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talk/slides.tex
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@ -13,7 +13,8 @@ labelformat=brace, position=top]{subcaption}
% \setbeameroption{show notes on second screen} %
\addbibresource{thesis.bib}
\graphicspath{ {figs/} }
\usepackage{animate}
\newfontfamily\DejaSans{DejaVu Sans}
\usetheme{Antibes}
% \usepackage{eulerpx}
\usepackage{ifdraft}
@ -30,6 +31,9 @@ labelformat=brace, position=top]{subcaption}
\setbeamertemplate{footline}[frame number]
\setbeamertemplate{note page}[plain]
\setbeamertemplate{bibliography item}{\insertbiblabel} %% Remove book
%% symbol from references and add
%% number
\sisetup{separate-uncertainty = true}
% Macros
@ -125,8 +129,7 @@ labelformat=brace, position=top]{subcaption}
\hypersetup{pageanchor=false}
\maketitle
\hypersetup{pageanchor=true}
\pagenumbering{arabic}
\hypersetup{pageanchor=true} \pagenumbering{arabic}
\begin{frame}
\tableofcontents
@ -206,8 +209,7 @@ labelformat=brace, position=top]{subcaption}
\end{column}
\pause
\begin{column}{.5\textwidth}
\begin{block}{Task: calculate
\(\abs{\mathcal{M}}^2\)}
\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
@ -217,8 +219,7 @@ labelformat=brace, position=top]{subcaption}
\item simplify with trigonometric identities
\end{enumerate}
\end{block}
\pause Here: Quark masses
neglected.
\pause Here: Quark masses neglected.
\end{column}
\end{columns}
\end{frame}
@ -256,10 +257,11 @@ labelformat=brace, position=top]{subcaption}
\end{figure}
\end{frame}
\begin{frame}{Comparison with \sherpa}
\begin{frame}{Comparison with \sherpa~\cite{Bothmann:2019yzt}}
\begin{itemize}
\item<1-> choose \result{xs/python/eta} and \result{xs/python/ecm} and
integrate XS
\item<1-> choose \result{xs/python/eta} and \result{xs/python/ecm}
and integrate XS
\begin{equation}
\label{eq:total-crossec}
\sigma = {\frac{\pi\alpha^2Z^4}{3\ecm^2}}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
@ -274,9 +276,8 @@ labelformat=brace, position=top]{subcaption}
\plot[scale=.5]{xs/total_xs}
\end{minipage}
\begin{minipage}[c]{0.3\textwidth}
\caption{\label{fig:totxs} The cross section
of the process for a pseudo-rapidity
integrated over \([-\eta, \eta]\).}
\caption{\label{fig:totxs} The cross section of the process for
a pseudo-rapidity integrated over \([-\eta, \eta]\).}
\end{minipage}
\end{figure}
\end{frame}
@ -284,13 +285,13 @@ labelformat=brace, position=top]{subcaption}
\section{Monte Carlo Methods}
\note[itemize]{
\item Gradually bring in knowledge through distribution.
}
\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
\(f\colon \vb{x}\in\Omega\subset\mathbb{R}^n\mapsto\mathbb{R}\)
\ldots
\item<+-> \ldots\ how do we answer questions about \(f\)?
\end{itemize}
\;\;\onslide<+->{\(\implies\) Sample it at random points.}
@ -308,8 +309,7 @@ labelformat=brace, position=top]{subcaption}
\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
}
\item some sort of importance sampling, volume: stratified sampling }
\begin{frame}
\begin{itemize}
\item<+-> we have:
@ -334,9 +334,9 @@ labelformat=brace, position=top]{subcaption}
\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}
\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}
@ -369,7 +369,12 @@ labelformat=brace, position=top]{subcaption}
\end{figure}
\end{frame}
\begin{frame}{Vegas}
\note[itemize]{
\item proposed by G. Peter Lepage (slac) 1976
\item own implementation!!!
