From 4c443140d1848491945d35f29ae550ab8faa8c07 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sat, 28 Jan 2017 22:17:55 +0000 Subject: [PATCH 01/28] title page --- .gitignore | 1 + LWB/propagators/figures/eth-logo.pdf | Bin 0 -> 13648 bytes LWB/propagators/header.tex | 150 +++++++++++++++++++++++ LWB/propagators/main.tex | 166 ++++++++++++++++++++++++++ LWB/propagators/sections/abstract.tex | 5 + LWB/propagators/sections/title.tex | 52 ++++++++ 6 files changed, 374 insertions(+) create mode 100644 LWB/propagators/figures/eth-logo.pdf create mode 100644 LWB/propagators/header.tex create mode 100644 LWB/propagators/main.tex create mode 100644 LWB/propagators/sections/abstract.tex create mode 100644 LWB/propagators/sections/title.tex diff --git a/.gitignore b/.gitignore index bbaaa01..30805ca 100644 --- a/.gitignore +++ b/.gitignore @@ -30,3 +30,4 @@ *.vrb *.xdy *.tdo +*.loa diff --git a/LWB/propagators/figures/eth-logo.pdf b/LWB/propagators/figures/eth-logo.pdf new file mode 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nf}t`H2!t0V`hQ30-*Vuck;Vw*?__`>JP=-7W@c$+8QlK|6qQX) literal 0 HcmV?d00001 diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex new file mode 100644 index 0000000..5269639 --- /dev/null +++ b/LWB/propagators/header.tex @@ -0,0 +1,150 @@ +\documentclass[a4paper,10pt]{scrartcl} + +\usepackage{standalone} + +% Load ``float'' before ``hyperref`` before ''algorithm`` +% Note: ''algorithm`` would load ''float`` by itself +\usepackage{float} +\usepackage[pagebackref,hyperindex=true]{hyperref} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Math +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{amsopn} +\usepackage{braket} +\usepackage{bbm} +\usepackage{dsfont} +% \usepackage{mathabx} + + +% Various new commands that ease typesetting math even further +% \newcommand{\assign}{\ensuremath{\coloneq}} +% \newcommand{\rassign}{\ensuremath{\eqcolon}} +% TODO: remove +% \newcommand{\assign}{\ensuremath{:=}} +% \newcommand{\rassign}{\ensuremath{=:}} +% \newcommand{\seteq}{\ensuremath{\overset{!}{=}}} +% \newcommand{\of}[1]{\ensuremath{\left( #1 \right)}} +% \newcommand{\ofs}[1]{\ensuremath{\left( #1 \right)}} +% \newcommand{\norm}[1]{\ensuremath{\| #1 \|}} +% \newcommand{\tmop}[1]{\ensuremath{\operatorname{#1}}} +% \newcommand{\id}{\ensuremath{\mathds{1}}} +% % \newcommand{\id}{\ensuremath{I}} +% \newcommand{\kron}[1]{\ensuremath{\delta_{#1}}} +% \newcommand{\conj}[1]{\ensuremath{\overline{#1}}} +% \renewcommand{\vec}[1]{\ensuremath{\underline{#1}}} +% \newcommand{\mat}[1]{\ensuremath{\mathbf{#1}}} +% \newcommand{\inv}{\ensuremath{{}^{-1}}} +% \newcommand{\T}{\ensuremath{{}^{\textnormal{T}}}} +% \renewcommand{\H}{\ensuremath{{}^{\textnormal{H}}}} +% \newcommand{\Tinv}{\ensuremath{{}^{\textnormal{-T}}}} +% \newcommand{\Hinv}{\ensuremath{{}^{\textnormal{-H}}}} +% \newcommand{\tr}{\ensuremath{\textnormal{Tr}}} +% \newcommand{\ft}[1]{\ensuremath{\mathcal{F}\left(#1\right)}} +% \newcommand{\ift}[1]{\ensuremath{\mathcal{F}^{-1}\left(#1\right)}} +% \newcommand{\fft}[1]{\ensuremath{\mathtt{FFT}\left(#1\right)}} +% \newcommand{\ifft}[1]{\ensuremath{\mathtt{IFFT}\left(#1\right)}} +% \newcommand{\dotp}[2]{\ensuremath{\left\langle #1 , #2 \right\rangle}} +% \newcommand{\bigO}[1]{\ensuremath{\mathcal{O}\left( #1 \right)}} +% \newcommand{\laplace}{\ensuremath{\operatorname{\Delta}}} +% \newcommand{\diff}[3][]{\frac{\mathrm{d}^{#1}#2}{\mathrm{d}#3^{#1}}} +% \newcommand{\pdiff}[3][]{\frac{\partial^{#1}#2}{\partial #3^{#1}}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Graphics +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{graphicx} +\usepackage{subfig} +\usepackage{asymptote} +\usepackage{tikz} +\usetikzlibrary{topaths,calc} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows,shapes,backgrounds} +\definecolor{gray}{gray}{0.55} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Math +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{algorithm} +\usepackage{algorithmic} +\usepackage{color} +\usepackage{relsize} +\usepackage[scaled=0.8]{beramono} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Code +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{listings} +% C++ Code macro +\newcommand{\cpp}[1] { + \lstset{language=c++, + basicstyle=\smaller, + basewidth=0.58em, + columns=fixed, + tabsize=2, + fontadjust=true, + frame=l, + xleftmargin=4.2pt, + numbers=left, + stepnumber=2, + breaklines=true, + breakindent=0pt, + prebreak=\mbox{\tiny$\searrow$}, + postbreak=\mbox{{\color{gray}$\cdots$}}, + numberstyle=\color{gray}, + commentstyle=\color{gray}, + stringstyle=\textit, + showstringspaces=false, + } + \lstinputlisting{#1} +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% More Packages +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[utf8x]{inputenc} +\usepackage{cancel} +\usepackage{url} +\usepackage{placeins} +\usepackage{microtype} +\usepackage{amsthm} +\newtheorem{theorem}{Theorem} +\newtheorem{definition}{Definition} + +% Links in pdf +\usepackage{color} +\newcommand{\blue}{ \color{blue} } +\definecolor{linkcol}{rgb}{0,0,0.4} +\definecolor{citecol}{rgb}{0.5,0,0} + +\hypersetup +{ +colorlinks=true, +linkcolor=linkcol, +citecolor=citecol, +urlcolor=linkcol +} + +% nicer backref links +\renewcommand*{\backref}[1]{} +\renewcommand*{\backrefalt}[4]{% +\ifcase #1 % +(Not cited.)% +\or +(Cited on page~#2.)% +\else +(Cited on pages~#2.)% +\fi} +\renewcommand*{\backrefsep}{, } +\renewcommand*{\backreftwosep}{ and~} +\renewcommand*{\backreflastsep}{ and~} + +% TODO: set some indent +% \parindent 0cm + +\newcommand{\clearemptydoublepage}{\newpage{\pagestyle{empty}\cleardoublepage}} diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex new file mode 100644 index 0000000..bf39154 --- /dev/null +++ b/LWB/propagators/main.tex @@ -0,0 +1,166 @@ +\input{./header.tex} + +\begin{document} + +\input{sections/title.tex} + +\clearemptydoublepage + +\mbox{} +\vfill +\input{sections/abstract.tex} +\vfill +\tableofcontents + +\clearpage + +\listoffigures +\listofalgorithms + +\clearemptydoublepage + +% \input{./introduction.tex} +% \input{./wavefunction_propagation.tex} +% \input{./wavepackets.tex} +% \input{./observables.tex} +% \input{./wavepacket_propagation.tex} +% \input{./results.tex} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% TODO: move to separate file when done + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\clearpage +\section{Introduction} + +Motivation - why is time propagation important? +Background + +\subsection{Quantum Time Propagation and Operator Splitting} + +how does one analytically propagate? +operators don't commute +operator splitting to the rescue + +The search for accurate time propagation schemes has become an area of research by itself, see papers of A and B + + +\clearpage +\section{Time evolution schemes} + +present a selection of propagators + +\subsection{Common building blocks for Quantum Time Propagators} +\subsection{Hagedorn Propagator} +\subsection{Semiclassical Propagator} +\subsection{Magnus Propagator} +\subsection{Pre764 Propagator} +\subsection{McL Propagators} + + + +\clearpage +\section{Implementation in C++} + +What was the main challenge? What was controllable (which code could be changed), what not? +Quick alternating o + +The challenges were twofold: +First, identifying of all the common basic building blocks and expressing all algorithms in terms of these components +Second, implementing all basic building blocks in a generic but efficient manner +Third, minimize the overhead for switching between function calls + + - encapsulate the functionality in a clean interface without loosing flexibility and + - write efficient/fast code + +Tell how each of these challenges were overcome + + +Motivation of software design choices + + - Why class structure/inheritance (explain hierarchy) + - CRTP / static polymorphism + - recognizing IP + - callback function + - implementation of intSplit (TU,UT, partial specialization) + - polymorphism through enable_if + + + + +\clearpage +\section{Benchmark / Case Study / Example} + + +\clearpage +\section{Conclusion} + +final observations +suggestions for future work + + + + + + + + + + + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +\clearemptydoublepage + +\appendix +% \input{./deriv_ew.tex} +% \input{./color_code.tex} +% \input{./ack.tex} + +\clearemptydoublepage + +\bibliographystyle{plain} +% \bibliography{mt,bt,chem,wp,own} + +\end{document} diff --git a/LWB/propagators/sections/abstract.tex b/LWB/propagators/sections/abstract.tex new file mode 100644 index 0000000..6ece1b1 --- /dev/null +++ b/LWB/propagators/sections/abstract.tex @@ -0,0 +1,5 @@ +\begin{abstract} + This article explains the central design decisions and most important code details related to the efficient implementation of time propagation for Hagedorn quantum wave packets in the WaveBlocks framework. + A short introduction to operator splitting and a summary of some of the most important time propagation schemes for quantum wave packets are also given. + Finally, some benchmarking results are presented in the context of an example of a ???Ddim Nlevel BLUB potential???. +\end{abstract} diff --git a/LWB/propagators/sections/title.tex b/LWB/propagators/sections/title.tex new file mode 100644 index 0000000..45eb953 --- /dev/null +++ b/LWB/propagators/sections/title.tex @@ -0,0 +1,52 @@ +\begin{titlepage} + \begin{center} + + \hfill + + \vspace{2cm} + + {\huge \textsc{Time Propagation\\ of Quantum Wave Packets\\[10pt]}} + {\Large \textsc{an efficient implementation in C++\\[10pt]}} + + \vspace{1cm} + + {\Large{Bachelor Thesis}} + + \vspace{1cm} + + {\emph{written by}} \\ + { + Matthias Untergassmair + } + + \vspace{1cm} + + {\emph{advised by}} \\ + { + Raoul Bourquin + } + + \vspace{1cm} + + {\emph{supervised by}} \\ + { + Dr. Vasile Gr\u{a}dinaru\\ + Prof. Dr. Ralf Hiptmair + } + + \vfill + + Seminar for Applied Mathematics\\ + ETH Zurich + + \vspace{.5cm} + + \emph{{Fall semester 2016}} + + \vspace{.5cm} + + \includegraphics[width=0.5\linewidth]{figures/eth-logo.pdf} + + \end{center} + +\end{titlepage} From 3ec70678adb3b0d84e2c1e838aebf7d3260d9842 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Mon, 30 Jan 2017 00:43:38 +0000 Subject: [PATCH 02/28] Semiclassical done --- LWB/propagators/Makefile | 18 ++++++ LWB/propagators/header.tex | 27 ++++++++- LWB/propagators/main.tex | 121 ++++++++++++++++++++++++++++++++++++- 3 files changed, 162 insertions(+), 4 deletions(-) create mode 100644 LWB/propagators/Makefile diff --git a/LWB/propagators/Makefile b/LWB/propagators/Makefile new file mode 100644 index 0000000..1224dff --- /dev/null +++ b/LWB/propagators/Makefile @@ -0,0 +1,18 @@ +PDFLATEX = pdflatex +BIBTEX = bibtex +SECTIONS = $(wildcard sections/*.tex) + +all: main.pdf + +main.pdf: main.tex header.tex $(SECTIONS) # figs + $(PDFLATEX) main.tex + $(BIBTEX) main.aux + $(PDFLATEX) main.tex + $(PDFLATEX) main.tex + +# figs: +# $(MAKE) -C figures + +clean: + $(MAKE) clean -C figures + rm -f *.aux *.bbl *.blg *.log *.out *.toc diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex index 5269639..615a107 100644 --- a/LWB/propagators/header.tex +++ b/LWB/propagators/header.tex @@ -70,7 +70,7 @@ % Math %%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{algorithm} -\usepackage{algorithmic} +% \usepackage{algorithmic} \usepackage{color} \usepackage{relsize} \usepackage[scaled=0.8]{beramono} @@ -148,3 +148,28 @@ % \parindent 0cm \newcommand{\clearemptydoublepage}{\newpage{\pagestyle{empty}\cleardoublepage}} + + +% Mycommands + +\usepackage{bm} + +\newcommand*{\im}{\ensuremath{\imath}} +\newcommand*{\eps}{\ensuremath{\varepsilon}} +\newcommand*{\del}{\ensuremath{\partial}} +\newcommand*{\bvec}[1]{\bm{#1}} +\newcommand*{\op}[1]{\hat{\mathcal{#1}}} +\newcommand*{\R}{\ensuremath{\mathbb{R}}} +\newcommand*{\C}{\ensuremath{\mathbb{C}}} +\newcommand*{\K}{\ensuremath{\mathcal{K}}} +\newcommand*{\upic}{\ensuremath{u[\Pi,\bm{c}]}} +\newcommand*{\dt}{\ensuremath{\delta t}} +\newcommand*{\Dt}{\ensuremath{\Delta t}} + +\usepackage{algorithm} +\usepackage{verbatim} +% \usepackage{physics} +\usepackage{amssymb} +\usepackage{bm} +% \usepackage[noend]{algpseudocode} +\usepackage{algpseudocode} diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index bf39154..8cb5a0d 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -39,15 +39,59 @@ \clearpage \section{Introduction} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\cite{GH_convsemiclassical} +Semiclassical time-dependent Schrödinger equation +% +\begin{align} + \im \eps^2 \; \del_t \psi (\bvec{x},t) = \op{H} (\eps) \; \psi (\bvec{x},t) +\end{align} +% +with wave function $\psi(\bvec{x},t)$ depending on the spatial variables $\bvec{x} = (x_1,\dots,x_D) \in \R^D$ and the time variable $t \in \R$. + +Hamiltonian +\begin{align} + \op{H} = \op{T} + \op{V} = \op{T} + \op{U} + \op{W} +\end{align} +\begin{align} + \op{T} &= - \frac{\eps^2}{2} \Delta_x \\ + \op{V} &= V(x) +\end{align} + +Small semiclassical parameter $\eps$ plays an important role in the stability of numerical schemes + +Classical dynamics in the limit $\eps \rightarrow 0$, quantum mechanics for $\eps = 1$. + +In particular, as underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\sim \eps^{-2})$ for Lie-Trotter splitting, Strang splitting and other related methods. + Motivation - why is time propagation important? Background +Describe what is going to happen in the rest of the paper +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \subsection{Quantum Time Propagation and Operator Splitting} + how does one analytically propagate? operators don't commute operator splitting to the rescue + +The Hamiltonian is split into three parts +where $\op{U}(\bvec{q}(t),\bvec{x})$ is the second order Taylor approximation of the potential $\op{V}$ around $\bvec{q}(t)$ and $\op{W}(\bvec{q}(t),\bvec{x})$ is the corresponding remainder. + +$\op{V} = V(\bvec{x}) = U(\bvec{q},\bvec{x}) + W(\bvec{q},\bvec{x})$ + +\begin{align} + U(\bvec{q},\bvec{x}) &:= V(\bvec{q}) + \nabla V(\bvec{q}) (\bvec{x}-\bvec{q}) + + \frac{1}{2} (\bvec{x}-\bvec{q})^T \nabla^2 V(\bvec{q}) (\bvec{x}-\bvec{q}) \\ + W(\bvec{q},\bvec{x}) &:= V(\bvec{x}) - U(\bvec{q},\bvec{x}) +\end{align} + +The kinetic part $\op{T}$ and the quadratic part of the potential $\op{U}$ can be integrated exactly (see \cite{FGL_semiclassical_dynamics}). + + The search for accurate time propagation schemes has become an area of research by itself, see papers of A and B @@ -57,10 +101,80 @@ \section{Time evolution schemes} present a selection of propagators \subsection{Common building blocks for Quantum Time Propagators} + + \subsection{Hagedorn Propagator} +% +\begin{algorithm}[h] + \caption{Single timestep with Hagedorn propagator} + \label{alg:hagedorn} + \begin{algorithmic} + \Procedure{Hagedorn.propagate}{$\Dt$} + \State \Call{stepT}{$\frac{\Dt}{2}$} + \State \Call{stepU}{$\Dt$} + \State \Call{stepW}{$\Dt$} + \State \Call{stepT}{$\frac{\Dt}{2}$} + \EndProcedure + \end{algorithmic} +\end{algorithm} + + \subsection{Semiclassical Propagator} +\label{sub:semiclassical_propagator} +% +The central idea of the semiclassical splitting, as introduced in \cite{GH_convsemiclassical}, +is to reduce the errors in $\Pi$ and $S$ by splitting the propagation with operators +$\op{T}$ and $\op{U}$ into many smaller, alternating steps. +Not only does this reduce the dominating error\footnote{for small $\eps$, +the main source of error lies in the updating of $\Pi$ and $S$}, +but it also does not require any additional, expensive evaluations of the operator $\op{W}$. +The extra computational cost caused by a higher order splitting for updating $\Pi$ and $S$ +is negligible compared to the propagation with $\op{W}$ which requires numerical integration. +\par\medskip +% +In addition, due to the numerical properties of the semiclassical splitting, +it even allows to take larger timesteps $\Dt$ than conventional +splitting methods like the YL-splitting, while maintaining the same error. +\par\medskip +% +Finally and most importantly, the error is no longer proportional to $1/\eps^2$ but instead +scales linearly in the semiclassical parameter $\eps$, +meaning that a smaller $\eps$ will now reduce the error instead of increasing it. +The error scales with $\eps (\Delta t)^2$ for the semiclassical splitting using the Y-splitting, +but the dependency on the timestep $\Delta t$ can be improved even further by using different +splittings which effectively corresponds to higher order coefficient pairs $w_T$ and $w_U$. +\par\medskip +% +The pseudo-algorithm for the semiclassical propagator is given in \ref{alg:semiclassical} and +further details can be found in the original paper \cite{GH_convsemiclassical}. +% +\begin{algorithm}[h] + \caption{Single timestep with Semiclassical propagator} + \label{alg:semiclassical} + \begin{algorithmic} + \Procedure{Semiclassical.propagate}{$\Dt$} + \State + \State $M := \lceil 1 + \frac{\sqrt{\Dt}}{\eps^{3/4}} \rceil$ + \Comment{Divide $\Dt$ into smaller steps} + \State + \State \Call{intSplit}{$\frac{\Dt}{2}, \frac{M}{2}, \{ w_T, w_U \}$} + \Comment{$M/2$ split steps with $T+U$} + \State \Call{stepW}{$\Dt$} + \Comment{Single step with $W$} + \State \Call{intSplit}{$\frac{\Dt}{2}, \frac{M}{2}, \{ w_T, w_U \}$} + \Comment{$M/2$ split steps with $T+U$} + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} + + \subsection{Magnus Propagator} + + \subsection{Pre764 Propagator} + + \subsection{McL Propagators} @@ -68,6 +182,7 @@ \subsection{McL Propagators} \clearpage \section{Implementation in C++} +\cite{libwaveblocks} What was the main challenge? What was controllable (which code could be changed), what not? Quick alternating o @@ -85,11 +200,11 @@ \section{Implementation in C++} Motivation of software design choices - Why class structure/inheritance (explain hierarchy) - - CRTP / static polymorphism + - CRTP / static polymorphism \cite{C_CRTP} - recognizing IP - callback function - implementation of intSplit (TU,UT, partial specialization) - - polymorphism through enable_if + - polymorphism through enable-if @@ -161,6 +276,6 @@ \section{Conclusion} \clearemptydoublepage \bibliographystyle{plain} -% \bibliography{mt,bt,chem,wp,own} +\bibliography{references,mt,bt,chem,wp,own} \end{document} From 07de1f08c3159699b998738281e6c83a5744e81d Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Mon, 30 Jan 2017 23:27:21 +0000 Subject: [PATCH 03/28] Hagedorn Propagator --- LWB/propagators/header.tex | 26 +++-- LWB/propagators/main.tex | 205 +++++++++++++++++++++++++++++++++---- 2 files changed, 203 insertions(+), 28 deletions(-) diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex index 615a107..ced3091 100644 --- a/LWB/propagators/header.tex +++ b/LWB/propagators/header.tex @@ -154,21 +154,29 @@ \usepackage{bm} -\newcommand*{\im}{\ensuremath{\imath}} -\newcommand*{\eps}{\ensuremath{\varepsilon}} -\newcommand*{\del}{\ensuremath{\partial}} -\newcommand*{\bvec}[1]{\bm{#1}} -\newcommand*{\op}[1]{\hat{\mathcal{#1}}} -\newcommand*{\R}{\ensuremath{\mathbb{R}}} +% \newcommand*{\op}[1]{\hat{\mathcal{#1}}} \newcommand*{\C}{\ensuremath{\mathbb{C}}} +\newcommand*{\Dt}{\ensuremath{\Delta t}} \newcommand*{\K}{\ensuremath{\mathcal{K}}} -\newcommand*{\upic}{\ensuremath{u[\Pi,\bm{c}]}} +\newcommand*{\R}{\ensuremath{\mathbb{R}}} +\newcommand*{\bmat}[1]{\bm{#1}} +\newcommand*{\bvec}[1]{\bm{#1}} +\newcommand*{\conj}[1]{\overline{#1}} +\newcommand*{\del}{\ensuremath{\partial}} +\newcommand*{\dif}{\ensuremath{d}} \newcommand*{\dt}{\ensuremath{\delta t}} -\newcommand*{\Dt}{\ensuremath{\Delta t}} +\newcommand*{\eps}{\ensuremath{\varepsilon}} +\newcommand*{\im}{\ensuremath{\imath}} +\newcommand*{\opH}{\mathcal{H}} +\newcommand*{\opT}{\mathcal{T}} +\newcommand*{\opU}{\mathcal{U}} +\newcommand*{\opV}{\mathcal{V}} +\newcommand*{\opW}{\mathcal{W}} +\newcommand*{\upic}{\ensuremath{u[\Pi,\bm{c}]}} \usepackage{algorithm} \usepackage{verbatim} -% \usepackage{physics} +\usepackage{physics} \usepackage{amssymb} \usepackage{bm} % \usepackage[noend]{algpseudocode} diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 8cb5a0d..81134b1 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -44,18 +44,19 @@ \section{Introduction} Semiclassical time-dependent Schrödinger equation % \begin{align} - \im \eps^2 \; \del_t \psi (\bvec{x},t) = \op{H} (\eps) \; \psi (\bvec{x},t) + \label{math:tdse} + \im \eps \; \del_t \psi (\bvec{x},t) = \opH (\eps) \; \psi (\bvec{x},t) \end{align} % with wave function $\psi(\bvec{x},t)$ depending on the spatial variables $\bvec{x} = (x_1,\dots,x_D) \in \R^D$ and the time variable $t \in \R$. Hamiltonian \begin{align} - \op{H} = \op{T} + \op{V} = \op{T} + \op{U} + \op{W} + \opH = \opT + \opV = \opT + \opU + \opW \end{align} \begin{align} - \op{T} &= - \frac{\eps^2}{2} \Delta_x \\ - \op{V} &= V(x) + \opT &= - \sum_{j=1}^D \frac{\eps^2}{2m_j} \frac{\del^2}{\del x_j^2} \\ + \opV &= V(x) \end{align} Small semiclassical parameter $\eps$ plays an important role in the stability of numerical schemes @@ -68,10 +69,32 @@ \section{Introduction} Background Describe what is going to happen in the rest of the paper + + +Hagedorn wavepackets explained in \cite{FGL_semiclassical_dynamics} + +The solution to the Schrödinger equation can be approximated as a finite linear combination if Hagedorn function \cite{FGL_semiclassical_dynamics} +\begin{align} + \label{math:hagedornwp} + \psi(x,t) \approx u(x,t) + = e^{\im S(t)/\eps} \sum_{k \in \K} c_k(t) \varphi_k^\eps [ q(t),p(t),Q(t),P(t) ] (x) +\end{align} + +$\K$ finite multi index set + +As the idea behind the splitting the Hamiltonian operator is so central to the time propagation approach used here, we repeat the most important findings from \cite{FGL_semiclassical_dynamics}: +\begin{itemize} + \item The free linear Schrödinger equation ($\opV=0$) can be solved exactly and the wavepacket remains in Hagedorn wavepacket form \ref{math:hagedornwp}. + For time propagation, only the parameters $\bvec{\Pi},S$ need to be updated, the coefficients $\{c_k\}_{k \in \K}$ remain unchanged. + \item The Schrödinger equation \ref{math:tdse} can be solved exactly in a pure quadratic potential, i.e. in a potential $\opV=U(\bvec{x})$ with $\opT=0$. + Again, time propagation only affects the parameters $\bvec{\Pi},S$, not the coefficients $\{c_k\}_{k \in \K}$. + \item In the case of an arbitrary potential $\opV = W(\bvec{x})$ that is not quadratic, a set of Galerkin functions can be propagated by adapting the coefficients $\{c_k\}_{k \in \K}$ without changing the parameters $\bvec{\Pi},S$. +\end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Quantum Time Propagation and Operator Splitting} +\subsection{Quantum Time Propagation, Operator Splitting and Hagedorn Propagator} +check propositions in hagedorn paper. how does one analytically propagate? operators don't commute @@ -79,9 +102,9 @@ \subsection{Quantum Time Propagation and Operator Splitting} The Hamiltonian is split into three parts -where $\op{U}(\bvec{q}(t),\bvec{x})$ is the second order Taylor approximation of the potential $\op{V}$ around $\bvec{q}(t)$ and $\op{W}(\bvec{q}(t),\bvec{x})$ is the corresponding remainder. +where $\opU(\bvec{q}(t),\bvec{x})$ is the second order Taylor approximation of the potential $\opV$ around $\bvec{q}(t)$ and $\opW(\bvec{q}(t),\bvec{x})$ is the corresponding remainder. -$\op{V} = V(\bvec{x}) = U(\bvec{q},\bvec{x}) + W(\bvec{q},\bvec{x})$ +$\opV = V(\bvec{x}) = U(\bvec{q},\bvec{x}) + W(\bvec{q},\bvec{x})$ \begin{align} U(\bvec{q},\bvec{x}) &:= V(\bvec{q}) + \nabla V(\bvec{q}) (\bvec{x}-\bvec{q}) @@ -89,7 +112,7 @@ \subsection{Quantum Time Propagation and Operator Splitting} W(\bvec{q},\bvec{x}) &:= V(\bvec{x}) - U(\bvec{q},\bvec{x}) \end{align} -The kinetic part $\op{T}$ and the quadratic part of the potential $\op{U}$ can be integrated exactly (see \cite{FGL_semiclassical_dynamics}). +The kinetic part $\opT$ and the quadratic part of the potential $\opU$ can be integrated exactly (see \cite{FGL_semiclassical_dynamics}). The search for accurate time propagation schemes has become an area of research by itself, see papers of A and B @@ -101,19 +124,85 @@ \section{Time evolution schemes} present a selection of propagators \subsection{Common building blocks for Quantum Time Propagators} +% + +General notes: what variables are accessible? + + +The time stepping with operators $\opT$ and $\opV = U(\op{x})$ follows directly from the propositions in \cite{FGL_semiclassical_dynamics} and is outlined in the algorithms \ref{alg:stepT} and \ref{alg:stepU} respectively. + +As pointed out in the introduction, the time propagation for non-quadratic potentials $W(\vec{x})$ can be achieved by updating the coefficients $\{ c_k \}_{k \in \K}$. +The update rule is +% +\begin{align} + \bvec{c}(t) = \exp \left( - \frac{\im t}{\eps} \bmat{F} \right) \bvec{c}(0) +\end{align} +% +where the matrix $\bmat{F} = \{ f_{k,l} \}_{k,l \in \K}$ has entries +% +\begin{align} + f_{k,l} = \matrixel{\varphi_k}{W}{\varphi_l} + = \int_{\R^N} \conj{\varphi_k(\bvec{x})} W(\bvec{x}) \varphi_l(\bvec{x}) \; \dif \bvec{x} +\end{align} + + +\begin{algorithm}[h] + \caption{Propagate with Kinetic Energy Operator $\opT$} + \label{alg:stepT} + \begin{algorithmic} + \Procedure{stepT}{$\eta$} + \State $q = q + \eta M^{-1} p$ + \State $Q = Q + \eta M^{-1} P$ + \State $S = S + \frac{\eta}{2} p^T M^{-1} p$ + \EndProcedure + \end{algorithmic} +\end{algorithm} +% +\begin{algorithm}[h] + \caption{Propagate with (Quadratic) Potential Energy Operator} + \label{alg:stepU} + \begin{algorithmic} + \Procedure{stepU}{$\eta$} + \State $p = p - \eta \nabla V (q)$ + \State $P = P - \eta \nabla^2 V (q) Q$ + \State $S = S - \eta V (q)$ + \EndProcedure + \end{algorithmic} +\end{algorithm} + + + + + + + + + + + \subsection{Hagedorn Propagator} +\label{sub:hagedorn_propagator} % +The Hagedorn propagator, shown in algorithm \ref{alg:hagedorn}, is one of the simplest propagators that can be built by exploiting the numerical aspects discussed above. +As pointed out in \cite{FGL_semiclassical_dynamics}, this simple time stepping scheme has many beneficial properties like the preservation of the $L^2$ norm of the wavepacket, time reversibility and stability in the classical limit $\eps \rightarrow 0$. +Also, in the limit of $\K$ approaching the full basis set, the variational approximation used for the propagation with the non-quadratic part $\opW$ becomes exact. \begin{algorithm}[h] \caption{Single timestep with Hagedorn propagator} \label{alg:hagedorn} \begin{algorithmic} - \Procedure{Hagedorn.propagate}{$\Dt$} + \Procedure{Hagedorn.Propagate}{$\Dt$} + \State \State \Call{stepT}{$\frac{\Dt}{2}$} + \Comment{Step of size $\Dt/2$ with $\opT$} \State \Call{stepU}{$\Dt$} + \Comment{Step of size $\Dt$ with $\opU$} \State \Call{stepW}{$\Dt$} + \Comment{Step of size $\Dt$ with $\opW$} \State \Call{stepT}{$\frac{\Dt}{2}$} + \Comment{Step of size $\Dt/2$ with $\opT$} + \State \EndProcedure \end{algorithmic} \end{algorithm} @@ -123,13 +212,9 @@ \subsection{Semiclassical Propagator} \label{sub:semiclassical_propagator} % The central idea of the semiclassical splitting, as introduced in \cite{GH_convsemiclassical}, -is to reduce the errors in $\Pi$ and $S$ by splitting the propagation with operators -$\op{T}$ and $\op{U}$ into many smaller, alternating steps. -Not only does this reduce the dominating error\footnote{for small $\eps$, -the main source of error lies in the updating of $\Pi$ and $S$}, -but it also does not require any additional, expensive evaluations of the operator $\op{W}$. -The extra computational cost caused by a higher order splitting for updating $\Pi$ and $S$ -is negligible compared to the propagation with $\op{W}$ which requires numerical integration. +is to split the propagation with operators $\opT$ and $\opU$ into many smaller, alternating steps, thereby reducing the dominating error\footnote{for small $\eps$, +the main source of error lies in the updating of $\Pi$ and $S$} in the update of $\bvec{\Pi}$ and $S$. +While there is some additional computational cost caused by a higher number of updates for the parameters $\bvec{\Pi}$ and $S$, the extra effort is usually negligible compared to the propagation with $\opW$ which requires numerical evaluation of multi-dimensional integrals. \par\medskip % In addition, due to the numerical properties of the semiclassical splitting, @@ -152,7 +237,7 @@ \subsection{Semiclassical Propagator} \caption{Single timestep with Semiclassical propagator} \label{alg:semiclassical} \begin{algorithmic} - \Procedure{Semiclassical.propagate}{$\Dt$} + \Procedure{Semiclassical.Propagate}{$\Dt$} \State \State $M := \lceil 1 + \frac{\sqrt{\Dt}}{\eps^{3/4}} \rceil$ \Comment{Divide $\Dt$ into smaller steps} @@ -170,13 +255,95 @@ \subsection{Semiclassical Propagator} \subsection{Magnus Propagator} - +\label{sub:magnus_propagator} +% +TODO: add comments to algorithm +\begin{algorithm}[h] + \caption{Single timestep with Magnus propagator} + \label{alg:magnus} + \begin{algorithmic} + \Procedure{Magnus.Propagate}{$\Dt$} + \State + \State $h_1 = \frac{3-\sqrt{3}}{6} \Dt$, $h_2 = \frac{\sqrt{3}}{6} \Dt$ + \Comment Gauss-Legendre coefficients on $[0,\Dt]$ + \State $M_{k} = 1+\sqrt{h_{k} \eps^{-3/8}}, \quad k=1,2$ + \State + \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} + \State $\bmat{A}_1 = - \im \frac{\Dt}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} + \State \Call{intSplit}{$h_2, M_2, \{w_T, w_U\}$} + \State $\bmat{A}_2 = - \im \frac{\Dt}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} + \State $\bmat{\Sigma} = \frac{1}{2} (\bmat{A}_1 + \bmat{A}_2) + \frac{\sqrt{3}}{12} (\bmat{A}_1 \cdot \bmat{A}_1 - \bmat{A}_2 \cdot \bmat{A}_1)$ + \State $\bvec{c} = \exp{ \left( \bmat{\Sigma} \right) } \bvec{c}$ + \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} \subsection{Pre764 Propagator} +\label{sub:pre764_propagator} +% +TODO: add comments to algorithm +\begin{algorithm}[h] + \caption{Single timestep with Pre764 propagator} + \label{alg:pre764} + \begin{algorithmic} + \Procedure{Pre764.PrePropagate}{$\Dt$} + \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \For{$j=0,...,v-1$} \Comment $v=6$ for Pre(7,6,4) + \State \Call{intSplit}{$-Z_j \Dt, M, \{w_T, w_U\}$} + \State \Call{stepW}{$\upic, -Y_j \Dt$} + \EndFor + \EndProcedure + \\\hrulefill + \Procedure{Pre764.Propagate}{$\Dt$} + \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \For{$j=0,...,k-1$} \Comment $k=4$ for Pre(7,6,4) + \State \Call{stepW}{$\alpha_j \Dt$} + \State \Call{intSplit}{$\beta_j \Dt, M, \{w_T, w_U\}$} + \EndFor + \EndProcedure + \\\hrulefill + \Procedure{Pre764.PostPropagate}{$\Dt$} + \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \For{$j=v-1,...,0$} \Comment $v=6$ for Pre(7,6,4) + \State \Call{stepW}{$Y_j \Dt$} + \State \Call{intSplit}{$Z_j \Dt, M, \{w_T, w_U\}$} + \EndFor + \EndProcedure + \end{algorithmic} +\end{algorithm} \subsection{McL Propagators} - +\label{sub:mcl_propagator} +% +TODO: add comments to algorithm +TODO: Add 84 algorithm +TODO: compute M, compute dt +\begin{algorithm}[h] + \caption{Single timestep with McL propagators} + \label{alg:mcl} + \begin{algorithmic} + \Procedure{McL42.Propagate}{$\Dt$} + \State \Call{intSplit}{$A_0 \dt, M, \{w_T, w_U\}$} + \State \Call{stepW}{$B_0 \dt$} + \State \Call{intSplit}{$A_1 \dt, M, \{w_T, w_U\}$} + \State \Call{stepW}{$B_1 \dt$} + \State \Call{intSplit}{$A_2 \dt, M, \{w_T, w_U\}$} + \EndProcedure + \\\hrulefill + \Procedure{McL84.Propagate}{$\Dt$} + \State \Call{intSplit}{$A_0 \dt, M, \{w_T, w_U\}$} + \State \Call{stepW}{$B_0 \dt$} + \State \Call{intSplit}{$A_1 \dt, M, \{w_T, w_U\}$} + \State \Call{stepW}{$B_1 \dt$} + \State \Call{intSplit}{$A_2 \dt, M, \{w_T, w_U\}$} + \State \Call{stepW}{$B_2 \dt$} + \State \Call{intSplit}{$A_3 \dt, M, \{w_T, w_U\}$} + \EndProcedure + \end{algorithmic} +\end{algorithm} \clearpage From 8269d974da6c99729ab894bb08277b8e6b095db2 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Thu, 2 Feb 2017 00:29:44 +0100 Subject: [PATCH 04/28] Magnus Propagator --- LWB/propagators/main.tex | 37 ++++++++++++++++++++++++++++++++++--- references/propagators.bib | 30 ++++++++++++++++++++++++++++++ 2 files changed, 64 insertions(+), 3 deletions(-) create mode 100644 references/propagators.bib diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 81134b1..cfa6093 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -39,6 +39,7 @@ \clearpage \section{Introduction} +Summary of the structure of the remaining report %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \cite{GH_convsemiclassical} Semiclassical time-dependent Schrödinger equation @@ -257,24 +258,54 @@ \subsection{Semiclassical Propagator} \subsection{Magnus Propagator} \label{sub:magnus_propagator} % -TODO: add comments to algorithm +As was noted by Magnus in \cite{Magnus1954}, the solution to a differential equation of the form +% +\begin{align} + y'(t) = a(t) y(t) \qquad t \ge 0 +\end{align} +% +can be written as +% +\begin{align} + y(t) = e^{\sigma (t)} y_0 +\end{align} +% +where $\sigma (t)$ is an infinite sum of iterated integrals and commutators, also known as the Magnus series. +\par\medskip +% +In order to approximate this solution, one can take only a finite number of terms from this series whereby a truncation error is committed +(additional error sources in this method are the discretization of integrals and the approximation of matrix exponentials). +% +This method has several beneficial numerical properties that are described in \cite{Iserles1999}. +In particular, for solutions that evolve within a Lie group, the same holds for the approximate numerical solution calculated through a truncated Magnus series. +It was also shown in the same paper that the Magnus series can compete with - and in fact may even outplay - classical schemes like Runge-Kutta or Gauss-Legendre. +Moreover, the numerical stability and good performance of the Magnus propagator are not limited to problems where the solution evolves within a Lie Group, but also apply to various problems when this is not the case. + \begin{algorithm}[h] \caption{Single timestep with Magnus propagator} \label{alg:magnus} \begin{algorithmic} \Procedure{Magnus.Propagate}{$\Dt$} \State - \State $h_1 = \frac{3-\sqrt{3}}{6} \Dt$, $h_2 = \frac{\sqrt{3}}{6} \Dt$ + \State $h_1 = \frac{3-\sqrt{3}}{6} \Dt$, $h_2 = \frac{2\sqrt{3}}{6} \Dt$ \Comment Gauss-Legendre coefficients on $[0,\Dt]$ \State $M_{k} = 1+\sqrt{h_{k} \eps^{-3/8}}, \quad k=1,2$ + \Comment number of timesteps for splitting \State \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} + \Comment advance till $\frac{3-\sqrt{3}}{6} \Dt$ \State $\bmat{A}_1 = - \im \frac{\Dt}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} + \Comment temporarily store interaction matrix \State \Call{intSplit}{$h_2, M_2, \{w_T, w_U\}$} + \Comment advance till $\frac{3+\sqrt{3}}{6} \Dt$ \State $\bmat{A}_2 = - \im \frac{\Dt}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} + \Comment temporarily store interaction matrix \State $\bmat{\Sigma} = \frac{1}{2} (\bmat{A}_1 + \bmat{A}_2) + \frac{\sqrt{3}}{12} (\bmat{A}_1 \cdot \bmat{A}_1 - \bmat{A}_2 \cdot \bmat{A}_1)$ + \Comment compute $\sigma (t)$ \State $\bvec{c} = \exp{ \left( \bmat{\Sigma} \right) } \bvec{c}$ + \Comment update coefficients with Magnus approximation \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} + \Comment advance till $\frac{6}{6} \Dt$ \State \EndProcedure \end{algorithmic} @@ -443,6 +474,6 @@ \section{Conclusion} \clearemptydoublepage \bibliographystyle{plain} -\bibliography{references,mt,bt,chem,wp,own} +\bibliography{references,mt,bt,chem,wp,own,propagators} \end{document} diff --git a/references/propagators.bib b/references/propagators.bib new file mode 100644 index 0000000..e42ddc1 --- /dev/null +++ b/references/propagators.bib @@ -0,0 +1,30 @@ +@Article{Iserles1999, + author="Iserles, A. + and Marthinsen, A. + and N{\o}rsett, S. P.", + title="On the Implementation of the Method of Magnus Series for Linear Differential Equations", + journal="BIT Numerical Mathematics", + year="1999", + volume="39", + number="2", + pages="281--304", + issn="1572-9125", + doi="10.1023/A:1022393913721", + url="http://dx.doi.org/10.1023/A:1022393913721" +} + + +@article {Magnus1954, + author = {Magnus, Wilhelm}, + title = {On the exponential solution of differential equations for a linear operator}, + journal = {Communications on Pure and Applied Mathematics}, + volume = {7}, + number = {4}, + publisher = {Wiley Subscription Services, Inc., A Wiley Company}, + issn = {1097-0312}, + url = {http://dx.doi.org/10.1002/cpa.3160070404}, + doi = {10.1002/cpa.3160070404}, + pages = {649--673}, + year = {1954} +} + From de187a9e4251a2365b223572e8bec75f10854cb4 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Thu, 2 Feb 2017 13:48:21 +0100 Subject: [PATCH 05/28] Preprocessing Propagator --- LWB/propagators/main.tex | 72 +++++++++++++++++++++++++++++++------- references/propagators.bib | 55 +++++++++++++++++++++++++++-- 2 files changed, 111 insertions(+), 16 deletions(-) diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index cfa6093..3bcb6ff 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -132,7 +132,7 @@ \subsection{Common building blocks for Quantum Time Propagators} The time stepping with operators $\opT$ and $\opV = U(\op{x})$ follows directly from the propositions in \cite{FGL_semiclassical_dynamics} and is outlined in the algorithms \ref{alg:stepT} and \ref{alg:stepU} respectively. -As pointed out in the introduction, the time propagation for non-quadratic potentials $W(\vec{x})$ can be achieved by updating the coefficients $\{ c_k \}_{k \in \K}$. +As pointed out in the introduction, the time propagation for non-quadratic potentials $W(\bvec{x})$ can be achieved by updating the coefficients $\{ c_k \}_{k \in \K}$. The update rule is % \begin{align} @@ -151,6 +151,7 @@ \subsection{Common building blocks for Quantum Time Propagators} \caption{Propagate with Kinetic Energy Operator $\opT$} \label{alg:stepT} \begin{algorithmic} + \State \Procedure{stepT}{$\eta$} \State $q = q + \eta M^{-1} p$ \State $Q = Q + \eta M^{-1} P$ @@ -163,6 +164,7 @@ \subsection{Common building blocks for Quantum Time Propagators} \caption{Propagate with (Quadratic) Potential Energy Operator} \label{alg:stepU} \begin{algorithmic} + \State \Procedure{stepU}{$\eta$} \State $p = p - \eta \nabla V (q)$ \State $P = P - \eta \nabla^2 V (q) Q$ @@ -193,6 +195,7 @@ \subsection{Hagedorn Propagator} \caption{Single timestep with Hagedorn propagator} \label{alg:hagedorn} \begin{algorithmic} + \State \Procedure{Hagedorn.Propagate}{$\Dt$} \State \State \Call{stepT}{$\frac{\Dt}{2}$} @@ -231,13 +234,14 @@ \subsection{Semiclassical Propagator} splittings which effectively corresponds to higher order coefficient pairs $w_T$ and $w_U$. \par\medskip % -The pseudo-algorithm for the semiclassical propagator is given in \ref{alg:semiclassical} and +The steps for the semiclassical propagator are shown in algorithm \ref{alg:semiclassical} and further details can be found in the original paper \cite{GH_convsemiclassical}. % \begin{algorithm}[h] \caption{Single timestep with Semiclassical propagator} \label{alg:semiclassical} \begin{algorithmic} + \State \Procedure{Semiclassical.Propagate}{$\Dt$} \State \State $M := \lceil 1 + \frac{\sqrt{\Dt}}{\eps^{3/4}} \rceil$ @@ -267,24 +271,30 @@ \subsection{Magnus Propagator} can be written as % \begin{align} + \label{math:magnussolution} y(t) = e^{\sigma (t)} y_0 \end{align} % where $\sigma (t)$ is an infinite sum of iterated integrals and commutators, also known as the Magnus series. +The idea behind the Magnus approximation was extensively studied using the Baker-Campbell-Hausdorff (BCH) formula and rooted trees techniques, more details can be found in \cite{Blanes2006}, \cite{Blanes2000}, \cite{Iserles1999}. \par\medskip % -In order to approximate this solution, one can take only a finite number of terms from this series whereby a truncation error is committed +In order to approximate the solution from equation \ref{math:magnussolution}, one can take only a finite number of terms from this series whereby a truncation error is committed (additional error sources in this method are the discretization of integrals and the approximation of matrix exponentials). % -This method has several beneficial numerical properties that are described in \cite{Iserles1999}. +This method of the Magnus Propagator has several beneficial numerical properties that are described in \cite{Iserles1999}. In particular, for solutions that evolve within a Lie group, the same holds for the approximate numerical solution calculated through a truncated Magnus series. It was also shown in the same paper that the Magnus series can compete with - and in fact may even outplay - classical schemes like Runge-Kutta or Gauss-Legendre. +Although the method is not a symplectic scheme in the usual sense, \cite{Iserles1999} has shown that in practical applications it conserves the Hamiltonian energy just as well as symplectic integrators. +\par\medskip Moreover, the numerical stability and good performance of the Magnus propagator are not limited to problems where the solution evolves within a Lie Group, but also apply to various problems when this is not the case. +Algorithm \ref{alg:magnus} shows the \emph{MG4} method as presented in \cite{Iserles1999} and implemented in C++ in the scope of this work. \begin{algorithm}[h] \caption{Single timestep with Magnus propagator} \label{alg:magnus} \begin{algorithmic} + \State \Procedure{Magnus.Propagate}{$\Dt$} \State \State $h_1 = \frac{3-\sqrt{3}}{6} \Dt$, $h_2 = \frac{2\sqrt{3}}{6} \Dt$ @@ -294,16 +304,16 @@ \subsection{Magnus Propagator} \State \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} \Comment advance till $\frac{3-\sqrt{3}}{6} \Dt$ - \State $\bmat{A}_1 = - \im \frac{\Dt}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} + \State $\bmat{A}_1 = - \frac{\im}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} \Comment temporarily store interaction matrix \State \Call{intSplit}{$h_2, M_2, \{w_T, w_U\}$} \Comment advance till $\frac{3+\sqrt{3}}{6} \Dt$ - \State $\bmat{A}_2 = - \im \frac{\Dt}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} + \State $\bmat{A}_2 = - \frac{\im}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} \Comment temporarily store interaction matrix - \State $\bmat{\Sigma} = \frac{1}{2} (\bmat{A}_1 + \bmat{A}_2) + \frac{\sqrt{3}}{12} (\bmat{A}_1 \cdot \bmat{A}_1 - \bmat{A}_2 \cdot \bmat{A}_1)$ + \State $\bmat{\Sigma} = \frac{1}{2} \Dt (\bmat{A}_1 + \bmat{A}_2) + \frac{\sqrt{3}}{12} (\Dt)^2 (\bmat{A}_2 \cdot \bmat{A}_1 - \bmat{A}_1 \cdot \bmat{A}_2)$ \Comment compute $\sigma (t)$ - \State $\bvec{c} = \exp{ \left( \bmat{\Sigma} \right) } \bvec{c}$ - \Comment update coefficients with Magnus approximation + \State $\bvec{c} = \exp \left( \bmat{\Sigma} \right) \bvec{c}$ + \Comment update coefficients \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} \Comment advance till $\frac{6}{6} \Dt$ \State @@ -311,36 +321,72 @@ \subsection{Magnus Propagator} \end{algorithmic} \end{algorithm} -\subsection{Pre764 Propagator} +\subsection{Processing Propagators} \label{sub:pre764_propagator} % -TODO: add comments to algorithm +The idea of processing propagators is to carry out the time evolution with a Hamiltonian that is slightly perturbed from the initial one. +To achieve this, a pre and post processing step is applied. \\ +More formally, it holds that +% +\begin{align} + e^{- \Dt \opH (\Dt)} = e^P e^{- \Dt K} e^{-P} +\end{align} +% +where the pre and post processing steps are applied via the multiplication with matrices $e^P$ and $e^{-P}$ (also referred to as \emph{processors}). +While the processors only need to be applied once at the very beginning and end of the propagation (in order to return to the original Hamiltonian), the multiplication with the \emph{kernel} $e^{-\Dt K}$ is repeated $M$ times, once for every timestep. \\ +As a consequence, one wants to choose the processor $e^P$ in such a way that the evaluation of the kernel $e^{-\Dt K}$ is as simple as possible, requiring a minimum of expensive function evaluations. +\par\medskip +% +Unlike most commonly used integration schemes, the family of processing propagators are symplectic integrators. +They are not suited for every kind of Hamiltonian, and the work of Blanes, Casas and Ros in \cite{Blanes1999} gives a method for finding the neccessary conditions that need to be satisfied in order for processing methods to be applicable. +\par\medskip +% +Algorithm \ref{alg:pre764} presents the \emph{Pre764} processing method which is a ?sixth? order processing method that was derived in \cite{Blanes1999}. +% \begin{algorithm}[h] \caption{Single timestep with Pre764 propagator} \label{alg:pre764} \begin{algorithmic} + \State \Procedure{Pre764.PrePropagate}{$\Dt$} + \State \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \Comment compute number of time steps \For{$j=0,...,v-1$} \Comment $v=6$ for Pre(7,6,4) - \State \Call{intSplit}{$-Z_j \Dt, M, \{w_T, w_U\}$} + \State \Call{intSplit}{$-Z_j \Dt, M, \{w_T, w_U\}$} + \Comment $M$ alternating steps with $\opT$ and $\opU$ \State \Call{stepW}{$\upic, -Y_j \Dt$} + \Comment single step with $\opW$ \EndFor + \State \EndProcedure \\\hrulefill + \State \Procedure{Pre764.Propagate}{$\Dt$} + \State \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \Comment compute number of time steps \For{$j=0,...,k-1$} \Comment $k=4$ for Pre(7,6,4) \State \Call{stepW}{$\alpha_j \Dt$} + \Comment single step with $\opW$ \State \Call{intSplit}{$\beta_j \Dt, M, \{w_T, w_U\}$} + \Comment $M$ alternating steps with $\opT$ and $\opU$ \EndFor + \State \EndProcedure \\\hrulefill + \State \Procedure{Pre764.PostPropagate}{$\Dt$} + \State \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \Comment compute number of time steps \For{$j=v-1,...,0$} \Comment $v=6$ for Pre(7,6,4) - \State \Call{stepW}{$Y_j \Dt$} + \State \Call{stepW}{$Y_j \Dt$} + \Comment single step with $\opW$ \State \Call{intSplit}{$Z_j \Dt, M, \{w_T, w_U\}$} + \Comment $M$ alternating steps with $\opT$ and $\opU$ \EndFor + \State \EndProcedure \end{algorithmic} \end{algorithm} diff --git a/references/propagators.bib b/references/propagators.bib index e42ddc1..663647b 100644 --- a/references/propagators.bib +++ b/references/propagators.bib @@ -1,4 +1,4 @@ -@Article{Iserles1999, +@article{Iserles1999, author="Iserles, A. and Marthinsen, A. and N{\o}rsett, S. P.", @@ -13,8 +13,7 @@ @Article{Iserles1999 url="http://dx.doi.org/10.1023/A:1022393913721" } - -@article {Magnus1954, +@article{Magnus1954, author = {Magnus, Wilhelm}, title = {On the exponential solution of differential equations for a linear operator}, journal = {Communications on Pure and Applied Mathematics}, @@ -28,3 +27,53 @@ @article {Magnus1954 year = {1954} } +@article{Blanes2006, + title = "Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems ", + journal = "Applied Numerical Mathematics ", + volume = "56", + number = "12", + pages = "1519 - 1537", + year = "2006", + note = "", + issn = "0168-9274", + doi = "http://dx.doi.org/10.1016/j.apnum.2005.11.004", + url = "//www.sciencedirect.com/science/article/pii/S0168927405002163", + author = "S. Blanes and P.C. Moan", + keywords = "Time-dependent differential equations", + keywords = "Initial value problems", + keywords = "Numerical methods", + keywords = "Magnus expansion " +} + + + +@Article{Blanes2000, + author="S. Blanes, F. Casas and J. Ros", + title="Improved High Order Integrators Based on the Magnus Expansion", + journal="BIT Numerical Mathematics", + year="2000", + volume="40", + number="3", + pages="434--450", + issn="1572-9125", + doi="10.1023/A:1022311628317", + url="http://dx.doi.org/10.1023/A:1022311628317" +} + + + + +@article{Blanes1999, + author = {S. Blanes and F. Casas and J. Ros}, + title = {Symplectic Integration with Processing: A General Study}, + journal = {SIAM Journal on Scientific Computing}, + volume = {21}, + number = {2}, + pages = {711-727}, + year = {1999}, + doi = {10.1137/S1064827598332497}, + URL = { http://dx.doi.org/10.1137/S1064827598332497 }, + eprint = { http://dx.doi.org/10.1137/S1064827598332497 } +} + + From f756d7454e9f7c6a5171d2227c00e950b26c709b Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Thu, 2 Feb 2017 17:51:32 +0100 Subject: [PATCH 06/28] McL Propagators --- LWB/propagators/header.tex | 3 ++ LWB/propagators/main.tex | 65 +++++++++++++++++++++++++++----------- references/propagators.bib | 17 +++++++--- 3 files changed, 61 insertions(+), 24 deletions(-) diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex index ced3091..53d0e66 100644 --- a/LWB/propagators/header.tex +++ b/LWB/propagators/header.tex @@ -172,6 +172,9 @@ \newcommand*{\opU}{\mathcal{U}} \newcommand*{\opV}{\mathcal{V}} \newcommand*{\opW}{\mathcal{W}} +\newcommand*{\opA}{\mathcal{A}} +\newcommand*{\opB}{\mathcal{B}} +\newcommand*{\opX}{\mathcal{X}} \newcommand*{\upic}{\ensuremath{u[\Pi,\bm{c}]}} \usepackage{algorithm} diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 3bcb6ff..92fe91c 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -39,6 +39,13 @@ \clearpage \section{Introduction} +All based on composition methods + +methods that compose elementary flows/operators to build an approximation of high order + +In general / in the practical implementation, propagation with $\opT$ and $\opU$ is usually replaced by a series of smaller, alternating steps with the two operators + + Summary of the structure of the remaining report %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \cite{GH_convsemiclassical} @@ -93,7 +100,7 @@ \section{Introduction} \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Quantum Time Propagation, Operator Splitting and Hagedorn Propagator} +\subsection{Quantum Time Propagation, Operator Splitting} check propositions in hagedorn paper. @@ -395,29 +402,49 @@ \subsection{Processing Propagators} \subsection{McL Propagators} \label{sub:mcl_propagator} % -TODO: add comments to algorithm -TODO: Add 84 algorithm -TODO: compute M, compute dt +McLachlan (see \cite{McLachlan1995}) has investigated various ?symplectic? schemes for computing the effect of operators of the form $\opX = \opA + \epsilon \opB$ where $\opA$ describes an exactly solvable system and $\opB$ is a perturbation. +The factor $\epsilon$ is a small parameter that indicates the limited influence of $\opB$, not to be confused with the semiclassical parameter $\eps$. +\par\medskip +The \emph{McL} integration schemes are characterised by pairs of weighing coefficients $a_i$ and $b_i$ that allow to represent the exponential $e^{\opX}$ as a product +% +\begin{align} + e^\opX = \prod_i e^{b_i t \epsilon \opB} e^{a_i t \opA} +\end{align} +% +McLachlan also goes through the process of deriving the coefficients for the following schemes +\begin{align*} + &\text{\emph{McL(2s,2)}} &&\text{with error of order } \epsilon (\Dt)^{2s} + \epsilon^2 (\Dt)^2 \\ + &\text{\emph{McL(6,4)}} &&\text{with error of order } \epsilon (\Dt)^{6} + \epsilon^2 (\Dt)^4 \\ + &\text{\emph{McL(8,4)}} &&\text{with error of order } \epsilon (\Dt)^{8} + \epsilon^2 (\Dt)^4 +\end{align*} +% +Note that the error depends on two small parameters, the timestep $\Dt$ and the parameter $\epsilon$ that quantifies the influence of $\opB$. +\par\medskip +% +In our case, $\opX = \opH = \opT + \opV + \opW$, the computationally expensive operator $\opW$ takes the role of $\epsilon \opB$, while the remaining Hamiltonian $\opT + \opU$ corresponds to $\opA$. +Our preferred method will therefore be of the form $ABA$ because this minimizes the number of steps with $\opB = \opW$. +\par\medskip +% +The source code of this project contains the C++ implementation of the propagators \emph{McL(4,2)} (asymmetric) and \emph{McL(8,4)} (symmetric) that require only two respectively three evaluations of $\opW$ per step. + \begin{algorithm}[h] \caption{Single timestep with McL propagators} \label{alg:mcl} \begin{algorithmic} + \State \Procedure{McL42.Propagate}{$\Dt$} - \State \Call{intSplit}{$A_0 \dt, M, \{w_T, w_U\}$} - \State \Call{stepW}{$B_0 \dt$} - \State \Call{intSplit}{$A_1 \dt, M, \{w_T, w_U\}$} - \State \Call{stepW}{$B_1 \dt$} - \State \Call{intSplit}{$A_2 \dt, M, \{w_T, w_U\}$} - \EndProcedure - \\\hrulefill - \Procedure{McL84.Propagate}{$\Dt$} - \State \Call{intSplit}{$A_0 \dt, M, \{w_T, w_U\}$} - \State \Call{stepW}{$B_0 \dt$} - \State \Call{intSplit}{$A_1 \dt, M, \{w_T, w_U\}$} - \State \Call{stepW}{$B_1 \dt$} - \State \Call{intSplit}{$A_2 \dt, M, \{w_T, w_U\}$} - \State \Call{stepW}{$B_2 \dt$} - \State \Call{intSplit}{$A_3 \dt, M, \{w_T, w_U\}$} + \State + \State \Call{intSplit}{$A_0 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_0 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_1 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_1 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_2 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \EndProcedure \end{algorithmic} \end{algorithm} diff --git a/references/propagators.bib b/references/propagators.bib index 663647b..aa36bf1 100644 --- a/references/propagators.bib +++ b/references/propagators.bib @@ -45,8 +45,6 @@ @article{Blanes2006 keywords = "Magnus expansion " } - - @Article{Blanes2000, author="S. Blanes, F. Casas and J. Ros", title="Improved High Order Integrators Based on the Magnus Expansion", @@ -60,9 +58,6 @@ @Article{Blanes2000 url="http://dx.doi.org/10.1023/A:1022311628317" } - - - @article{Blanes1999, author = {S. Blanes and F. Casas and J. Ros}, title = {Symplectic Integration with Processing: A General Study}, @@ -76,4 +71,16 @@ @article{Blanes1999 eprint = { http://dx.doi.org/10.1137/S1064827598332497 } } +@Article{McLachlan1995, + author="Robert I. McLachlan", + title="Composition methods in the presence of small parameters", + journal="BIT Numerical Mathematics", + year="1995", + volume="35", + number="2", + pages="258--268", + issn="1572-9125", + doi="10.1007/BF01737165", + url="http://dx.doi.org/10.1007/BF01737165" +} From d6dca6aa5bc7164812566f404e478fd461711f4f Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Thu, 2 Feb 2017 23:41:58 +0100 Subject: [PATCH 07/28] Building Blocks --- LWB/propagators/header.tex | 1 + LWB/propagators/main.tex | 377 +++----------------- LWB/propagators/sections/buildingblocks.tex | 170 +++++++++ LWB/propagators/sections/propagators.tex | 267 ++++++++++++++ 4 files changed, 481 insertions(+), 334 deletions(-) create mode 100644 LWB/propagators/sections/buildingblocks.tex create mode 100644 LWB/propagators/sections/propagators.tex diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex index 53d0e66..7f9c7d1 100644 --- a/LWB/propagators/header.tex +++ b/LWB/propagators/header.tex @@ -176,6 +176,7 @@ \newcommand*{\opB}{\mathcal{B}} \newcommand*{\opX}{\mathcal{X}} \newcommand*{\upic}{\ensuremath{u[\Pi,\bm{c}]}} +\newcommand*{\proc}[1]{\textsc{#1}} \usepackage{algorithm} \usepackage{verbatim} diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 92fe91c..640ad3f 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -36,19 +36,22 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{comment} \clearpage +\end{comment} + \section{Introduction} +\begin{comment} All based on composition methods methods that compose elementary flows/operators to build an approximation of high order -In general / in the practical implementation, propagation with $\opT$ and $\opU$ is usually replaced by a series of smaller, alternating steps with the two operators +Summary of the structure of the remaining report +Describe what is going to happen in the rest of the paper -Summary of the structure of the remaining report -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\cite{GH_convsemiclassical} +%%% Mathematical properties Semiclassical time-dependent Schrödinger equation % \begin{align} @@ -58,7 +61,11 @@ \section{Introduction} % with wave function $\psi(\bvec{x},t)$ depending on the spatial variables $\bvec{x} = (x_1,\dots,x_D) \in \R^D$ and the time variable $t \in \R$. +how does one analytically propagate? +operators don't commute +operator splitting to the rescue Hamiltonian + \begin{align} \opH = \opT + \opV = \opT + \opU + \opW \end{align} @@ -67,17 +74,9 @@ \section{Introduction} \opV &= V(x) \end{align} -Small semiclassical parameter $\eps$ plays an important role in the stability of numerical schemes - -Classical dynamics in the limit $\eps \rightarrow 0$, quantum mechanics for $\eps = 1$. - -In particular, as underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\sim \eps^{-2})$ for Lie-Trotter splitting, Strang splitting and other related methods. - -Motivation - why is time propagation important? -Background - -Describe what is going to happen in the rest of the paper - +For the rest of this report, it will be used that $\opU = U(\bvec{x})$ and $\opW = W(\bvec{x})$ and they will denote the quadratic and non-quadratic splitting of $\opV$ respectively. +\par\medskip +% Hagedorn wavepackets explained in \cite{FGL_semiclassical_dynamics} @@ -92,21 +91,22 @@ \section{Introduction} As the idea behind the splitting the Hamiltonian operator is so central to the time propagation approach used here, we repeat the most important findings from \cite{FGL_semiclassical_dynamics}: \begin{itemize} - \item The free linear Schrödinger equation ($\opV=0$) can be solved exactly and the wavepacket remains in Hagedorn wavepacket form \ref{math:hagedornwp}. + \item The free linear Schrödinger equation ($\opV=0$) can be solved exactly and the wavepacket remains in Hagedorn wavepacket form~\ref{math:hagedornwp}. For time propagation, only the parameters $\bvec{\Pi},S$ need to be updated, the coefficients $\{c_k\}_{k \in \K}$ remain unchanged. - \item The Schrödinger equation \ref{math:tdse} can be solved exactly in a pure quadratic potential, i.e. in a potential $\opV=U(\bvec{x})$ with $\opT=0$. + \item The Schrödinger equation~\ref{math:tdse} can be solved exactly in a pure quadratic potential, i.e. in a potential $\opV=U(\bvec{x})$ with $\opT=0$. Again, time propagation only affects the parameters $\bvec{\Pi},S$, not the coefficients $\{c_k\}_{k \in \K}$. \item In the case of an arbitrary potential $\opV = W(\bvec{x})$ that is not quadratic, a set of Galerkin functions can be propagated by adapting the coefficients $\{c_k\}_{k \in \K}$ without changing the parameters $\bvec{\Pi},S$. \end{itemize} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Quantum Time Propagation, Operator Splitting} + +Small semiclassical parameter $\eps$ plays an important role in the stability of numerical schemes + +Classical dynamics in the limit $\eps \rightarrow 0$, quantum mechanics for $\eps = 1$. -check propositions in hagedorn paper. +In particular, as underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\sim \eps^{-2})$ for Lie-Trotter splitting, Strang splitting and other related methods. -how does one analytically propagate? -operators don't commute -operator splitting to the rescue +Motivation - why is time propagation important? +Background The Hamiltonian is split into three parts @@ -125,65 +125,22 @@ \subsection{Quantum Time Propagation, Operator Splitting} The search for accurate time propagation schemes has become an area of research by itself, see papers of A and B +General notes about Operator Splitting? -\clearpage -\section{Time evolution schemes} - -present a selection of propagators - -\subsection{Common building blocks for Quantum Time Propagators} -% - -General notes: what variables are accessible? - - -The time stepping with operators $\opT$ and $\opV = U(\op{x})$ follows directly from the propositions in \cite{FGL_semiclassical_dynamics} and is outlined in the algorithms \ref{alg:stepT} and \ref{alg:stepU} respectively. - -As pointed out in the introduction, the time propagation for non-quadratic potentials $W(\bvec{x})$ can be achieved by updating the coefficients $\{ c_k \}_{k \in \K}$. -The update rule is -% -\begin{align} - \bvec{c}(t) = \exp \left( - \frac{\im t}{\eps} \bmat{F} \right) \bvec{c}(0) -\end{align} -% -where the matrix $\bmat{F} = \{ f_{k,l} \}_{k,l \in \K}$ has entries -% -\begin{align} - f_{k,l} = \matrixel{\varphi_k}{W}{\varphi_l} - = \int_{\R^N} \conj{\varphi_k(\bvec{x})} W(\bvec{x}) \varphi_l(\bvec{x}) \; \dif \bvec{x} -\end{align} - - -\begin{algorithm}[h] - \caption{Propagate with Kinetic Energy Operator $\opT$} - \label{alg:stepT} - \begin{algorithmic} - \State - \Procedure{stepT}{$\eta$} - \State $q = q + \eta M^{-1} p$ - \State $Q = Q + \eta M^{-1} P$ - \State $S = S + \frac{\eta}{2} p^T M^{-1} p$ - \EndProcedure - \end{algorithmic} -\end{algorithm} +Notation and Conventions % -\begin{algorithm}[h] - \caption{Propagate with (Quadratic) Potential Energy Operator} - \label{alg:stepU} - \begin{algorithmic} - \State - \Procedure{stepU}{$\eta$} - \State $p = p - \eta \nabla V (q)$ - \State $P = P - \eta \nabla^2 V (q) Q$ - \State $S = S - \eta V (q)$ - \EndProcedure - \end{algorithmic} -\end{algorithm} - - +In this short section, some +Consistent with previous work on libwaveblocks and existing code +$N$ number of levels +$D$ dimension +$M$ number of timesteps steps +TODO: check: are q and p vectors? $\bm{q,p}$ \\ +$q^n, p^n \in V_N, \; Q^n, P^n \in M_{N \times N}, \; S^n \in \mathbb{R}, \bm{c}^n \in (c_k^n)_{k \in \K}$ +TODO: hom/inhom for all algorithms!!! +\end{comment} @@ -191,268 +148,15 @@ \subsection{Common building blocks for Quantum Time Propagators} +\input{sections/buildingblocks.tex} +\input{sections/propagators.tex} -\subsection{Hagedorn Propagator} -\label{sub:hagedorn_propagator} -% -The Hagedorn propagator, shown in algorithm \ref{alg:hagedorn}, is one of the simplest propagators that can be built by exploiting the numerical aspects discussed above. -As pointed out in \cite{FGL_semiclassical_dynamics}, this simple time stepping scheme has many beneficial properties like the preservation of the $L^2$ norm of the wavepacket, time reversibility and stability in the classical limit $\eps \rightarrow 0$. -Also, in the limit of $\K$ approaching the full basis set, the variational approximation used for the propagation with the non-quadratic part $\opW$ becomes exact. -\begin{algorithm}[h] - \caption{Single timestep with Hagedorn propagator} - \label{alg:hagedorn} - \begin{algorithmic} - \State - \Procedure{Hagedorn.Propagate}{$\Dt$} - \State - \State \Call{stepT}{$\frac{\Dt}{2}$} - \Comment{Step of size $\Dt/2$ with $\opT$} - \State \Call{stepU}{$\Dt$} - \Comment{Step of size $\Dt$ with $\opU$} - \State \Call{stepW}{$\Dt$} - \Comment{Step of size $\Dt$ with $\opW$} - \State \Call{stepT}{$\frac{\Dt}{2}$} - \Comment{Step of size $\Dt/2$ with $\opT$} - \State - \EndProcedure - \end{algorithmic} -\end{algorithm} - - -\subsection{Semiclassical Propagator} -\label{sub:semiclassical_propagator} -% -The central idea of the semiclassical splitting, as introduced in \cite{GH_convsemiclassical}, -is to split the propagation with operators $\opT$ and $\opU$ into many smaller, alternating steps, thereby reducing the dominating error\footnote{for small $\eps$, -the main source of error lies in the updating of $\Pi$ and $S$} in the update of $\bvec{\Pi}$ and $S$. -While there is some additional computational cost caused by a higher number of updates for the parameters $\bvec{\Pi}$ and $S$, the extra effort is usually negligible compared to the propagation with $\opW$ which requires numerical evaluation of multi-dimensional integrals. -\par\medskip -% -In addition, due to the numerical properties of the semiclassical splitting, -it even allows to take larger timesteps $\Dt$ than conventional -splitting methods like the YL-splitting, while maintaining the same error. -\par\medskip -% -Finally and most importantly, the error is no longer proportional to $1/\eps^2$ but instead -scales linearly in the semiclassical parameter $\eps$, -meaning that a smaller $\eps$ will now reduce the error instead of increasing it. -The error scales with $\eps (\Delta t)^2$ for the semiclassical splitting using the Y-splitting, -but the dependency on the timestep $\Delta t$ can be improved even further by using different -splittings which effectively corresponds to higher order coefficient pairs $w_T$ and $w_U$. -\par\medskip -% -The steps for the semiclassical propagator are shown in algorithm \ref{alg:semiclassical} and -further details can be found in the original paper \cite{GH_convsemiclassical}. -% -\begin{algorithm}[h] - \caption{Single timestep with Semiclassical propagator} - \label{alg:semiclassical} - \begin{algorithmic} - \State - \Procedure{Semiclassical.Propagate}{$\Dt$} - \State - \State $M := \lceil 1 + \frac{\sqrt{\Dt}}{\eps^{3/4}} \rceil$ - \Comment{Divide $\Dt$ into smaller steps} - \State - \State \Call{intSplit}{$\frac{\Dt}{2}, \frac{M}{2}, \{ w_T, w_U \}$} - \Comment{$M/2$ split steps with $T+U$} - \State \Call{stepW}{$\Dt$} - \Comment{Single step with $W$} - \State \Call{intSplit}{$\frac{\Dt}{2}, \frac{M}{2}, \{ w_T, w_U \}$} - \Comment{$M/2$ split steps with $T+U$} - \State - \EndProcedure - \end{algorithmic} -\end{algorithm} - - -\subsection{Magnus Propagator} -\label{sub:magnus_propagator} -% -As was noted by Magnus in \cite{Magnus1954}, the solution to a differential equation of the form -% -\begin{align} - y'(t) = a(t) y(t) \qquad t \ge 0 -\end{align} -% -can be written as -% -\begin{align} - \label{math:magnussolution} - y(t) = e^{\sigma (t)} y_0 -\end{align} -% -where $\sigma (t)$ is an infinite sum of iterated integrals and commutators, also known as the Magnus series. -The idea behind the Magnus approximation was extensively studied using the Baker-Campbell-Hausdorff (BCH) formula and rooted trees techniques, more details can be found in \cite{Blanes2006}, \cite{Blanes2000}, \cite{Iserles1999}. -\par\medskip -% -In order to approximate the solution from equation \ref{math:magnussolution}, one can take only a finite number of terms from this series whereby a truncation error is committed -(additional error sources in this method are the discretization of integrals and the approximation of matrix exponentials). -% -This method of the Magnus Propagator has several beneficial numerical properties that are described in \cite{Iserles1999}. -In particular, for solutions that evolve within a Lie group, the same holds for the approximate numerical solution calculated through a truncated Magnus series. -It was also shown in the same paper that the Magnus series can compete with - and in fact may even outplay - classical schemes like Runge-Kutta or Gauss-Legendre. -Although the method is not a symplectic scheme in the usual sense, \cite{Iserles1999} has shown that in practical applications it conserves the Hamiltonian energy just as well as symplectic integrators. -\par\medskip -Moreover, the numerical stability and good performance of the Magnus propagator are not limited to problems where the solution evolves within a Lie Group, but also apply to various problems when this is not the case. -Algorithm \ref{alg:magnus} shows the \emph{MG4} method as presented in \cite{Iserles1999} and implemented in C++ in the scope of this work. - -\begin{algorithm}[h] - \caption{Single timestep with Magnus propagator} - \label{alg:magnus} - \begin{algorithmic} - \State - \Procedure{Magnus.Propagate}{$\Dt$} - \State - \State $h_1 = \frac{3-\sqrt{3}}{6} \Dt$, $h_2 = \frac{2\sqrt{3}}{6} \Dt$ - \Comment Gauss-Legendre coefficients on $[0,\Dt]$ - \State $M_{k} = 1+\sqrt{h_{k} \eps^{-3/8}}, \quad k=1,2$ - \Comment number of timesteps for splitting - \State - \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} - \Comment advance till $\frac{3-\sqrt{3}}{6} \Dt$ - \State $\bmat{A}_1 = - \frac{\im}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} - \Comment temporarily store interaction matrix - \State \Call{intSplit}{$h_2, M_2, \{w_T, w_U\}$} - \Comment advance till $\frac{3+\sqrt{3}}{6} \Dt$ - \State $\bmat{A}_2 = - \frac{\im}{\eps^2} \cdot$ \Call{buildF}{\mbox{}} - \Comment temporarily store interaction matrix - \State $\bmat{\Sigma} = \frac{1}{2} \Dt (\bmat{A}_1 + \bmat{A}_2) + \frac{\sqrt{3}}{12} (\Dt)^2 (\bmat{A}_2 \cdot \bmat{A}_1 - \bmat{A}_1 \cdot \bmat{A}_2)$ - \Comment compute $\sigma (t)$ - \State $\bvec{c} = \exp \left( \bmat{\Sigma} \right) \bvec{c}$ - \Comment update coefficients - \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} - \Comment advance till $\frac{6}{6} \Dt$ - \State - \EndProcedure - \end{algorithmic} -\end{algorithm} - -\subsection{Processing Propagators} -\label{sub:pre764_propagator} -% -The idea of processing propagators is to carry out the time evolution with a Hamiltonian that is slightly perturbed from the initial one. -To achieve this, a pre and post processing step is applied. \\ -More formally, it holds that -% -\begin{align} - e^{- \Dt \opH (\Dt)} = e^P e^{- \Dt K} e^{-P} -\end{align} -% -where the pre and post processing steps are applied via the multiplication with matrices $e^P$ and $e^{-P}$ (also referred to as \emph{processors}). -While the processors only need to be applied once at the very beginning and end of the propagation (in order to return to the original Hamiltonian), the multiplication with the \emph{kernel} $e^{-\Dt K}$ is repeated $M$ times, once for every timestep. \\ -As a consequence, one wants to choose the processor $e^P$ in such a way that the evaluation of the kernel $e^{-\Dt K}$ is as simple as possible, requiring a minimum of expensive function evaluations. -\par\medskip -% -Unlike most commonly used integration schemes, the family of processing propagators are symplectic integrators. -They are not suited for every kind of Hamiltonian, and the work of Blanes, Casas and Ros in \cite{Blanes1999} gives a method for finding the neccessary conditions that need to be satisfied in order for processing methods to be applicable. -\par\medskip -% -Algorithm \ref{alg:pre764} presents the \emph{Pre764} processing method which is a ?sixth? order processing method that was derived in \cite{Blanes1999}. -% -\begin{algorithm}[h] - \caption{Single timestep with Pre764 propagator} - \label{alg:pre764} - \begin{algorithmic} - \State - \Procedure{Pre764.PrePropagate}{$\Dt$} - \State - \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ - \Comment compute number of time steps - \For{$j=0,...,v-1$} \Comment $v=6$ for Pre(7,6,4) - \State \Call{intSplit}{$-Z_j \Dt, M, \{w_T, w_U\}$} - \Comment $M$ alternating steps with $\opT$ and $\opU$ - \State \Call{stepW}{$\upic, -Y_j \Dt$} - \Comment single step with $\opW$ - \EndFor - \State - \EndProcedure - \\\hrulefill - \State - \Procedure{Pre764.Propagate}{$\Dt$} - \State - \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ - \Comment compute number of time steps - \For{$j=0,...,k-1$} \Comment $k=4$ for Pre(7,6,4) - \State \Call{stepW}{$\alpha_j \Dt$} - \Comment single step with $\opW$ - \State \Call{intSplit}{$\beta_j \Dt, M, \{w_T, w_U\}$} - \Comment $M$ alternating steps with $\opT$ and $\opU$ - \EndFor - \State - \EndProcedure - \\\hrulefill - \State - \Procedure{Pre764.PostPropagate}{$\Dt$} - \State - \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ - \Comment compute number of time steps - \For{$j=v-1,...,0$} \Comment $v=6$ for Pre(7,6,4) - \State \Call{stepW}{$Y_j \Dt$} - \Comment single step with $\opW$ - \State \Call{intSplit}{$Z_j \Dt, M, \{w_T, w_U\}$} - \Comment $M$ alternating steps with $\opT$ and $\opU$ - \EndFor - \State - \EndProcedure - \end{algorithmic} -\end{algorithm} - - -\subsection{McL Propagators} -\label{sub:mcl_propagator} -% -McLachlan (see \cite{McLachlan1995}) has investigated various ?symplectic? schemes for computing the effect of operators of the form $\opX = \opA + \epsilon \opB$ where $\opA$ describes an exactly solvable system and $\opB$ is a perturbation. -The factor $\epsilon$ is a small parameter that indicates the limited influence of $\opB$, not to be confused with the semiclassical parameter $\eps$. -\par\medskip -The \emph{McL} integration schemes are characterised by pairs of weighing coefficients $a_i$ and $b_i$ that allow to represent the exponential $e^{\opX}$ as a product -% -\begin{align} - e^\opX = \prod_i e^{b_i t \epsilon \opB} e^{a_i t \opA} -\end{align} -% -McLachlan also goes through the process of deriving the coefficients for the following schemes -\begin{align*} - &\text{\emph{McL(2s,2)}} &&\text{with error of order } \epsilon (\Dt)^{2s} + \epsilon^2 (\Dt)^2 \\ - &\text{\emph{McL(6,4)}} &&\text{with error of order } \epsilon (\Dt)^{6} + \epsilon^2 (\Dt)^4 \\ - &\text{\emph{McL(8,4)}} &&\text{with error of order } \epsilon (\Dt)^{8} + \epsilon^2 (\Dt)^4 -\end{align*} -% -Note that the error depends on two small parameters, the timestep $\Dt$ and the parameter $\epsilon$ that quantifies the influence of $\opB$. -\par\medskip -% -In our case, $\opX = \opH = \opT + \opV + \opW$, the computationally expensive operator $\opW$ takes the role of $\epsilon \opB$, while the remaining Hamiltonian $\opT + \opU$ corresponds to $\opA$. -Our preferred method will therefore be of the form $ABA$ because this minimizes the number of steps with $\opB = \opW$. -\par\medskip -% -The source code of this project contains the C++ implementation of the propagators \emph{McL(4,2)} (asymmetric) and \emph{McL(8,4)} (symmetric) that require only two respectively three evaluations of $\opW$ per step. - -\begin{algorithm}[h] - \caption{Single timestep with McL propagators} - \label{alg:mcl} - \begin{algorithmic} - \State - \Procedure{McL42.Propagate}{$\Dt$} - \State - \State \Call{intSplit}{$A_0 \Dt, M, \{w_T, w_U\}$} - \Comment $\opT + \opU = A$ - \State \Call{stepW}{$B_0 \Dt$} - \Comment $\opW = B$ - \State \Call{intSplit}{$A_1 \Dt, M, \{w_T, w_U\}$} - \Comment $\opT + \opU = A$ - \State \Call{stepW}{$B_1 \Dt$} - \Comment $\opW = B$ - \State \Call{intSplit}{$A_2 \Dt, M, \{w_T, w_U\}$} - \Comment $\opT + \opU = A$ - \State - \EndProcedure - \end{algorithmic} -\end{algorithm} \clearpage \section{Implementation in C++} +\begin{comment} \cite{libwaveblocks} What was the main challenge? What was controllable (which code could be changed), what not? Quick alternating o @@ -474,23 +178,28 @@ \section{Implementation in C++} - CRTP / static polymorphism \cite{C_CRTP} - recognizing IP - callback function - - implementation of intSplit (TU,UT, partial specialization) + - implementation of IntSplit (TU,UT, partial specialization) - polymorphism through enable-if \clearpage +\end{comment} \section{Benchmark / Case Study / Example} +\begin{comment} \clearpage +\end{comment} \section{Conclusion} +\begin{comment} final observations suggestions for future work +\end{comment} diff --git a/LWB/propagators/sections/buildingblocks.tex b/LWB/propagators/sections/buildingblocks.tex new file mode 100644 index 0000000..ba336c8 --- /dev/null +++ b/LWB/propagators/sections/buildingblocks.tex @@ -0,0 +1,170 @@ +\clearpage +\section{Common building blocks for time evolution schemes} +% +This section will briefly introduce the most important, generic building blocks that are needed for building the propagators in section~\ref{sec:propagators}. +For each of them, it will be assumed that the following wave packet variables are implicitly accessible, without passing them as an argument to the procedure: the parameter pack $\bvec{\Pi} = (\bvec{q},\bvec{p},\bvec{Q},\bvec{P})$, the phase factor $S$, the coefficients $\{ c_k \}_{k \in \K}$, the potential function $V(\bvec{q})$ (including Jacobian and Hessian thereof) and the potential remainder $W(\bvec{q})$. +\par\medskip +% +Many of the building blocks depend on the number of energy levels and on whether the wave packet is homogeneous or inhomogeneous. +If any of these cases require a special implementation, it is denoted in the signature of the corresponding algorithm. + + +\subsection{Evolve} +\label{subsec:evolve} +The convenience function \proc{Evolve} (algorithm~\ref{alg:Evolve}) is a wrapper that will carry out the \proc{PrePropagate}, \proc{Propagate} and \proc{PostPropagate} functions. +The concrete implementation of the methods \proc{PrePropagate}, \proc{Propagate} and \proc{PostPropagate} is delegated to the derived classes (see section~\ref{sec:implementation} for more details), +but will generally be a combination of the basic building blocks that follow in this section. +% +\begin{algorithm}[h] + \caption{Evolve the wave packet for a time period $\tau$} + \label{alg:Evolve} + \begin{algorithmic} + \State + \Procedure{Evolve}{$\tau, \Dt$} + \State + \State $M = \tau/\Dt$ + \State \Call{PrePropagate}{$\Dt$} + \For{$1 \le m \le M$} + \State \Call{Propagate}{$\Dt$} + \EndFor + \State \Call{PostPropagate}{$\Dt$} + \EndProcedure + \State + \end{algorithmic} +\end{algorithm} + + +\subsection{StepT, StepU and IntSplit} +\label{subsec:tuintsplit} +% +%%% StepT & StepU +The time stepping with operators $\opT$ and $\opU = U(\bvec{x})$ follows directly from the propositions in \cite{FGL_semiclassical_dynamics} and is outlined in the algorithms~\ref{alg:StepT} and~\ref{alg:StepU} respectively. +Both functions take a variable $\eta$ as an argument which describes the size of the time step. \\ +In the inhomogeneous implementation of the algorithms, the subscript $n$ is used to denote the relative parameters at energy level $n$. +\par\medskip +% +\begin{algorithm}[h] + \caption{Propagate with Kinetic Energy Operator $\opT$} + \label{alg:StepT} + \begin{algorithmic} + \State + \Procedure{StepT[homogeneous]}{$\eta$} + \State + \State $\bvec{q} = \bvec{q} + \eta \bmat{M}^{-1} \bvec{p}$ + \Comment update $\bvec{q}$ + \State $\bmat{Q} = \bmat{Q} + \eta \bmat{M}^{-1} \bmat{P}$ + \Comment update $\bmat{Q}$ + \State $\bvec{S} = \bvec{S} + \frac{\eta}{2} \bvec{p}^T \bmat{M}^{-1} \bvec{p}$ + \Comment update $\bmat{S}$ + \State + \EndProcedure + \\\hrulefill + \State + \Procedure{StepT[inhomogeneous]}{$\eta$} + \State + \For{$n=1,...,N$} + \Comment loop over all energy levels + \State $\bvec{q}_n = \bvec{q}_n + \eta \bmat{M}^{-1} \bvec{p}_n$ + \Comment update $\bvec{q}$ for level $n$ + \State $\bmat{Q}_n = \bmat{Q}_n + \eta \bmat{M}^{-1} \bmat{P}_n$ + \Comment update $\bmat{Q}$ for level $n$ + \State $\bvec{S}_n = \bvec{S}_n + \frac{\eta}{2} \bvec{p}_n^T \bmat{M}^{-1} \bvec{p}_n$ + \Comment update $\bmat{S}$ for level $n$ + \EndFor + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} +% +\begin{algorithm}[h] + \caption{Propagate with (Quadratic) Potential Energy Operator} + \label{alg:StepU} + \begin{algorithmic} + \State + \Procedure{StepU[homogeneous]}{$\eta$} + \State + \State $\bvec{p} = \bvec{p} - \eta \nabla V (\bvec{q})$ + \Comment update $\bvec{p}$ + \State $\bvec{P} = \bvec{P} - \eta \nabla^2 V (\bvec{q}) \bvec{Q}$ + \Comment update $\bvec{P}$ + \State $\bvec{S} = \bvec{S} - \eta V (\bvec{q})$ + \Comment update $\bvec{S}$ + \State + \EndProcedure + \\\hrulefill + \State + \Procedure{StepU[inhomogeneous]}{$\eta$} + \State + \For{$n=1,...,N$} + \Comment loop over all energy levels + \State $\bvec{p}_n = \bvec{p}_n - \eta \nabla V (\bvec{q}_n)$ + \Comment update $\bvec{p}$ for level $n$ + \State $\bvec{P}_n = \bvec{P}_n - \eta \nabla^2 V (\bvec{q}_n) \bvec{Q}_n$ + \Comment update $\bvec{P}$ for level $n$ + \State $\bvec{S}_n = \bvec{S}_n - \eta V (\bvec{q}_n)$ + \Comment update $\bvec{S}$ for level $n$ + \EndFor + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} +% +%%% IntSplit +In the practical implementations used here, propagation with $\opT$ and $\opU$ is usually replaced by a series of smaller, alternating steps with the two operators. +In order to facilitate the use of such an alternating pattern, the helper function \proc{IntSplit} (algorithm~\ref{alg:intsplit}) is introduced. +This function splits the timestep $\Dt$ into $M$ smaller timesteps of size $\dt$ and uses the pair of weight arrays $\{ w_T, w_U \}$ to alternatingly apply operators $\opT$ and $\opU$ on each of the sub-intervals. +% +\begin{algorithm}[h] + \caption{Split a given time interval into $M$ sub-intervals and alternatingly apply weighted steps with $\opT$ and $\opU$ on each of them} + \label{alg:intsplit} + \begin{algorithmic} + \State + \Procedure{IntSplit}{$\Dt, M, \{w_T,w_U\}$} + \State + \State $\dt = \Dt/M$ + \Comment step size for splitting + \For{$n = 1,\dots,M$} + \Comment split interval in M substeps + \For{$x_{w_T}$ in $w_T$ and $x_{w_U}$ in $w_U$} + \Comment alternatingly apply $\opT$ and $\opU$ + \State \Call{StepT}{$x_{w_T} \dt$} + \State \Call{StepU}{$x_{w_U} \dt$} + \EndFor + \EndFor + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} + + +\subsection{StepW and building of the interaction matrix} +% +As pointed out in the introduction, the time propagation for non-quadratic potentials $W(\bvec{x})$ (see algorithm~\ref{alg:StepW}) can be achieved by updating only the coefficients $\{ c_k \}_{k \in \K}$. +The update rule is +% +\begin{align} + \bvec{c}(t) = \exp \left( - \frac{\im t}{\eps} \bmat{F} \right) \bvec{c}(0) +\end{align} +% +where the matrix $\bmat{F} = \{ f_{k,l} \}_{k,l \in \K}$ has entries +% +\begin{align} + f_{k,l} = \matrixel{\varphi_k}{W}{\varphi_l} + = \int_{\R^N} \conj{\varphi_k(\bvec{x})} W(\bvec{x}) \varphi_l(\bvec{x}) \; \dif \bvec{x} +\end{align} +% +The calculation of such interaction matrices $\bmat{F}$ makes heavy use of the work on inner products that has been done in previous student projects for the libwaveblocks library \cite{libwaveblocks}. +For the further treatment in this report, it is assumed that an efficient implementation of \proc{BuildF} is readily available. +% +\begin{algorithm}[h] + \caption{Propagate with (Non-Quadratic) Potential Energy Operator} + \label{alg:StepW} + \begin{algorithmic} + \State + \Procedure{StepW}{$\upic,\Dt$} + \State $\bm{\Sigma} = - \im \frac{\Dt}{\eps^2} \cdot$ \Call{BuildF}{$\Pi$} + \State $\bm{c} = \exp{\bm{\Sigma}} \bm{c}$ + \EndProcedure + \State + \end{algorithmic} +\end{algorithm} diff --git a/LWB/propagators/sections/propagators.tex b/LWB/propagators/sections/propagators.tex new file mode 100644 index 0000000..71930f2 --- /dev/null +++ b/LWB/propagators/sections/propagators.tex @@ -0,0 +1,267 @@ +\clearpage +\section{Propagators} +\label{sec:propagators} + +This section gives a short overview of the numerical properties of the propagators that were implemented in the context of this project. +Where appropriate, references are given to find a more detailed mathematical analysis of the propagators. +\par\medskip +The methods are called in the sequence determined by the \proc{Evolve} function (\ref{alg:evolve}) and generally are expressed in terms of the basic building blocks that were presented in the previous section. +If the methods \proc{PrePropagate} and \proc{PostPropagate} are omitted, it means that they are empty. \\ +Whenever convenient, the direct succession of functions \proc{StepT} and \proc{StepU} was replaced by a call to \proc{IntSplit} (see explanation in \ref{subsec:tuintsplit}) for better accuracy. + +\subsection{Hagedorn Propagator} +\label{sub:hagedorn_propagator} +% +The Hagedorn propagator, shown in algorithm \ref{alg:hagedorn}, is one of the simplest propagators that can be built by exploiting the numerical aspects discussed above. +As pointed out in \cite{FGL_semiclassical_dynamics}, this simple time stepping scheme has many beneficial properties like the preservation of the $L^2$ norm of the wavepacket, time reversibility and stability