}{\sigma^2}}{\textcolor<.->{blue}{N}} \\ + \sigma^2 &= \VAR{\frac{F}{\Rho}} = \int_{\textcolor<+(3)->{blue}{\Omega}} \qty[I - + \frac{f(\vb{x})}{\textcolor<+->{blue}{\rho(\vb{x})}}]^2 + \textcolor<.->{blue}{\rho(\vb{x})} \textcolor<+->{blue}{\dd{\vb{x}}} \approx \frac{1}{N - 1}\sum_i \qty[I - + \frac{f(\vb{x_i})}{\rho(\vb{x_i})}]^2 \label{eq:varI-approx} + \end{align} + \end{itemize} +\end{frame} + +\begin{frame}{Change of Variables} + Choose \(\rho(\vb{x}) = \frac{1}{\abs{\Omega}}\) + \onslide<2->{\(\implies I=\frac{\abs{\Omega}}{N}\sum_{i=1}^N + f(\vb{x_i})=\abs{\Omega}\cdot\bar{f}\) and + \(\VAR{\frac{F}{P}}\approx\frac{\abs{\Omega}^2}{N-1}\sum_{i}\qty[f(\vb{x}_i) + - \bar{f}]^2\)} + \begin{block}{Results} + \begin{itemize} + \item<3-> integrating \(\dv{\sigma}{\theta}\) with target error of + \(\sigma = \SI{1e-3}{\pico\barn}\) takes + \result{xs/python/xs_mc_N} samples + \item<4-> integrating \(\dv{\sigma}{\eta}\) takes just + \result{xs/python/xs_mc_eta_N} samples + \end{itemize} + \end{block} + \begin{figure}[hb] + \centering \onslide<3->{ + \begin{subfigure}[c]{.4\textwidth} + \centering \plot[scale=.6]{xs/xs_integrand} + \end{subfigure} + } \onslide<4->{ + \begin{subfigure}[c]{.4\textwidth} + \centering \plot[scale=.6]{xs/xs_integrand_eta} + \end{subfigure} + } + \end{figure} +\end{frame} + +\begin{frame}{Vegas} + \begin{columns} + \begin{column}{.5\textwidth} + \begin{block}{Idea} + \begin{enumerate} + \item subdivide integration volume into grid, take equal + number of samples in each hypercube \(\iff\) define \(\rho\) + as step function + \item iteratively approximate optimal \(\rho = f(\vb{x})/I\) + with step function + \end{enumerate} + \end{block} + \begin{block}{Result} + Total function evaluations: \result{xs/python/xs_mc_θ_vegas_N} + \end{block} + \end{column} + \begin{column}{.5\textwidth} + \begin{figure}[ht] + \centering \plot[scale=.6]{xs/xs_integrand_vegas} + \caption{\(2\pi\dv{\sigma}{\theta}\) scaled to increments + found by \vegas} + \end{figure} + \end{column} + \end{columns} +\end{frame} \end{document} -massless limit diff --git a/talk/vortrag.org b/talk/vortrag.org index 1ec5973..fdc494b 100644 --- a/talk/vortrag.org +++ b/talk/vortrag.org @@ -326,6 +326,7 @@ What the heck should be in there. Let's draft up an outline. - spending time on tooling is OK - have to put more time into detailed diagnosis - event generators are marvelously complex + - should have introduced the term importance sampling properly ** Outlook - more effects - multi channel mc