}
\begin{frame}{\vegas\ Algorithm \cite{Lepage:19781an}}
\begin{columns}
\begin{column}{.5\textwidth}
\begin{block}{Idea}
@ -379,10 +384,14 @@ labelformat=brace, position=top]{subcaption}
as step function
\item iteratively approximate optimal \(\rho = f(\vb{x})/I\)
with step function
\item this is quite efficient when \(n\geq 4\)
\end{enumerate}
\end{block}
\pause
\begin{block}{Result}
Total function evaluations: \result{xs/python/xs_mc_θ_vegas_N}
Total function evaluations:
\result{xs/python/xs_mc_θ_vegas_N}\\
(for same accuracy as before)
\end{block}
\end{column}
\begin{column}{.5\textwidth}
@ -394,4 +403,200 @@ labelformat=brace, position=top]{subcaption}
\end{column}
\end{columns}
\end{frame}
\subsection{Sampling}
\note[itemize]{
\item prop. to density
\item generalization to n dim is easy
\item idea -> cumulative propability the same
}
\begin{frame}
\begin{itemize}[<+->]
\item we have: \(f\colon x\in\Omega\mapsto\mathbb{R}_{>0}\)
(choose \(\Omega = [0, 1]\)) and uniformly random samples \(\{x_i\}\)
\item we seek: a sample \(\{y_i\}\) distributed according to \(f\)
\end{itemize}
\begin{block}<+->{Basic Idea}
\begin{itemize}[<+->]
\item<.-> let \(x\) be sample of uniform distribution, solve
\[\int_{0}^{y}f(x')\dd{x'} = x\cdot\int_0^1f(x')\dd{x'} =
x\cdot A\] for y to obtain sample of \(f/A\)
\item let \(F\) be the antiderivative of \(f\), then
\(y=F^{-1}(x\cdot A + F(0))\)
\begin{itemize}
\item sometimes analytical form available
\item otherwise tackle that numerically
\end{itemize}
\end{itemize}
\end{block}
\end{frame}
\begin{frame}{Hit or Miss}
\centering
\animategraphics[loop,scale=.4,autoplay,palindrome]{5}{pi/pi-}{0}{9}
\end{frame}
\begin{frame}{Hit or Miss}
\begin{block}{Basic Idea}
\begin{itemize}[<+->]
\item take samples \({x_i}\) distributed according to \(g/B\),
where \(B=\int_0^1g(x)\dd{x}\) and
\(\forall x\in\Omega\colon g(x)\geq f(x)\)
\item accept each sample with the probability~\(f(x_i)/g(x_i)\)
(importance sampling)
\item total probability of accepting a sample: \(\mathfrak{e} =
A/B < 1\) (efficiency)
\item simplest choice \(g=\max_{x\in\Omega}f(x)=f_{\text{max}}\)
\item again: efficiency gain through reduction of variance
\end{itemize}
\end{block}
\begin{block}<+->{Results with \(g=f_{\text{max}}\) }
\begin{itemize}[<+->]
\item<.-> sampling \(\dv{\sigma}{\cos\theta}\):
\result{xs/python/naive_th_samp}
\item sampling \(\dv{\sigma}{\cos\theta}\):
\result{xs/python/eta_eff}
\end{itemize}
\end{block}
\end{frame}
\begin{frame}{Hit or Miss}
\begin{columns}
\begin{column}{.4\textwidth}
\begin{block}<+->{Results with \(g=a\cdot x^2 + b\)} Modest
efficiency gain: \result{xs/python/tuned_th_samp}
\end{block}
\begin{itemize}
\item<+-> Of course, we can use \vegas\ to provide a better \(g\).