in the classical limit $\eps \rightarrow 0$. +Also, in the limit of $\K$ approaching the full basis set, the variational approximation used for the propagation with the non-quadratic part $\opW$ becomes exact. +\begin{algorithm}[h] + \caption{Single timestep with Hagedorn propagator} + \label{alg:hagedorn} + \begin{algorithmic} + \State + \Procedure{Hagedorn.Propagate}{$\Dt$} + \State + \State \Call{stepT}{$\frac{\Dt}{2}$} + \Comment{Step of size $\Dt/2$ with $\opT$} + \State \Call{stepU}{$\Dt$} + \Comment{Step of size $\Dt$ with $\opU$} + \State \Call{stepW}{$\Dt$} + \Comment{Step of size $\Dt$ with $\opW$} + \State \Call{stepT}{$\frac{\Dt}{2}$} + \Comment{Step of size $\Dt/2$ with $\opT$} + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} + + +\subsection{Semiclassical Propagator} +\label{sub:semiclassical_propagator} +% +The central idea of the semiclassical splitting, as introduced in \cite{GH_convsemiclassical}, +is to split the propagation with operators $\opT$ and $\opU$ into many smaller, alternating steps, thereby reducing the dominating error\footnote{for small $\eps$, +the main source of error lies in the updating of $\Pi$ and $S$} in the update of $\bvec{\Pi}$ and $S$. +While there is some additional computational cost caused by a higher number of updates for the parameters $\bvec{\Pi}$ and $S$, the extra effort is usually negligible compared to the propagation with $\opW$ which requires numerical evaluation of multi-dimensional integrals. +\par\medskip +% +In addition, due to the numerical properties of the semiclassical splitting, +it even allows to take larger timesteps $\Dt$ than conventional +splitting methods like the YL-splitting, while maintaining the same error. +\par\medskip +% +Finally and most importantly, the error is no longer proportional to $1/\eps^2$ but instead +scales linearly in the semiclassical parameter $\eps$, +meaning that a smaller $\eps$ will now reduce the error instead of increasing it. +The error scales with $\eps (\Delta t)^2$ for the semiclassical splitting using the Y-splitting, +but the dependency on the timestep $\Delta t$ can be improved even further by using different +splittings which effectively corresponds to higher order coefficient pairs $w_T$ and $w_U$. +\par\medskip +% +The steps for the semiclassical propagator are shown in algorithm \ref{alg:semiclassical} and +further details can be found in the original paper \cite{GH_convsemiclassical}. +% +\begin{algorithm}[h] + \caption{Single timestep with Semiclassical propagator} + \label{alg:semiclassical} + \begin{algorithmic} + \State + \Procedure{Semiclassical.Propagate}{$\Dt$} + \State + \State $M := \lceil 1 + \frac{\sqrt{\Dt}}{\eps^{3/4}} \rceil$ + \Comment{Divide $\Dt$ into smaller steps} + \State + \State \Call{intSplit}{$\frac{\Dt}{2}, \frac{M}{2}, \{ w_T, w_U \}$} + \Comment{$M/2$ split steps with $T+U$} + \State \Call{stepW}{$\Dt$} + \Comment{Single step with $W$} + \State \Call{intSplit}{$\frac{\Dt}{2}, \frac{M}{2}, \{ w_T, w_U \}$} + \Comment{$M/2$ split steps with $T+U$} + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} + + +\subsection{Magnus Propagator} +\label{sub:magnus_propagator} +% +As was noted by Magnus in \cite{Magnus1954}, the solution to a differential equation of the form +% +\begin{align} + y'(t) = a(t) y(t) \qquad t \ge 0 +\end{align} +% +can be written as +% +\begin{align} + \label{math:magnussolution} + y(t) = e^{\sigma (t)} y_0 +\end{align} +% +where $\sigma (t)$ is an infinite sum of iterated integrals and commutators, also known as the Magnus series. +The idea behind the Magnus approximation was extensively studied using the Baker-Campbell-Hausdorff (BCH) formula and rooted trees techniques, more details can be found in \cite{Blanes2006}, \cite{Blanes2000}, \cite{Iserles1999}. +\par\medskip +% +In order to approximate the solution from equation \ref{math:magnussolution}, one can take only a finite number of terms from this series whereby a truncation error is committed +(additional error sources in this method are the discretization of integrals and the approximation of matrix exponentials). +% +This method of the Magnus Propagator has several beneficial numerical properties that are described in \cite{Iserles1999}. +In particular, for solutions that evolve within a Lie group, the same holds for the approximate numerical solution calculated through a truncated Magnus series. +It was also shown in the same paper that the Magnus series can compete with - and in fact may even outplay - classical schemes like Runge-Kutta or Gauss-Legendre. +Although the method is not a symplectic scheme in the usual sense, \cite{Iserles1999} has shown that in practical applications it conserves the Hamiltonian energy just as well as symplectic integrators. +\par\medskip +Moreover, the numerical stability and good performance of the Magnus propagator are not limited to problems where the solution evolves within a Lie Group, but also apply to various problems when this is not the case. +Algorithm \ref{alg:magnus} shows the \emph{MG4} method as presented in \cite{Iserles1999} and implemented in C++ in the scope of this work. + +\begin{algorithm}[h] + \caption{Single timestep with Magnus propagator} + \label{alg:magnus} + \begin{algorithmic} + \State + \Procedure{Magnus.Propagate}{$\Dt$} + \State + \State $h_1 = \frac{3-\sqrt{3}}{6} \Dt$, $h_2 = \frac{2\sqrt{3}}{6} \Dt$ + \Comment Gauss-Legendre coefficients on $[0,\Dt]$ + \State $M_{k} = 1+\sqrt{h_{k} \eps^{-3/8}}, \quad k=1,2$ + \Comment number of timesteps for splitting + \State + \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} + \Comment advance till $\frac{3-\sqrt{3}}{6} \Dt$ + \State $\bmat{A}_1 = - \frac{\im}{\eps^2} \cdot$ \Call{BuildF}{\mbox{}} + \Comment temporarily store interaction matrix + \State \Call{intSplit}{$h_2, M_2, \{w_T, w_U\}$} + \Comment advance till $\frac{3+\sqrt{3}}{6} \Dt$ + \State $\bmat{A}_2 = - \frac{\im}{\eps^2} \cdot$ \Call{BuildF}{\mbox{}} + \Comment temporarily store interaction matrix + \State $\bmat{\Sigma} = \frac{1}{2} \Dt (\bmat{A}_1 + \bmat{A}_2) + \frac{\sqrt{3}}{12} (\Dt)^2 (\bmat{A}_2 \cdot \bmat{A}_1 - \bmat{A}_1 \cdot \bmat{A}_2)$ + \Comment compute $\sigma (t)$ + \State $\bvec{c} = \exp \left( \bmat{\Sigma} \right) \bvec{c}$ + \Comment update coefficients + \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} + \Comment advance till $\frac{6}{6} \Dt$ + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} + +\subsection{Processing Propagators} +\label{sub:pre764_propagator} +% +The idea of processing propagators is to carry out the time evolution with a Hamiltonian that is slightly perturbed from the initial one. +To achieve this, a pre and post processing step is applied. \\ +More formally, it holds that +% +\begin{align} + e^{- \Dt \opH (\Dt)} = e^P e^{- \Dt K} e^{-P} +\end{align} +% +where the pre and post processing steps are applied via the multiplication with matrices $e^P$ and $e^{-P}$ (also referred to as \emph{processors}). +While the processors only need to be applied once at the very beginning and end of the propagation (in order to return to the original Hamiltonian), the multiplication with the \emph{kernel} $e^{-\Dt K}$ is repeated $M$ times, once for every timestep. \\ +As a consequence, one wants to choose the processor $e^P$ in such a way that the evaluation of the kernel $e^{-\Dt K}$ is as simple as possible, requiring a minimum of expensive function evaluations. +\par\medskip +% +Unlike most commonly used integration schemes, the family of processing propagators are symplectic integrators. +They are not suited for every kind of Hamiltonian, and the work of Blanes, Casas and Ros in \cite{Blanes1999} gives a method for finding the neccessary conditions that need to be satisfied in order for processing methods to be applicable. +\par\medskip +% +Algorithm \ref{alg:pre764} presents the \emph{Pre764} processing method which is a ?sixth? order processing method that was derived in \cite{Blanes1999}. +% +\begin{algorithm}[h] + \caption{Single timestep with Pre764 propagator} + \label{alg:pre764} + \begin{algorithmic} + \State + \Procedure{Pre764.PrePropagate}{$\Dt$} + \State + \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \Comment compute number of time steps + \For{$j=0,...,v-1$} \Comment $v=6$ for Pre(7,6,4) + \State \Call{intSplit}{$-Z_j \Dt, M, \{w_T, w_U\}$} + \Comment $M$ alternating steps with $\opT$ and $\opU$ + \State \Call{stepW}{$\upic, -Y_j \Dt$} + \Comment single step with $\opW$ + \EndFor + \State + \EndProcedure + \\\hrulefill + \State + \Procedure{Pre764.Propagate}{$\Dt$} + \State + \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \Comment compute number of time steps + \For{$j=0,...,k-1$} \Comment $k=4$ for Pre(7,6,4) + \State \Call{stepW}{$\alpha_j \Dt$} + \Comment single step with $\opW$ + \State \Call{intSplit}{$\beta_j \Dt, M, \{w_T, w_U\}$} + \Comment $M$ alternating steps with $\opT$ and $\opU$ + \EndFor + \State + \EndProcedure + \\\hrulefill + \State + \Procedure{Pre764.PostPropagate}{$\Dt$} + \State + \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \Comment compute number of time steps + \For{$j=v-1,...,0$} \Comment $v=6$ for Pre(7,6,4) + \State \Call{stepW}{$Y_j \Dt$} + \Comment single step with $\opW$ + \State \Call{intSplit}{$Z_j \Dt, M, \{w_T, w_U\}$} + \Comment $M$ alternating steps with $\opT$ and $\opU$ + \EndFor + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} + + +\subsection{McLachlan Propagators} +\label{sub:mcl_propagator} +% +McLachlan (see \cite{McLachlan1995}) has investigated various ?symplectic? schemes for computing the effect of operators of the form $\opX = \opA + \epsilon \opB$ where $\opA$ describes an exactly solvable system and $\opB$ is a perturbation. +The factor $\epsilon$ is a small parameter that indicates the limited influence of $\opB$, not to be confused with the semiclassical parameter $\eps$. +\par\medskip +The \emph{McL} integration schemes are characterised by pairs of weighing coefficients $a_i$ and $b_i$ that allow to represent the exponential $e^{\opX}$ as a product +% +\begin{align} + e^\opX = \prod_i e^{b_i t \epsilon \opB} e^{a_i t \opA} +\end{align} +% +McLachlan also goes through the process of deriving the coefficients for the following schemes +\begin{align*} + &\text{\emph{McL(2s,2)}} &&\text{with error of order } \epsilon (\Dt)^{2s} + \epsilon^2 (\Dt)^2 \\ + &\text{\emph{McL(6,4)}} &&\text{with error of order } \epsilon (\Dt)^{6} + \epsilon^2 (\Dt)^4 \\ + &\text{\emph{McL(8,4)}} &&\text{with error of order } \epsilon (\Dt)^{8} + \epsilon^2 (\Dt)^4 +\end{align*} +% +Note that the error depends on two small parameters, the timestep $\Dt$ and the parameter $\epsilon$ that quantifies the influence of $\opB$. +\par\medskip +% +In our case, $\opX = \opH = \opT + \opV + \opW$, the computationally expensive operator $\opW$ takes the role of $\epsilon \opB$, while the remaining Hamiltonian $\opT + \opU$ corresponds to $\opA$. +Our preferred method will therefore be of the form $ABA$ because this minimizes the number of steps with $\opB = \opW$. +\par\medskip +% +The source code of this project contains the C++ implementation of the propagators \emph{McL(4,2)} (asymmetric) and \emph{McL(8,4)} (symmetric) that require only two respectively three evaluations of $\opW$ per step. + +\begin{algorithm}[h] + \caption{Single timestep with McLachlan propagators} + \label{alg:mcl} + \begin{algorithmic} + \State + \Procedure{McL42.Propagate}{$\Dt$} + \State + \State \Call{intSplit}{$A_0 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_0 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_1 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_1 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_2 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} From d81734f447331b4af282c6cef6e9fdd7fed38a20 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Fri, 3 Feb 2017 00:48:27 +0100 Subject: [PATCH 08/28] Introduction --- LWB/propagators/main.tex | 48 +++------------------ LWB/propagators/sections/buildingblocks.tex | 2 +- LWB/propagators/sections/introduction.tex | 16 +++++++ LWB/propagators/sections/propagators.tex | 1 - 4 files changed, 23 insertions(+), 44 deletions(-) create mode 100644 LWB/propagators/sections/introduction.tex diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 640ad3f..923565b 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -17,50 +17,15 @@ \listoffigures \listofalgorithms -\clearemptydoublepage - -% \input{./introduction.tex} -% \input{./wavefunction_propagation.tex} -% \input{./wavepackets.tex} -% \input{./observables.tex} -% \input{./wavepacket_propagation.tex} -% \input{./results.tex} - - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -% TODO: move to separate file when done - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{comment} +\clearemptydoublepage \input{sections/introduction.tex} \clearpage -\end{comment} - -\section{Introduction} +\section{Time Propagation of Quantum Wave Packets} +\label{sec:timeprop} +% -\begin{comment} All based on composition methods - methods that compose elementary flows/operators to build an approximation of high order -Summary of the structure of the remaining report -Describe what is going to happen in the rest of the paper - - -%%% Mathematical properties -Semiclassical time-dependent Schrödinger equation -% -\begin{align} - \label{math:tdse} - \im \eps \; \del_t \psi (\bvec{x},t) = \opH (\eps) \; \psi (\bvec{x},t) -\end{align} -% -with wave function $\psi(\bvec{x},t)$ depending on the spatial variables $\bvec{x} = (x_1,\dots,x_D) \in \R^D$ and the time variable $t \in \R$. - how does one analytically propagate? operators don't commute operator splitting to the rescue @@ -140,7 +105,6 @@ \section{Introduction} TODO: hom/inhom for all algorithms!!! -\end{comment} @@ -148,8 +112,8 @@ \section{Introduction} -\input{sections/buildingblocks.tex} -\input{sections/propagators.tex} +% \clearpage \input{sections/buildingblocks.tex} +% \clearpage \input{sections/propagators.tex} diff --git a/LWB/propagators/sections/buildingblocks.tex b/LWB/propagators/sections/buildingblocks.tex index ba336c8..2a9a5be 100644 --- a/LWB/propagators/sections/buildingblocks.tex +++ b/LWB/propagators/sections/buildingblocks.tex @@ -1,5 +1,5 @@ -\clearpage \section{Common building blocks for time evolution schemes} +\label{sec:buildingblocks} % This section will briefly introduce the most important, generic building blocks that are needed for building the propagators in section~\ref{sec:propagators}. For each of them, it will be assumed that the following wave packet variables are implicitly accessible, without passing them as an argument to the procedure: the parameter pack $\bvec{\Pi} = (\bvec{q},\bvec{p},\bvec{Q},\bvec{P})$, the phase factor $S$, the coefficients $\{ c_k \}_{k \in \K}$, the potential function $V(\bvec{q})$ (including Jacobian and Hessian thereof) and the potential remainder $W(\bvec{q})$. diff --git a/LWB/propagators/sections/introduction.tex b/LWB/propagators/sections/introduction.tex new file mode 100644 index 0000000..c940998 --- /dev/null +++ b/LWB/propagators/sections/introduction.tex @@ -0,0 +1,16 @@ +\section*{Introduction} +% +With the growing understanding for quantum mechanics and quantum dynamics, one can attempt to solve increasingly complex systems and computational problems. +Without doubt, time propagation is at the core of simulating quantum systems and getting a better understanding for them. +\par\medskip +% +The aim of this project was to give a summary of some state of the art quantum time propagators and provide an efficient C++ implementation for evolving Hagedorn wave packets in time. +The produced code is an addition to the C++ WaveBlocks Framework whose functionality has already been extended in the context of preceding student projects. +The entire WaveBlocks project, which is partially based on an existing Python project with the same name, can be found at \cite{libwaveblocks}. +\par\medskip +% +In this report, section \ref{sec:timeprop} repeats some of the most important mathematical ideas underlying the time propagation of wave packets and introduces the notations that will be used throughout the rest of the document. +Based on these mathematical observations, section \ref{sec:buildingblocks} prepares a toolbox of essential building blocks that will help in the construction of propagators as well as for the final understanding of the source code. +In section \ref{sec:propagators}, a series of numerical propagators for quantum wave packets are presented and expressed in terms of the components provided earlier. +Some detailed insights into the most important technical ideas of the C++ implementation are given in section \ref{sec:implementation}. +Finally, section \ref{sec:benchmark} ... diff --git a/LWB/propagators/sections/propagators.tex b/LWB/propagators/sections/propagators.tex index 71930f2..66a82da 100644 --- a/LWB/propagators/sections/propagators.tex +++ b/LWB/propagators/sections/propagators.tex @@ -1,4 +1,3 @@ -\clearpage \section{Propagators} \label{sec:propagators} From af465a1545e3595c21ea9f1fb8606120027222fb Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Fri, 3 Feb 2017 02:11:25 +0100 Subject: [PATCH 09/28] Operator Splitting --- LWB/propagators/main.tex | 107 ++---------------- .../sections/operatorsplitting.tex | 67 +++++++++++ 2 files changed, 76 insertions(+), 98 deletions(-) create mode 100644 LWB/propagators/sections/operatorsplitting.tex diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 923565b..ac8b0da 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -18,108 +18,19 @@ \listofalgorithms \clearemptydoublepage \input{sections/introduction.tex} -\clearpage -\section{Time Propagation of Quantum Wave Packets} -\label{sec:timeprop} -% - -All based on composition methods -methods that compose elementary flows/operators to build an approximation of high order - -how does one analytically propagate? -operators don't commute -operator splitting to the rescue -Hamiltonian - -\begin{align} - \opH = \opT + \opV = \opT + \opU + \opW -\end{align} -\begin{align} - \opT &= - \sum_{j=1}^D \frac{\eps^2}{2m_j} \frac{\del^2}{\del x_j^2} \\ - \opV &= V(x) -\end{align} - -For the rest of this report, it will be used that $\opU = U(\bvec{x})$ and $\opW = W(\bvec{x})$ and they will denote the quadratic and non-quadratic splitting of $\opV$ respectively. -\par\medskip -% - -Hagedorn wavepackets explained in \cite{FGL_semiclassical_dynamics} - -The solution to the Schrödinger equation can be approximated as a finite linear combination if Hagedorn function \cite{FGL_semiclassical_dynamics} -\begin{align} - \label{math:hagedornwp} - \psi(x,t) \approx u(x,t) - = e^{\im S(t)/\eps} \sum_{k \in \K} c_k(t) \varphi_k^\eps [ q(t),p(t),Q(t),P(t) ] (x) -\end{align} - -$\K$ finite multi index set - -As the idea behind the splitting the Hamiltonian operator is so central to the time propagation approach used here, we repeat the most important findings from \cite{FGL_semiclassical_dynamics}: -\begin{itemize} - \item The free linear Schrödinger equation ($\opV=0$) can be solved exactly and the wavepacket remains in Hagedorn wavepacket form~\ref{math:hagedornwp}. - For time propagation, only the parameters $\bvec{\Pi},S$ need to be updated, the coefficients $\{c_k\}_{k \in \K}$ remain unchanged. - \item The Schrödinger equation~\ref{math:tdse} can be solved exactly in a pure quadratic potential, i.e. in a potential $\opV=U(\bvec{x})$ with $\opT=0$. - Again, time propagation only affects the parameters $\bvec{\Pi},S$, not the coefficients $\{c_k\}_{k \in \K}$. - \item In the case of an arbitrary potential $\opV = W(\bvec{x})$ that is not quadratic, a set of Galerkin functions can be propagated by adapting the coefficients $\{c_k\}_{k \in \K}$ without changing the parameters $\bvec{\Pi},S$. -\end{itemize} - - -Small semiclassical parameter $\eps$ plays an important role in the stability of numerical schemes - -Classical dynamics in the limit $\eps \rightarrow 0$, quantum mechanics for $\eps = 1$. - -In particular, as underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\sim \eps^{-2})$ for Lie-Trotter splitting, Strang splitting and other related methods. - -Motivation - why is time propagation important? -Background - - -The Hamiltonian is split into three parts -where $\opU(\bvec{q}(t),\bvec{x})$ is the second order Taylor approximation of the potential $\opV$ around $\bvec{q}(t)$ and $\opW(\bvec{q}(t),\bvec{x})$ is the corresponding remainder. - -$\opV = V(\bvec{x}) = U(\bvec{q},\bvec{x}) + W(\bvec{q},\bvec{x})$ - -\begin{align} - U(\bvec{q},\bvec{x}) &:= V(\bvec{q}) + \nabla V(\bvec{q}) (\bvec{x}-\bvec{q}) - + \frac{1}{2} (\bvec{x}-\bvec{q})^T \nabla^2 V(\bvec{q}) (\bvec{x}-\bvec{q}) \\ - W(\bvec{q},\bvec{x}) &:= V(\bvec{x}) - U(\bvec{q},\bvec{x}) -\end{align} - -The