\end{itemize}
\end{column}
\begin{column}{.6\textwidth}
\begin{figure}[ht]
\centering \plot[scale=.8]{xs_sampling/upper_bound}
\caption{The distribution \(\dv{\sigma}{\cos\theta}\) and an
upper bound of the form \(a + b\cdot x^2\).}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Stratified Sampling}
\begin{block}{Basic Idea}
\begin{itemize}
\item subdivide sampling volume \(\Omega\) into \(K\) subvolumes
\(\Omega_i\)
\item let \(A_i = \int_{\Omega_i}f(x)\dd{x}\)
\item take \(N_i=A_i \cdot N\) samples in each subvolume
\item efficiency is given by:
\(\mathfrak{e} = \frac{\sum_i A_i}{\sum_i A_i/\mathfrak{e}_i}\)
\end{itemize}
\(\implies\) can optimize in each subvolume independently
\end{block}
How do choose the \(\Omega_i\)? \pause {\huge\vegas! :-)}
\end{frame}
\note[itemize]{
\item no need to know the jacobian ;)
}
\begin{frame}{Observables}
\begin{itemize}
\item we want: distributions of other observables
\item turns out: simpliy piping samples \(\{x_i\}\) through a map
\(\gamma\colon\Omega\mapsto\mathbb{R}\) is enough
\end{itemize}
\begin{figure}[p]
\centering
\begin{subfigure}[b]{.49\textwidth}
\centering \plot[scale=.5]{xs_sampling/histo_sherpa_eta}
\caption{histogram of the pseudo-rapidity
(\(\eta\)).}
\end{subfigure}
\begin{subfigure}[b]{.49\textwidth}
\centering \plot[scale=.5]{xs_sampling/histo_sherpa_pt}
\caption{histogram of the transverse momentum
(\(\pt\))}
\end{subfigure}
\end{figure}
\end{frame}
\begin{frame}[allowframebreaks]
\frametitle{References}
\printbibliography
\end{frame}
\appendix
\section{Appendix}
\subsection{More on \vegas}
\begin{frame}{\vegas Details}
\begin{columns}
\begin{column}{.6\textwidth}
\begin{block}{Algorithm 1D}
\begin{enumerate}
\item start with \(N\) evenly spaced increments
\(\{[x_i, x_{i+1}]\}_{i\in\overline{1,N}}\)
\item calculate the integral weights
\(w_i = \abs{\int_{x_i}^{x_{i+1}}f(x)\dd{x}}\) and define
\(W=\sum_iw_i\)
\begin{itemize}
\item this is done with ordinary MC integration
\end{itemize}
\item calculate subdivide the \(i\)-th increment into
\(K\frac{w_i}{W}\) increments (round up), where
\(K = \mathcal{O}(1000)\)
\item amalgamate the new increments into \(N\) groups \(=\)
new increments
\end{enumerate}
\end{block}
\end{column}
\pause
\begin{column}{.4\textwidth}
\begin{block}{Advantages}
\begin{itemize}
\item number of \(f\) evaluations independent of number of
hypercubes
\item adaption itself is adaptive
\item \textcolor{red}{the advantages only show if \(n\)
``high''.}
\end{itemize}
\end{block}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
\begin{figure}[ht]
\centering \plot[scale=.9]{xs/xs_integrand_vegas}
\caption{\(2\pi\dv{\sigma}{\theta}\) scaled to increments found by
\vegas}
\end{figure}
\end{frame}
\begin{frame}{\vegas\ + Hit or Miss}
\begin{figure}[ht]
\centering
\begin{subfigure}{.49\textwidth}
\centering
\plot[scale=.8]{xs_sampling/vegas_strat_dist}
\caption[The distribution for \(\cos\theta\), derived from the
differential cross-section and the \vegas-weighted
distribution]{\label{fig:vegasdist} The distribution for
\(\cos\theta\) and the \vegas-weighted
distribution.}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\centering
\plot[scale=.8]{xs_sampling/vegas_rho}
\caption[The weighting distribution generated by
\vegas.]{\label{fig:vegasrho} The weighting distribution generated
by \vegas. It is clear, that it closely follows the original
distribution.}
\end{subfigure}
\caption{\label{fig:vegas-weighting} \vegas-weighted distribution
and weighting distribution.}
\end{figure}
\end{frame}
\end{document}