kinetic part $\opT$ and the quadratic part of the potential $\opU$ can be integrated exactly (see \cite{FGL_semiclassical_dynamics}). - - -The search for accurate time propagation schemes has become an area of research by itself, see papers of A and B - -General notes about Operator Splitting? - -Notation and Conventions -% -In this short section, some -Consistent with previous work on libwaveblocks and existing code -$N$ number of levels -$D$ dimension -$M$ number of timesteps steps - -TODO: check: are q and p vectors? $\bm{q,p}$ \\ -$q^n, p^n \in V_N, \; Q^n, P^n \in M_{N \times N}, \; S^n \in \mathbb{R}, \bm{c}^n \in (c_k^n)_{k \in \K}$ - -TODO: hom/inhom for all algorithms!!! - - - - - - - - -% \clearpage \input{sections/buildingblocks.tex} -% \clearpage \input{sections/propagators.tex} +\clearpage \input{sections/operatorsplitting.tex} +\clearpage \input{sections/buildingblocks.tex} +\clearpage \input{sections/propagators.tex} \clearpage \section{Implementation in C++} +\subsection{Callback function} +\subsection{IntSplit using alternating templates} +\subsection{Static Polymorphism through CRTP and enable\_if} + \begin{comment} \cite{libwaveblocks} What was the main challenge? What was controllable (which code could be changed), what not? @@ -159,8 +70,8 @@ \section{Benchmark / Case Study / Example} \section{Conclusion} \begin{comment} -final observations -suggestions for future work +% TODO: final observations +% TODO: suggestions for future work \end{comment} diff --git a/LWB/propagators/sections/operatorsplitting.tex b/LWB/propagators/sections/operatorsplitting.tex new file mode 100644 index 0000000..ca4109b --- /dev/null +++ b/LWB/propagators/sections/operatorsplitting.tex @@ -0,0 +1,67 @@ +\section{Operator Splitting for Time Propagation} +\label{sec:operatorsplitting} +% +In this report, we will use the variable $D$ for describing the number of dimensions and $N$ for the number of energy levels in the system. +The semiclassical time-dependent Schrödinger equation is of the form +% +\begin{align} + \label{math:tdse} + \im \eps \; \del_t \psi (\bvec{x},t) = \opH (\eps) \; \psi (\bvec{x},t) +\end{align} +% +with wave function $\psi(\bvec{x},t)$ depending on the spatial variables $\bvec{x} = (x_1,\dots,x_D) \in \R^D$ and the time variable $t \in \R$. +The parameter $\eps$, also known as the semiclassical parameter, determines the degree of influence of quantum dynamics, approaching classical dynamics in the limit $\eps \rightarrow 0$ and full quantum mechanics for $\eps = 1$. \\ +Moreover, $\eps$ plays an important role in the stability of numerical schemes. +As underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\sim \eps^{-2})$ for Lie-Trotter splitting, Strang splitting and other related methods, meaning that smaller $\eps$ would lead to a larger error. +\par\medskip +% +The solution to the above Schrödinger equation \ref{math:tdse} can be approximated as a finite linear combination of Hagedorn functions +\begin{align} + \label{math:hagedornwp} + \psi(\bvec{x},t) \approx u(\bvec{x},t) + = e^{\im S(t)/\eps} \sum_{k \in \K} c_k(t) \varphi_k^\eps [ \bvec{\Pi} ] (\bvec{x}) +\end{align} +where $\K$ represents a finite multi-index set and the notation for the parameter pack $\bvec{\Pi} = (\bvec{q}(t),\bvec{p}(t),\bvec{Q}(t),\bvec{P}(t))$ is introduced for convenience. +A more detailed analysis of Hagedorn functions is given in \cite{FGL_semiclassical_dynamics}. +\par\medskip +% +For the rest of this report, $\opU$ and $\opW$ will denote the quadratic and non-quadratic component of $\opV$ respectively and it will be used that +% +\begin{align} + \opV &= V(\bvec{x}) \\ + \opU &= U(\bvec{q},\bvec{x}) := V(\bvec{q}) + \nabla V(\bvec{q}) (\bvec{x}-\bvec{q}) + + \frac{1}{2} (\bvec{x}-\bvec{q})^T \nabla^2 V(\bvec{q}) (\bvec{x}-\bvec{q}) \\ + \opW &= W(\bvec{q},\bvec{x}) := V(\bvec{x}) - U(\bvec{q},\bvec{x}) +\end{align} +% +In other words, $\opU = U(\bvec{q},\bvec{x})$ is the second order Taylor expansion of the potential energy $\opV = V(\bvec{x})$ around $\bvec{q}$ (the coordinate $\bvec{q}$ will be omitted for simplicity of notation) and $\opW = W(\bvec{q},\bvec{x})$ is the corresponding remainder. +\par\medskip +% +By further recalling the form of the kinetic operator $\opT$ +\begin{align} + \opT &= - \sum_{j=1}^D \frac{\eps^2}{2m_j} \frac{\del^2}{\del x_j^2} +\end{align} +% +one can then split the Hamiltonian operator $\opH$ into the components +% +\begin{align} + \opH = \opT + \opV = \opT + \opU + \opW +\end{align} +% +This splitting is so central to the construction of the time propagators in the following sections that it is worthwile repeating the most important findings from \cite{FGL_semiclassical_dynamics}: +\begin{itemize} + \item The free linear Schrödinger equation ($\opV=0$) can be solved exactly and the wavepacket remains in Hagedorn wavepacket form (equation~\ref{math:hagedornwp}). + For time propagation, only the parameters $\bvec{\Pi},S$ need to be updated, the coefficients $\{c_k\}_{k \in \K}$ remain unchanged. + \item The Schrödinger equation~\ref{math:tdse} can be solved exactly in a pure quadratic potential, i.e. in a potential $\opV=U(\bvec{x})$ with $\opT=0$. + Again, time propagation only affects the parameters $\bvec{\Pi},S$, not the coefficients $\{c_k\}_{k \in \K}$. + \item In the case of an arbitrary potential $\opV = W(\bvec{x})$ that is not quadratic, a set of Galerkin functions can be propagated by adapting the coefficients $\{c_k\}_{k \in \K}$ without changing the parameters $\bvec{\Pi},S$. +\end{itemize} +% +The following sections will heavily make use of these properties in order to build composition methods for time integration of quantum wave packets. + +% TODO: describe how exact propagation works +% TODO: describe idea of operator splitting in a nutshell +% All based on composition methods +% methods that compose elementary flows/operators to build an approximation of high order +% TODO: operators don't commute +% TODO?: $q^n, p^n \in V_N, \; Q^n, P^n \in M_{N \times N}, \; S^n \in \mathbb{R}, \bm{c}^n \in (c_k^n)_{k \in \K}$ From 70cd593a303943128998285885ebb4c0de8dfeb3 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sat, 4 Feb 2017 03:44:49 +0100 Subject: [PATCH 10/28] Updated Title --- LWB/propagators/main.tex | 4 ++++ LWB/propagators/sections/title.tex | 6 +++--- 2 files changed, 7 insertions(+), 3 deletions(-) diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index ac8b0da..ba6b1eb 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -32,6 +32,10 @@ \subsection{IntSplit using alternating templates} \subsection{Static Polymorphism through CRTP and enable\_if} \begin{comment} +The main achievement is that the algorithms can be implemented in a stunningly simple way + +and that they are stunningly similar to the pseudo-algorithms of section \ref{sec:propagators} +even though they rely on highly optimized C++ code \cite{libwaveblocks} What was the main challenge? What was controllable (which code could be changed), what not? Quick alternating o diff --git a/LWB/propagators/sections/title.tex b/LWB/propagators/sections/title.tex index 45eb953..942b34a 100644 --- a/LWB/propagators/sections/title.tex +++ b/LWB/propagators/sections/title.tex @@ -5,12 +5,12 @@ \vspace{2cm} - {\huge \textsc{Time Propagation\\ of Quantum Wave Packets\\[10pt]}} - {\Large \textsc{an efficient implementation in C++\\[10pt]}} + {\Large \textsc{Time Propagation\\ of Hagedorn Quantum Wave Packets\\[5pt]}} + {\large \textsc{an efficient implementation in C++}} \vspace{1cm} - {\Large{Bachelor Thesis}} + {\large{Bachelor Thesis}} \vspace{1cm} From c41b09f0f5e521ced34fc9440673bbd31d330de8 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sat, 4 Feb 2017 21:51:43 +0100 Subject: [PATCH 11/28] Implementation: Intsplit --- LWB/propagators/header.tex | 17 ++ LWB/propagators/main.tex | 59 ++++--- LWB/propagators/sections/buildingblocks.tex | 6 +- LWB/propagators/sections/implementation.tex | 167 ++++++++++++++++++++ 4 files changed, 216 insertions(+), 33 deletions(-) create mode 100644 LWB/propagators/sections/implementation.tex diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex index 7f9c7d1..60f852f 100644 --- a/LWB/propagators/header.tex +++ b/LWB/propagators/header.tex @@ -102,6 +102,23 @@ } \lstinputlisting{#1} } + + \lstset{language=c++, + basicstyle=\smaller, + basewidth=0.58em, + columns=fixed, + tabsize=2, + fontadjust=true, + breaklines=true, + breakindent=0pt, + prebreak=\mbox{\tiny$\searrow$}, + postbreak=\mbox{{\color{gray}$\cdots$}}, + numberstyle=\color{gray}, + commentstyle=\color{gray}, + stringstyle=\textit, + showstringspaces=false, + basicstyle=\footnotesize + } %%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index ba6b1eb..49be59a 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -1,5 +1,17 @@ \input{./header.tex} + + + +\usepackage{tikz} +\usetikzlibrary{arrows,chains,matrix,positioning,scopes} + + + + + + + \begin{document} \input{sections/title.tex} @@ -24,61 +36,46 @@ -\clearpage -\section{Implementation in C++} -\subsection{Callback function} -\subsection{IntSplit using alternating templates} -\subsection{Static Polymorphism through CRTP and enable\_if} -\begin{comment} -The main achievement is that the algorithms can be implemented in a stunningly simple way -and that they are stunningly similar to the pseudo-algorithms of section \ref{sec:propagators} -even though they rely on highly optimized C++ code -\cite{libwaveblocks} -What was the main challenge? What was controllable (which code could be changed), what not? -Quick alternating o -The challenges were twofold: -First, identifying of all the common basic building blocks and expressing all algorithms in terms of these components -Second, implementing all basic building blocks in a generic but efficient manner -Third, minimize the overhead for switching between function calls +\clearpage \input{sections/implementation.tex} - - encapsulate the functionality in a clean interface without loosing flexibility and - - write efficient/fast code +\subsection{Callback function} +\label{subsec:callback} +encapsulate all the repetitive +and provide a +mention printing +prettyprint for clean printing + + +\subsection{Static Polymorphism through CRTP and enable\_if} +\label{subsec:poly} +inheritance: ugly implementational details are hidden in base class Propagators. +This was achieved by introducing a class hierarchy for the propagators, meaning that all +\cite{C_CRTP} -Tell how each of these challenges were overcome +Allows to implement the basic building blocks in a generic but efficient manner. -Motivation of software design choices - - Why class structure/inheritance (explain hierarchy) - - CRTP / static polymorphism \cite{C_CRTP} - - recognizing IP - - callback function - - implementation of IntSplit (TU,UT, partial specialization) - - polymorphism through enable-if \clearpage -\end{comment} \section{Benchmark / Case Study / Example} -\begin{comment} +IDEA: express cost of every propagator as a ratio to the hagedorn propagator \clearpage -\end{comment} \section{Conclusion} -\begin{comment} % TODO: final observations % TODO: suggestions for future work -\end{comment} diff --git a/LWB/propagators/sections/buildingblocks.tex b/LWB/propagators/sections/buildingblocks.tex index 2a9a5be..0cb75c0 100644 --- a/LWB/propagators/sections/buildingblocks.tex +++ b/LWB/propagators/sections/buildingblocks.tex @@ -1,7 +1,8 @@ \section{Common building blocks for time evolution schemes} \label{sec:buildingblocks} % -This section will briefly introduce the most important, generic building blocks that are needed for building the propagators in section~\ref{sec:propagators}. +In order to be able to create an efficient, still flexible, implementation of various time propagators, it is helpful to break them down in common, optimized building blocks in terms of which all (or most) propagators can be expressed. +This section will briefly introduce the most important, generic building blocks that were found neccessary in order to build the propagators in section~\ref{sec:propagators}. For each of them, it will be assumed that the following wave packet variables are implicitly accessible, without passing them as an argument to the procedure: the parameter pack $\bvec{\Pi} = (\bvec{q},\bvec{p},\bvec{Q},\bvec{P})$, the phase factor $S$, the coefficients $\{ c_k \}_{k \in \K}$, the potential function $V(\bvec{q})$ (including Jacobian and Hessian thereof) and the potential remainder $W(\bvec{q})$. \par\medskip % @@ -112,7 +113,8 @@ \subsection{StepT, StepU and IntSplit} %%% IntSplit In the practical implementations used here, propagation with $\opT$ and $\opU$ is usually replaced by a series of smaller, alternating steps with the two operators. In order to facilitate the use of such an alternating pattern, the helper function \proc{IntSplit} (algorithm~\ref{alg:intsplit}) is introduced. -This function splits the timestep $\Dt$ into $M$ smaller timesteps of size $\dt$ and uses the pair of weight arrays $\{ w_T, w_U \}$ to alternatingly apply operators $\opT$ and $\opU$ on each of the sub-intervals. +This function splits the timestep $\Dt$ into $M$ smaller timesteps of size $\dt$ and uses the pair of weight lists $\{ w_T, w_U \}$ to alternatingly apply operators $\opT$ and $\opU$ on each of the sub-intervals. +More detailed information on the individual parameters to this function and the practical implementation can be found in section \ref{subsec:intsplit} % \begin{algorithm}[h] \caption{Split a given time interval into $M$ sub-intervals and alternatingly apply weighted steps with $\opT$ and $\opU$ on each of them} diff --git a/LWB/propagators/sections/implementation.tex b/LWB/propagators/sections/implementation.tex new file mode 100644 index 0000000..ba619e3 --- /dev/null +++ b/LWB/propagators/sections/implementation.tex @@ -0,0 +1,167 @@ +\section{Implementation in C++} +% +This section will explain the most important design decisions and technical details on which the implementation of the presented algorithms is based. +While the first priority was clearly to make the code efficient and reduce computation time, another aim was also to create simple and readable code interfaces that make it easy to use and re-use components and extend the existing functionality. \\ +% +Even though these two requirements often seem to mutually exclude each other, in this case it was possible to meet both targets. +In fact, it is quite remarkable how similar the C++ implementations of the individual propagators look to their pseudo-algorithms as presented in section \ref{sec:propagators}. +\par\medskip +% +Also, it should be mentioned here that some of the most important bottlenecks of the code were not strictly related to the implementation of the propagators, such as for example the building of the $\bmat{F}$-matrix through numerical quadrature of $\matrixel{\varphi_i}{W}{\varphi_j}$. +In fact, the current implementation of the \proc{build\_matrix} routine for computing the inner product requires to always compute a full $N \times N$ sub-matrix, even if just one single entry is needed, which is a very severe inefficiency if the number of levels $N>1$. +\par\medskip +% +However, the aim of this project was to provide a fast implementation of the propagators by re-using existing functions, therefore possible performance enhancements through changing of the existing code was out of the scope of this project and could be targeted in future work. +Consequently, providing an efficient implementation for time propagation mainly translates into finding an efficient way for calling existing functions with minimal overhead and by making use of compiler optimizations where possible. +\par\medskip +% +First, in subsection \ref{subsec:callback}, it will be discussed how the \proc{Evolve} method (see algorithm \ref{alg:evolve}) was implemented in order to provide a minimal working interface to the user while still preserving all the flexibility that may be required for intermediate measurements or other interaction. +Secondly, subsection \ref{subsec:poly} outlines how C++ features were used in order to achieve class inheritance and polymorphism witout any additional runtime cost. +Finally, subsection \ref{subsec:intsplit} dives into the technical details of achieving inlined, highly optimized function calls for the \proc{IntSplit} method (see algorithm \ref{alg:intsplit}) that is used intensively in all the presented propagators. +\par\medskip +% +Some details about the code were not found worthy to be mentioned here, +but more information may be found in the Doxygen code documentation. + + +\subsection{IntSplit using alternating templates} +\label{subsec:instplit} +% +The \proc{IntSplit} function is merely a helper function that alternatingly applies \proc{StepT} and \proc{StepU}. +Steps with the operators $\opT$ and $\opU$ are usually much cheaper than steps with $\opW$ (which requires numerical integration for approximating the inner product $\matrixel{\varphi_i}{W}{\varphi_j}$), but are in many cases found to be one of the main soures of error. +Therefore the idea of sub-dividing the steps into many smaller intervals - an approach introduced in the semiclassical splitting \ref{subsec:semiclassical} - is applied in many other propagators as well. +Consequently, it is desirable to have a generic and efficient implementation available. +\par\medskip +% +The current \proc{IntSplit} algorithm described in \ref{alg:intsplit} takes three arguments: +the size of the time interval to be covered $\Dt$, the number $M$ of subintervals of size $\dt = \Dt/M$ in which the initial interval should be divided, and a pair of coefficient lists $\{ w_T, w_U \}$ denoting the weights with which the single steps of size $\dt$ should be scaled. +In the simplest case, the lists $w_T$ and $w_U$ are both only of length one (Lie-Trotter), but higher order splitting strategies can have much larger coefficient sets. +In general, it must hold that $\text{length}(w_T) = \text{length}(w_U)$ (for TU..TU schemes) or $\text{length}(w_T) = \text{length}(w_U)$ (for TU..UT schemes). +\par\medskip +% +The obvious implementation of using a dynamic container (like a C++ vector) for the coefficients and a simple \emph{for loop} over all its entries brings a two (minimal) drawbacks: First, the overhead of the loop itself and second, the fact that the number of coefficients is not known at compile time and thus no optimizations can be done. +Given that the functions \proc{StepT} and \proc{StepU} themselves are very cheap (just simple updates of parameters $\bvec{\Pi}$ and $S$), it is preferable to remove any kind of additional cost, even if only marginal. +\par\medskip +% +By making use of C++ arrays and templates, one can remove the issues mentioned above. \\ +% +Firstly, by replacing C++ vectors (dynamically allocated) through C++ arrays (statically allocated), the container size is known at compile time. +For convenience, a new struct \proc{SplitCoefs} was introduced that holds a pair of such arrays. +Secondly, and with the information about container size readily available at compile time, one can then implement two sets of templates \proc{TU} and \proc{UT} that execute a step with operator $\opT$ or $\opU$ respectively and then mutually invoke each other, until both arrays of coefficients are used up. +As a termination condition, when all coefficients have been used, a partially specialized instance of the templates will be called and interrupt the mutual invocation. \\ +% +One technical subtlety of this approach is, that the C++ language does not allow partially specialized function templates. This can, however, be overcome quite easily by declaring \proc{TU} and \proc{UT} as classes with a static function. +In contrast to function templates, class templates can be partially specialized. +By using static member functions, the actual class objects do not need to be created and all the function calls will be resolved at compile time. +\footnote{One limitation is the maximal template recursion depth. The default value on gcc compiler is 900 and can be increased if required. Typical template nesting depths for coefficient pairs should lie significantly below 100.} +All templates, no loops, and the resulting sequential code can then be further optimized by the compiler. +\par\medskip +% +Some minimal code snippets explaining the working principle of \proc{IntSplit} are shown in figure~\ref{fig:intsplit} +% +In order to prevent the inappropriate use of the \proc{IntSplit} function through incompatible coefficient lists $w_T$ and $w_U$, their sizes are checked in the code at compile time via static assertions. +\par\medskip +% +Also, as seen in section \ref{sec:blub}, a splitting of the Hamiltonian into three operators $\opT + \opU + \opW$ was used, but $\opW$ is significantly more costly to compute. +This justifies the approach of using templates that are hard-coded to call functions \proc{StepT} and \proc{StepU}. + +\begin{figure} + \begin{minipage}[c]{\textwidth} + \begin{minipage}[l][10cm][t]{.4\textwidth} + \begin{minipage}[t]{\textwidth} +\begin{lstlisting} +// Specialized Template +// N_T == S_T +// N_U == S_U +template +struct TU { + static void split( + array& w_T, + array& w_U, + float dt) + { /* terminate */ } +}; +\end{lstlisting} + \end{minipage} \\ + \vspace{5mm} + \begin{minipage}[t]{\textwidth} +\begin{lstlisting} +template +struct TU { + static void split( + array& w_T, + array& w_U, + float dt) + /* T */ + stepT(w_T(N_T)*dt); + /* UT... */ + UT + ::split(w_U,w_T,dt); + } +}; +\end{lstlisting} + \end{minipage} + \end{minipage} + \begin{minipage}[l][10cm][t]{.18\textwidth} + \vspace{25mm} + \resizebox{\textwidth}{!}{ +\begin{tikzpicture}[->,>=stealth',auto,node distance=3cm, + thick,main node/.style={circle,draw,font=\sffamily\Large\bfseries}] + \node (TU) at (0,0) {}; + \node (UT) at (2,0) {}; + \node (TUexit) at (0,2) {}; + \node (UTexit) at (2,2) {}; + \node (TUend) at (0,4) {}; + \node (UTend) at (2,4) {}; + \path[every node/.style={font=\sffamily\small}] + (TU) edge[bend right] node {} (UT) + (UT) edge[bend right] node {} (TU) + (TUexit) edge node {} (UTend) + (UTexit) edge node {} (TUend); +\end{tikzpicture} + } + \end{minipage} + \begin{minipage}[r][10cm][t]{.4\textwidth} + \begin{minipage}[t]{\textwidth} +\begin{lstlisting} +// Specialized Template +// N_U == S_U +// N_T == S_T +template +struct UT { + static void split( + array& w_U, + array& w_T, + float dt) + { /* terminate */ } +}; +\end{lstlisting} + \end{minipage} \\ + \vspace{5mm} + \begin{minipage}[t]{\textwidth} +\begin{lstlisting} +template +struct UT { + static void split( + array& w_U, + array& w_T, + float dt) + /* U */ + stepU(w_U(N_U)*dt); + /* TU... */ + TU + ::split(w_T,w_U,dt); + } +}; +\end{lstlisting} + \end{minipage} + \end{minipage} + + + \end{minipage} + \label{fig:intsplit} + \caption{Simplified C++ snippets for outlining the template invocation sequence in the intsplit function. The template parameters \proc{S\_T} and \proc{S\_U} denote the total size of the coefficient arrays \proc{w\_T} and \proc{w\_U} while the \proc{N\_T} and \proc{N\_U} are the indices of the current coefficient to be applied. The \proc{TU::split} and \proc{UT::split} methods first apply their own step and then invoke each other by gradually incrementing the incices \proc{N\_T} and \proc{N\_U}. \\ + When the conditions \proc{N\_T == S\_T} and \proc{N\_U == S\_U} are satisfied, the partially specialized templates become active and the recursion is terminated.} +\end{figure} From 79ea3a0f13431fbd58d99756eb09deb478aad418 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sat, 4 Feb 2017 22:23:52 +0100 Subject: [PATCH 12/28] Implementation: Callback --- LWB/propagators/main.tex | 11 ---------- LWB/propagators/sections/implementation.tex | 24 +++++++++++++++++++++ 2 files changed, 24 insertions(+), 11 deletions(-) diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 49be59a..948eb85 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -42,14 +42,6 @@ \clearpage \input{sections/implementation.tex} -\subsection{Callback function} -\label{subsec:callback} -encapsulate all the repetitive -and provide a -mention printing -prettyprint for clean printing - - \subsection{Static Polymorphism through CRTP and enable\_if} \label{subsec:poly} inheritance: ugly implementational details are hidden in base class Propagators. @@ -61,9 +53,6 @@ \subsection{Static Polymorphism through CRTP and enable\_if} - - - \clearpage \section{Benchmark / Case Study / Example} diff --git a/LWB/propagators/sections/implementation.tex b/LWB/propagators/sections/implementation.tex index ba619e3..ff511f5 100644 --- a/LWB/propagators/sections/implementation.tex +++ b/LWB/propagators/sections/implementation.tex @@ -24,6 +24,30 @@ \section{Implementation in C++} but more information may be found in the Doxygen code documentation. +\subsection{Evolve and callback function} +\label{subsec:callback} +% +The simple \proc{Evolve} method as described in algorithm \ref{alg:evolve} in the section about propagator building blocks provides a handy wrapper for calling the \proc{PreProcessing}, \proc{Processing} and \proc{PostProcessing} methods in sequence and dividing the propagation time into smaller steps. \\ +% +However, one may still want to interact with the wave packet during the simulation, for example for writing the wave packet norm to file every 100 steps, throwing errors when energy conservation is violated or to carry out any other task that requires intermediate simulation results. +In order not to loose any flexibility in this regard, an optional callback-function argument was added to the \proc{Evolve} function. +The callback function is passed as a lambda function taking two arguments, the current simulation time \proc{t} and the index of the current timestep \proc{m}. +If no callback function is passed, it defaults to an empty function. +\par\medskip +% +Furthermore, since the wave packet is passed to the \proc{Propagator} class \emph{by reference}, it is of course available in the callback function as well and does not need to be passed as an argument explicitly (provided that it is captured in the angular brackets \proc{[]} of the lambda function). \\ +\emph{Disclaimer} if the value of the wave packet is (intentionally or unintentionally) changed in the callback function, this will of course also affect the further results of the simulation. +Also, care must be taken when measuring energies of packets that are being propagated with propagators such as \emph{Pre764} which apply a pre-processor to the packet before starting the simulation. +In order to get the ``physically correct'' wave packet, the corresponding post-processor needs to be applied before making any meaningful measurement. +\par\medskip +% +The callback function is called once prior to simulation start and then in every time step until termination. +\par\medskip +% +Finally, a few helper functions were introduced in order to facilitate pretty printing of console output and some information about the currently propagated wave packet is printed at the beginning of the \proc{Evolve} function. +These console outputs can (and should) of course be removed for optimal performance, but in most cases it was found that some feedback about the current status of the simulation is highly valuable. + + \subsection{IntSplit using alternating templates} \label{subsec:instplit} % From 5b5280b4b91d77d88aa83db1d82f9736a44ed508 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sat, 4 Feb 2017 23:14:59 +0100 Subject: [PATCH 13/28] Implementation: Polymorphism --- LWB/propagators/Makefile | 1 + LWB/propagators/header.tex | 4 + LWB/propagators/main.tex | 101 +------------------- LWB/propagators/sections/benchmark.tex | 3 + LWB/propagators/sections/conclusion.tex | 4 + LWB/propagators/sections/implementation.tex | 24 +++++ references/propagators.bib | 9 ++ 7 files changed, 49 insertions(+), 97 deletions(-) create mode 100644 LWB/propagators/sections/benchmark.tex create mode 100644 LWB/propagators/sections/conclusion.tex diff --git a/LWB/propagators/Makefile b/LWB/propagators/Makefile index 1224dff..432f842 100644 --- a/LWB/propagators/Makefile +++ b/LWB/propagators/Makefile @@ -4,6 +4,7 @@ SECTIONS = $(wildcard sections/*.tex) all: main.pdf +.PHONY: main.pdf main.pdf: main.tex header.tex $(SECTIONS) # figs $(PDFLATEX) main.tex $(BIBTEX) main.aux diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex index 60f852f..62a9591 100644 --- a/LWB/propagators/header.tex +++ b/LWB/propagators/header.tex @@ -202,3 +202,7 @@ \usepackage{bm} % \usepackage[noend]{algpseudocode} \usepackage{algpseudocode} + + +\usepackage{tikz} +\usetikzlibrary{arrows,chains,matrix,positioning,scopes} diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 948eb85..3e48d05 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -1,27 +1,12 @@ \input{./header.tex} - - - -\usepackage{tikz} -\usetikzlibrary{arrows,chains,matrix,positioning,scopes} - - - - - - - \begin{document} \input{sections/title.tex} \clearemptydoublepage -\mbox{} -\vfill -\input{sections/abstract.tex} -\vfill +\mbox{} \vfill \input{sections/abstract.tex} \vfill \tableofcontents \clearpage @@ -33,90 +18,12 @@ \clearpage \input{sections/operatorsplitting.tex} \clearpage \input{sections/buildingblocks.tex} \clearpage \input{sections/propagators.tex} - - - - - - - \clearpage \input{sections/implementation.tex} +\clearpage \input{sections/benchmark.tex} +\clearpage \input{sections/conclusion.tex} -\subsection{Static Polymorphism through CRTP and enable\_if} -\label{subsec:poly} -inheritance: ugly implementational details are hidden in base class Propagators. -This was achieved by introducing a class hierarchy for the propagators, meaning that all -\cite{C_CRTP} - -Allows to implement the basic building blocks in a generic but efficient manner. - - - - -\clearpage -\section{Benchmark / Case Study / Example} - -IDEA: express cost of every propagator as a ratio to the hagedorn propagator - -\clearpage -\section{Conclusion} - -% TODO: final observations -% TODO: suggestions for future work - - - - - - - - - - - - - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\clearemptydoublepage +% \clearemptydoublepage \appendix -\appendix -% \input{./deriv_ew.tex} -% \input{./color_code.tex} -% \input{./ack.tex} \clearemptydoublepage diff --git a/LWB/propagators/sections/benchmark.tex b/LWB/propagators/sections/benchmark.tex new file mode 100644 index 0000000..d0356a7 --- /dev/null +++ b/LWB/propagators/sections/benchmark.tex @@ -0,0 +1,3 @@ +\section{Benchmark / Case Study / Example} + +% TODO: IDEA: express cost of every propagator as a ratio to the hagedorn propagator diff --git a/LWB/propagators/sections/conclusion.tex b/LWB/propagators/sections/conclusion.tex new file mode 100644 index 0000000..e9934bf --- /dev/null +++ b/LWB/propagators/sections/conclusion.tex @@ -0,0 +1,4 @@ +\section{Conclusion} + +% TODO: final observations +% TODO: suggestions for future work diff --git a/LWB/propagators/sections/implementation.tex b/LWB/propagators/sections/implementation.tex index ff511f5..9fd3109 100644 --- a/LWB/propagators/sections/implementation.tex +++ b/LWB/propagators/sections/implementation.tex @@ -22,6 +22,30 @@ \section{Implementation in C++} % Some details about the code were not found worthy to be mentioned here, but more information may be found in the Doxygen code documentation. +\par\medskip +% +The Hagedorn Propagator was already implemented in C++ as part of another project on the waveblocks library \cite{libwaveblocks}. +Many components of that code were re-used for the present implementation, but they were fundamentally re-organized into a new structure that has better support for many, diverse propagators. + + +\subsection{Static Polymorphism through CRTP and enable\_if} +\label{subsec:poly} +% +In the current software design, all propagators from section \ref{sec:propagators} inherit from a common base class \proc{Propagator}, see the UML diagram \ref{fig:uml}. +The strategy for the implementation was the following: +Provide an efficient and generic implementation of all the building blocks from section \ref{sec:buildingblocks} in a common base class (\proc{Propagator}) and then let the derived class decide which combination of building blocks to use for its own, specific implementation of the propagation methods. +This idea is essentially an application of the Strategy Pattern as introduced by Gamma et al. in \cite{Gamma1995}, one of the most important references for software design patterns. +\par\medskip +% +A popular C++ design pattern for achieving static polymorphism is the \emph{Curiously Recurring Template Pattern}, \emph{CRTP} for short, which is described in more detail in \cite{C_CRTP} and in earlier work on the WaveBlocks project \cite{libwaveblocks}. +\par\medskip +% +While CRTP is an excellent candidate for providing static polymorphism for derived classes, there was also the need for some polymorphism in the Propagator base class itself, since some of the building blocks need to have different implementations based on the dimensionality or homogeneity of the wave packet. +Providing a partially specialized class for every possible combination of parameters is often tedious and unneccessarily bloats the code. +Therefore, the C++11 \proc{enable\_if} keyword was used. +\proc{enable\_if} allows to define multiple versions of a function and (de)activate them based on a compile-time boolean value. +If the boolean is true, the function is enabled. Otherwise it is omitted. +This handy metafunction from the C++ \proc{type\_traits} header was used in order to conditionally enable functions based on whether the number of levels $N$ was equal or greater to one or to distinguish between functions for homogeneous or inhomogeneous wave functions, which effectively provides a form of compile time polymorphism. \subsection{Evolve and callback function} diff --git a/references/propagators.bib b/references/propagators.bib index aa36bf1..3cc3547 100644 --- a/references/propagators.bib +++ b/references/propagators.bib @@ -84,3 +84,12 @@ @Article{McLachlan1995 url="http://dx.doi.org/10.1007/BF01737165" } + +@book{Gamma1995, + author = {Gamma, Erich and Helm, Richard and Johnson, Ralph and Vlissides, John}, + title = {Design Patterns: Elements of Reusable Object-oriented Software}, + year = {1995}, + isbn = {0-201-63361-2}, + publisher = {Addison-Wesley Longman Publishing Co., Inc.}, + address = {Boston, MA, USA}, +} From db2df36b3814f0dde1ca232e85f0dc18923da634 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sun, 5 Feb 2017 00:53:16 +0100 Subject: [PATCH 14/28] Implementation: UML --- LWB/propagators/Makefile | 5 +- LWB/propagators/figures/uml.png | Bin 0 -> 99714 bytes LWB/propagators/main.tex | 9 ++ LWB/propagators/sections/buildingblocks.tex | 6 +- LWB/propagators/sections/implementation.tex | 13 ++- LWB/propagators/sections/propagators.tex | 2 +- LWB/propagators/uml.dot | 100 ++++++++++++++++++++ 7 files changed, 129 insertions(+), 6 deletions(-) create mode 100644 LWB/propagators/figures/uml.png create mode 100644 LWB/propagators/uml.dot diff --git a/LWB/propagators/Makefile b/LWB/propagators/Makefile index 432f842..1d18ca7 100644 --- a/LWB/propagators/Makefile +++ b/LWB/propagators/Makefile @@ -5,12 +5,15 @@ SECTIONS = $(wildcard sections/*.tex) all: main.pdf .PHONY: main.pdf -main.pdf: main.tex header.tex $(SECTIONS) # figs +main.pdf: main.tex header.tex $(SECTIONS) uml.png # figs 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\clearemptydoublepage \appendix diff --git a/LWB/propagators/sections/buildingblocks.tex b/LWB/propagators/sections/buildingblocks.tex index 0cb75c0..51c1cdf 100644 --- a/LWB/propagators/sections/buildingblocks.tex +++ b/LWB/propagators/sections/buildingblocks.tex @@ -78,7 +78,7 @@ \subsection{StepT, StepU and IntSplit} \end{algorithm} % \begin{algorithm}[h] - \caption{Propagate with (Quadratic) Potential Energy Operator} + \caption{Propagate with (Quadratic) Potential Energy Operator $\opU$} \label{alg:StepU} \begin{algorithmic} \State @@ -117,7 +117,7 @@ \subsection{StepT, StepU and IntSplit} More detailed information on the individual parameters to this function and the practical implementation can be found in section \ref{subsec:intsplit} % \begin{algorithm}[h] - \caption{Split a given time interval into $M$ sub-intervals and alternatingly apply weighted steps with $\opT$ and $\opU$ on each of them} + \caption{Split a time interval and alternatingly apply $\opT$ and $\opU$} \label{alg:intsplit} \begin{algorithmic} \State @@ -159,7 +159,7 @@ \subsection{StepW and building of the interaction matrix} For the further treatment in this report, it is assumed that an efficient implementation of \proc{BuildF} is readily available. % \begin{algorithm}[h] - \caption{Propagate with (Non-Quadratic) Potential Energy Operator} + \caption{Propagate with (Non-Quadratic) Potential Energy Operator $\opW$} \label{alg:StepW} \begin{algorithmic} \State diff --git a/LWB/propagators/sections/implementation.tex b/LWB/propagators/sections/implementation.tex index 9fd3109..6098c63 100644 --- a/LWB/propagators/sections/implementation.tex +++ b/LWB/propagators/sections/implementation.tex @@ -46,6 +46,17 @@ \subsection{Static Polymorphism through CRTP and enable\_if} \proc{enable\_if} allows to define multiple versions of a function and (de)activate them based on a compile-time boolean value. If the boolean is true, the function is enabled. Otherwise it is omitted. This handy metafunction from the C++ \proc{type\_traits} header was used in order to conditionally enable functions based on whether the number of levels $N$ was equal or greater to one or to distinguish between functions for homogeneous or inhomogeneous wave functions, which effectively provides a form of compile time polymorphism. +% +\begin{figure}[h] + \begin{center} + \includegraphics[height=10cm]{figures/uml.png} + \end{center} + \label{fig:uml} + \caption{Simplified UML diagram for the propagator implementation. + The derived propagator provides implementations for \proc{PrePropagate}, \proc{Propagate} and \proc{PostPropagate} which are called from the public \proc{Evolve} function. + Some functions and function parameters are omitted for better readability of the graph.} +\end{figure} + \subsection{Evolve and callback function} @@ -113,7 +124,7 @@ \subsection{IntSplit using alternating templates} Also, as seen in section \ref{sec:blub}, a splitting of the Hamiltonian into three operators $\opT + \opU + \opW$ was used, but $\opW$ is significantly more costly to compute. This justifies the approach of using templates that are hard-coded to call functions \proc{StepT} and \proc{StepU}. -\begin{figure} +\begin{figure}[h] \begin{minipage}[c]{\textwidth} \begin{minipage}[l][10cm][t]{.4\textwidth} \begin{minipage}[t]{\textwidth} diff --git a/LWB/propagators/sections/propagators.tex b/LWB/propagators/sections/propagators.tex index 66a82da..7744318 100644 --- a/LWB/propagators/sections/propagators.tex +++ b/LWB/propagators/sections/propagators.tex @@ -244,7 +244,7 @@ \subsection{McLachlan Propagators} The source code of this project contains the C++ implementation of the propagators \emph{McL(4,2)} (asymmetric) and \emph{McL(8,4)} (symmetric) that require only two respectively three evaluations of $\opW$ per step. \begin{algorithm}[h] - \caption{Single timestep with McLachlan propagators} + \caption{Single timestep with McL42 propagator} \label{alg:mcl} \begin{algorithmic} \State diff --git a/LWB/propagators/uml.dot b/LWB/propagators/uml.dot new file mode 100644 index 0000000..979d92d --- /dev/null +++ b/LWB/propagators/uml.dot @@ -0,0 +1,100 @@ +digraph Propagators { + + graph [dpi = 300]; + + fontname = "Bitstream Vera Sans" + fontsize = 48 + + node [ + fontname = "Bitstream Vera Sans" + fontsize = 8 + shape = "record" + ] + + edge [ + fontname = "Bitstream Vera Sans" + fontsize = 8 + ] + + PropagatorBase [ + label = "{Base Propagator|+ evolve(T,Dt,callback) \l|# virtual getName() \l# virtual propagate(Dt) \l# virtual pre_propagate(Dt) \l# virtual post_propagate(Dt) \l|# stepT(Dt) \l# stepU(Dt) \l# stepW(Dt) \l# buildF() \l# intSplit(Dt,M,\{w\}) \l- TU::split(Dt,\{w\}) \l- UT::split(Dt,\{w\}) \l|# Packet_t wpacket \l# Potential_t V \l# Matrix F \l}" + width = "2" + ] + + PropagatorDerived [ + label = "{Derived Propagator|# override getName() \l# override propagate(Dt) \l# override pre_propagate(Dt) \l# override post_propagate(Dt) \l}" + width = "2" + ] + + edge [ + arrowtail = "empty" + ] + + PropagatorBase -> PropagatorDerived [dir="back"] + + /* + subgraph clusterVectorialWavepackets { + label = "Vectorial Wavepackets" + + HomogeneousHaWp__Component [ + label = "{Component|- shape \l- coefficients \l}" + group = groupHomogeneousHaWp + ] + + HomogeneousHaWp [ + label = "{HomogeneousHaWp|- components \l- eps \l- parameters \l+ evaluate(grid) \l}" + group = groupHomogeneousHaWp + ] + + + InhomogeneousHaWp__Component [ + label = "{Component|- parameters \l- shape\l- coefficients \l}" + group = groupInhomogeneousHaWp + ] + + InhomogeneousHaWp [ + label = "{InhomogeneousHaWp|- components \l- eps \l+ evaluate(grid) \l}" + group = groupInhomogeneousHaWp + ] + + + HaWpGradient__Component [ + label = "{Component|- coefficients \l}" + group = groupHaWpGradient + ] + + HaWpGradient [ + label = "{HaWpGradient|- components \l- eps \l- parameters \l- shape \l+ evaluate(grid) \l}" + group = groupHaWpGradient + ] + + edge [ + arrowtail = "none" + headlabel = "N" + taillabel = "1" + ] + + HomogeneousHaWp -> HomogeneousHaWp__Component [dir="back"] + InhomogeneousHaWp -> InhomogeneousHaWp__Component [dir="back"] + + edge [ + arrowtail = "none" + headlabel = "D" + taillabel = "1" + ] + + HaWpGradient -> HaWpGradient__Component [dir="back"] + } + */ + + /* + edge [ + arrowtail = "empty" + ] + + AbstractScalarHaWpBasis -> HaWpGradient [dir="back"] + AbstractScalarHaWp -> HomogeneousHaWp__Component [dir="back"] + AbstractScalarHaWp -> InhomogeneousHaWp__Component [dir="back"] + AbstractScalarHaWp -> HaWpGradient__Component [dir="back"] + */ +} From 1827c0f5d15f33e6a187ac7ee1536b0b4fab6eb8 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sun, 5 Feb 2017 23:55:48 +0100 Subject: [PATCH 15/28] Results --- LWB/propagators/figures/drift_Hagedorn.png | Bin 0 -> 44777 bytes LWB/propagators/figures/drift_MG4.png | Bin 0 -> 44551 bytes LWB/propagators/figures/drift_McL42.png | Bin 0 -> 44513 bytes LWB/propagators/figures/drift_McL84.png | Bin 0 -> 38253 bytes LWB/propagators/figures/drift_Pre764.png | Bin 0 -> 44408 bytes .../figures/drift_Semiclassical.png | Bin 0 -> 44235 bytes LWB/propagators/figures/energy_Hagedorn.png | Bin 0 -> 76122 bytes LWB/propagators/figures/energy_MG4.png | Bin 0 -> 76131 bytes LWB/propagators/figures/energy_McL42.png | Bin 0 -> 76131 bytes LWB/propagators/figures/energy_McL84.png | Bin 0 -> 60693 bytes LWB/propagators/figures/energy_Pre764.png | Bin 0 -> 76652 bytes .../figures/energy_Semiclassical.png | Bin 0 -> 76131 bytes LWB/propagators/main.tex | 19 ++- LWB/propagators/sections/abstract.tex | 3 +- LWB/propagators/sections/appendix.tex | 81 +++++++++++ LWB/propagators/sections/benchmark.tex | 3 - LWB/propagators/sections/results.tex | 131 ++++++++++++++++++ 17 files changed, 231 insertions(+), 6 deletions(-) create mode 100644 LWB/propagators/figures/drift_Hagedorn.png create mode 100644 LWB/propagators/figures/drift_MG4.png create mode 100644 LWB/propagators/figures/drift_McL42.png create mode 100644 LWB/propagators/figures/drift_McL84.png create mode 100644 LWB/propagators/figures/drift_Pre764.png create mode 100644 LWB/propagators/figures/drift_Semiclassical.png create mode 100644 LWB/propagators/figures/energy_Hagedorn.png create mode 100644 LWB/propagators/figures/energy_MG4.png create mode 100644 LWB/propagators/figures/energy_McL42.png create mode 100644 LWB/propagators/figures/energy_McL84.png create mode 100644 LWB/propagators/figures/energy_Pre764.png create mode 100644 LWB/propagators/figures/energy_Semiclassical.png create mode 100644 LWB/propagators/sections/appendix.tex delete mode 100644 LWB/propagators/sections/benchmark.tex create mode 100644 LWB/propagators/sections/results.tex diff --git a/LWB/propagators/figures/drift_Hagedorn.png b/LWB/propagators/figures/drift_Hagedorn.png new file mode 100644 index 0000000000000000000000000000000000000000..b485d753f0fe49925f701bea6e7b88ccdb1cf36b GIT binary patch literal 44777 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zz{D4T>Q1{-5nAj=sdn{Ffsdl~%yhMxvG=ByX%vgYdSs`1r>N33&YrWNR+DTZqXJzh zNMb||PM7%fF*xYI<7SBU_m+~DoSdC?D6MU6b3g-EyINXI0jhzHzY^1NVEUAL5~+3_ z^SXfaWPY7vlm*|)8oTNg2^A$&$jkVJu91HvLwEL5d_y`^(dRM^Q@e80KQ9+uCc0vPx^Bo zkG(Oyyjika@EI-!3~?AP9jYa%QA$nvyEGfshsw&my&ABulfV8!^PQLR*&|QSYZv#7thS37<;%g(1@g#?BZPyB_s{{VjuF<}4z literal 0 HcmV?d00001 diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index b2ca6ae..7fe8231 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -19,7 +19,7 @@ \clearpage \input{sections/buildingblocks.tex} \clearpage \input{sections/propagators.tex} \clearpage \input{sections/implementation.tex} -\clearpage \input{sections/benchmark.tex} +\clearpage \input{sections/results.tex} \clearpage \input{sections/conclusion.tex} @@ -27,11 +27,26 @@ +% + +\subsection{Error Analysis / Convergence Study} + + + + +% \begin{figure} +% \label{fig:benchmarksplit} +% \end{figure} + + + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% \clearemptydoublepage \appendix +\clearemptydoublepage \input{sections/appendix.tex} \clearemptydoublepage diff --git a/LWB/propagators/sections/abstract.tex b/LWB/propagators/sections/abstract.tex index 6ece1b1..f3913b1 100644 --- a/LWB/propagators/sections/abstract.tex +++ b/LWB/propagators/sections/abstract.tex @@ -1,5 +1,6 @@ \begin{abstract} This article explains the central design decisions and most important code details related to the efficient implementation of time propagation for Hagedorn quantum wave packets in the WaveBlocks framework. A short introduction to operator splitting and a summary of some of the most important time propagation schemes for quantum wave packets are also given. - Finally, some benchmarking results are presented in the context of an example of a ???Ddim Nlevel BLUB potential???. + The correctness of the implemented algorithms is confirmed by showing energy conservation plots. + Finally, some benchmarking results are presented which compare different splitting schemes and dimensionalities. \end{abstract} diff --git a/LWB/propagators/sections/appendix.tex b/LWB/propagators/sections/appendix.tex new file mode 100644 index 0000000..6e783ce --- /dev/null +++ b/LWB/propagators/sections/appendix.tex @@ -0,0 +1,81 @@ +\appendix + +\begin{figure} + \centered + \begin{minipage}[c]{\textwidth} + \centered + \begin{center} + \large Hagedorn Propagator \\[1mm] + \normalsize Energy Evolution and Drift + \vspace{4mm} + \end{center} + \end{minipage} + \includegraphics[width=.45\textwidth]{figures/energy_Hagedorn.png} + \includegraphics[width=.45\textwidth]{figures/drift_Hagedorn.png} + \label{fig:energy_Hagedorn} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Hagedorn Propagator with \emph{Y4} splitting} +\end{figure} + +\begin{figure} + \centered + \begin{minipage}[c]{\textwidth} + \centered + \begin{center} + \large MG4 Propagator \\[1mm] + \normalsize Energy Evolution and Drift + \vspace{4mm} + \end{center} + \end{minipage} + \includegraphics[width=.45\textwidth]{figures/energy_MG4.png} + \includegraphics[width=.45\textwidth]{figures/drift_MG4.png} + \label{fig:energy_MG4} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the MG4 Propagator with \emph{Y4} splitting} +\end{figure} + +\begin{figure} + \centered + \begin{minipage}[c]{\textwidth} + \centered + \begin{center} + \large Pre764 Propagator \\[1mm] + \normalsize Energy Evolution and Drift + \vspace{4mm} + \end{center} + \end{minipage} + \includegraphics[width=.45\textwidth]{figures/energy_Pre764.png} + \includegraphics[width=.45\textwidth]{figures/drift_Pre764.png} + \label{fig:energy_Pre764} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Pre764 Propagator with \emph{Y4} splitting} +\end{figure} + +\begin{figure} + \centered + \begin{minipage}[c]{\textwidth} + \centered + \begin{center} + \large McL42 Propagator \\[1mm] + \normalsize Energy Evolution and Drift + \vspace{4mm} + \end{center} + \end{minipage} + \includegraphics[width=.45\textwidth]{figures/energy_McL42.png} + \includegraphics[width=.45\textwidth]{figures/drift_McL42.png} + \label{fig:energy_McL42} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the McL42 Propagator with \emph{Y4} splitting} +\end{figure} + +\begin{figure} + \centered + \begin{minipage}[c]{\textwidth} + \centered + \begin{center} + \large McL84 Propagator \\[1mm] + \normalsize Energy Evolution and Drift + \vspace{4mm} + \end{center} + \end{minipage} + \includegraphics[width=.45\textwidth]{figures/energy_McL84.png} + \includegraphics[width=.45\textwidth]{figures/drift_McL84.png} + \label{fig:energy_McL84} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the McL84 Propagator with \emph{Y4} splitting} +\end{figure} diff --git a/LWB/propagators/sections/benchmark.tex b/LWB/propagators/sections/benchmark.tex deleted file mode 100644 index d0356a7..0000000 --- a/LWB/propagators/sections/benchmark.tex +++ /dev/null @@ -1,3 +0,0 @@ -\section{Benchmark / Case Study / Example} - -% TODO: IDEA: express cost of every propagator as a ratio to the hagedorn propagator diff --git a/LWB/propagators/sections/results.tex b/LWB/propagators/sections/results.tex new file mode 100644 index 0000000..5a11920 --- /dev/null +++ b/LWB/propagators/sections/results.tex @@ -0,0 +1,131 @@ +\section{Results} + + +\subsection{Physical Correctness} +% +This short subsection analyzes the physical correctness of the implemented propagators. +To do this, a two dimensional wave packet was propagated with each of the presented propagators for a time $\tau = 10$ with timestep $\Dt = 0.01$. +All propagators used the \emph{Y4} splitting coefficients for the \proc{IntSplit} method. +\par\medskip +% +The harmonic potential was used as an example potential as this is easy to verify. +The generalized harmonic potential for $D$ dimensions is +\begin{align} + V(\bvec{x}) = + \frac{1}{2} \sum_{i=1}^D x_i^2 +\end{align} +% +where $x_i, \; 1 \le i \le D$ denote the single entries of the vector $\bvec{x}$. +\par\medskip +% +Figure \ref{fig:energy_Semiclassical} shows the time evolution of energy and the energy drift for the semiclassical propagator. The corresponding energy plots for the remaining propagators can be found in the appendix to this report. \\ +As is clearly shown by the graphs, the propagators conserve the total energy (apart from very small oscillations of the order $10^{-7}$ for the Hagedorn propagator and $10^{-12}$ for the remaining propagators that use \proc{IntSplit}). +Another interesting fact is that the \emph{McL84} operator seems to have a much longer oscillation period than all other propagators. +% +\begin{figure} + \centered + \begin{minipage}[c]{\textwidth} + \centered + \begin{center} + \large Semiclassical Propagator \\[1mm] + \normalsize Energy Evolution and Drift + \vspace{4mm} + \end{center} + \end{minipage} + \includegraphics[width=.45\textwidth]{figures/energy_Semiclassical.png} + \includegraphics[width=.45\textwidth]{figures/drift_Semiclassical.png} + \label{fig:energy_Semiclassical} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Semiclassical Propagator with \emph{Y4} splitting} +\end{figure} + +\subsection{Benchmark} +% +In order to benchmark the scaling of the code, three timing experiments were carried out: a simple comparison of runtimes for different propagators, an investigation of the computational cost associated with the usage of high order splitting schemes, and an analysis of the compute time as a function of the dimension $D$. +In all cases, the harmonic potential introduced at the beginning of this section was used. +\par\medskip +% +The runtime analysis was carried out on a Linux (Kernel 4.8.7, Fedora 24 Workstation) Quad-Core machine with an Intel Core i5-3210M Processor and +4GB of RAM. The C++ code was compiled with the GNU compiler, version 6.2.1, and using the \emph{-Ofast} optimization flag. + + + + +\subsubsection{Comparison of propagators} +% +A quick comparison of timings for the different propagators is shown in table \ref{tab:comparison}. +The same setup as for the energy analysis was used (2D wave packet, $\tau = 10$, step size $\Dt = 0.001$, \emph{Y4} time splitting coefficients). +It comes as quite a surprise, that the Semiclassical propagator is faster at computing the result than the Hagedorn propagator, which at first glance seems to have a simpler structure and involve fewer steps. +Also, the Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_i|W|\varphi_j}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. +% +\begin{table}[h] + \centering + \begin{tabular}{|l | r |} + \hline + \multicolumn{1}{|C}{\textbf{Propagator}} & + \multicolumn{1}{|C|}{\textbf{Timing [s]}} \\ + \hline + Hagedorn & 12.54 \\ + Semiclassical & 12.51 \\ + MG4 & 13.82 \\ + Pre764 & 17.10 \\ + McL42 & 22.82 \\ + McL84 & 23.52 \\ + \hline + \end{tabular} + \label{tab:comparison} + \caption{Runtimes for propagating a 2D wave packet to a time $\tau = 10$ with timestep $\Dt = 0.001$ with different propagators. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $1\%$ for the Python timings and below $0.1\%$ for the C++ timings.} +\end{table} + + +\subsubsection{Splitting Schemes} +% +As mentioned previously, the splitting coefficients $\{ w_T, w_U \}$ that are used as weights on the timestep $\dt$ in the \proc{IntSplit} method can have vastly different complexity, ranging from the \emph{Lie-Trotter} coefficients (one coefficient for propagation with $\opT$, one for propagation with $\opU$) up to the \emph{KL10} coefficients (34 coefficients for each operator). \\ +Higher order schemes are usually preferred in terms of accuracy, but they come at the price of longer computation time. +\par\medskip +% +Therefore, a numerical experiment was carried out in order to analyze how much computational time is consumed for coefficient pairs of different sizes. +To do this, the Semiclassical Propagator was used to propagate a two dimensional wave packet in a harmonic potential over a time of $\tau = 400$ and with stepsize $\Dt = 0.01$. +The same simulation was carried out for coefficient pairs of various different sizes and were run in Python as well as C++. +The results are shown in table \ref{tab:benchmarksplit}. % figure \ref{fig:benchmarksplit}. +\par\medskip +% +Looking at the Python timings only, the measurements suggest that in the case of the \emph{KL10} coefficient set, at least 80\% of the total runtime is spend in the \proc{IntSplit} function (since larger coefficient sets $\{ w_T, w_U \}$ only affect the number of steps with operators $\opT$ and $\opU$, but not with operator $\opW$). +\par\medskip +% +When bringing the timings in context with the C++ code, a comparison of absolute runtimes is of course not fair since C++ is intrinsically faster than a scripting language like Python and the C++ version has been optimized for speed in many different ways. +However, a very significant result in the context of the work on Propagators is how the speedup increases for larger coefficient pairs. +Using the \emph{KL10} coefficient set instead of a simple \emph{LT} set, the Python code takes more than six times longer. +The same comparison on C++ code shows that the code takes only 30\% longer for the large coefficient set than it takes for the minimal coefficient set. +This is a very encouraging result as it means that high order splitting coefficients come at almost no extra cost. +\par\medskip +% +\begin{table}[h] + \centering + \begin{tabular}{|l | r | r | r | r |} + \hline + \multicolumn{1}{|C}{\textbf{Splitting}} & + \multicolumn{1}{|C}{\textbf{No. of coefs}} & + \multicolumn{1}{|C}{\textbf{Python timing [s]}} & + \multicolumn{1}{|C}{\textbf{C++ timing [s]}} & + \multicolumn{1}{|C|}{\textbf{Speedup}} \\ + \hline + LT & 1 & 174.6 & 5.706 &\textbf{30.3} \\ + S2 & 2 & 202.9 & 5.766 &\textbf{35.6} \\ + Y4 & 4 & 258.6 & 5.861 &\textbf{44.1} \\ + PRKS6 & 7 & 341.0 & 5.995 &\textbf{56.9} \\ + Y61 & 8 & 366.0 & 6.004 &\textbf{61.0} \\ + KL6 & 10 & 421.4 & 6.124 &\textbf{68.8} \\ + BM63 & 15 & 557.1 & 6.467 &\textbf{86.2} \\ + KL8 & 18 & 634.8 & 6.637 &\textbf{95.6} \\ + KL10 & 34 & 1067.2 & 7.457 &\textbf{143.1} \\ + \hline + \end{tabular} + \label{tab:benchmarksplit} + \caption{Table comparing the computation time of Python code vs. C++ code for different sizes of the splitting coefficients $\{ w_T, w_U \}$. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $1\%$ for the Python timings and below $0.1\%$ for the C++ timings.} +\end{table} + + +\subsubsection{Scaling with dimensionality D} +% +The second code benchmark analyzed the C++ code only. The question that was addressed is how the compute time of the code scales with increasing dimension $D$ of the wave packet. + From ba19e94fce0d435d3165b0dbb271c171a9aa67c9 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Mon, 6 Feb 2017 00:38:00 +0100 Subject: [PATCH 16/28] minor fixes --- LWB/propagators/header.tex | 1 + LWB/propagators/main.tex | 9 --------- LWB/propagators/sections/appendix.tex | 10 ---------- LWB/propagators/sections/conclusion.tex | 14 ++++++++++++-- LWB/propagators/sections/propagators.tex | 2 ++ LWB/propagators/sections/results.tex | 18 ++++++++---------- LWB/propagators/uml.png | Bin 0 -> 99714 bytes 7 files changed, 23 insertions(+), 31 deletions(-) create mode 100644 LWB/propagators/uml.png diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex index 62a9591..b7fcf31 100644 --- a/LWB/propagators/header.tex +++ b/LWB/propagators/header.tex @@ -206,3 +206,4 @@ \usepackage{tikz} \usetikzlibrary{arrows,chains,matrix,positioning,scopes} +\usepackage{multicol} diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 7fe8231..0b6414b 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -25,15 +25,6 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - -% - -\subsection{Error Analysis / Convergence Study} - - - - % \begin{figure} % \label{fig:benchmarksplit} % \end{figure} diff --git a/LWB/propagators/sections/appendix.tex b/LWB/propagators/sections/appendix.tex index 6e783ce..6789473 100644 --- a/LWB/propagators/sections/appendix.tex +++ b/LWB/propagators/sections/appendix.tex @@ -1,9 +1,7 @@ \appendix \begin{figure} - \centered \begin{minipage}[c]{\textwidth} - \centered \begin{center} \large Hagedorn Propagator \\[1mm] \normalsize Energy Evolution and Drift @@ -17,9 +15,7 @@ \end{figure} \begin{figure} - \centered \begin{minipage}[c]{\textwidth} - \centered \begin{center} \large MG4 Propagator \\[1mm] \normalsize Energy Evolution and Drift @@ -33,9 +29,7 @@ \end{figure} \begin{figure} - \centered \begin{minipage}[c]{\textwidth} - \centered \begin{center} \large Pre764 Propagator \\[1mm] \normalsize Energy Evolution and Drift @@ -49,9 +43,7 @@ \end{figure} \begin{figure} - \centered \begin{minipage}[c]{\textwidth} - \centered \begin{center} \large McL42 Propagator \\[1mm] \normalsize Energy Evolution and Drift @@ -65,9 +57,7 @@ \end{figure} \begin{figure} - \centered \begin{minipage}[c]{\textwidth} - \centered \begin{center} \large McL84 Propagator \\[1mm] \normalsize Energy Evolution and Drift diff --git a/LWB/propagators/sections/conclusion.tex b/LWB/propagators/sections/conclusion.tex index e9934bf..ba91d94 100644 --- a/LWB/propagators/sections/conclusion.tex +++ b/LWB/propagators/sections/conclusion.tex @@ -1,4 +1,14 @@ \section{Conclusion} +% +In this project, a series of time propagators for Hagedorn wave packets was studied, analyzed and implemented. +The C++ code, which is the main outcome of the conduced work, is based on a clean software design that makes it very easy to add new Propagators and integrate them into the existing framework. +At the same time, as shown by the benchmarking results, the performance of the resulting code is by orders of magnitude better than that of the corresponding Python code. +In particular, the additional cost for using higher-order splitting coefficients is very contained thanks to a templated implementation of the \proc{IntSplit} method which is resolved at compile time and avoids the usage of any loops. +Given these performance improvements, more complex quantum mechanical simulations are now possible with this new framework. -% TODO: final observations -% TODO: suggestions for future work + +\vspace{5cm} + +\begin{center} + The code to the project can be found on Github at \cite{libwaveblocks}. +\end{center} diff --git a/LWB/propagators/sections/propagators.tex b/LWB/propagators/sections/propagators.tex index 7744318..44d800b 100644 --- a/LWB/propagators/sections/propagators.tex +++ b/LWB/propagators/sections/propagators.tex @@ -250,6 +250,8 @@ \subsection{McLachlan Propagators} \State \Procedure{McL42.Propagate}{$\Dt$} \State + \State $M = \lceil \left( \Dt^4 / \eps^3 \right)^{\frac{1}{8}} \rceil$ + \Comment compute number of substeps \State \Call{intSplit}{$A_0 \Dt, M, \{w_T, w_U\}$} \Comment $\opT + \opU = A$ \State \Call{stepW}{$B_0 \Dt$} diff --git a/LWB/propagators/sections/results.tex b/LWB/propagators/sections/results.tex index 5a11920..007002b 100644 --- a/LWB/propagators/sections/results.tex +++ b/LWB/propagators/sections/results.tex @@ -23,9 +23,7 @@ \subsection{Physical Correctness} Another interesting fact is that the \emph{McL84} operator seems to have a much longer oscillation period than all other propagators. % \begin{figure} - \centered \begin{minipage}[c]{\textwidth} - \centered \begin{center} \large Semiclassical Propagator \\[1mm] \normalsize Energy Evolution and Drift @@ -55,14 +53,14 @@ \subsubsection{Comparison of propagators} A quick comparison of timings for the different propagators is shown in table \ref{tab:comparison}. The same setup as for the energy analysis was used (2D wave packet, $\tau = 10$, step size $\Dt = 0.001$, \emph{Y4} time splitting coefficients). It comes as quite a surprise, that the Semiclassical propagator is faster at computing the result than the Hagedorn propagator, which at first glance seems to have a simpler structure and involve fewer steps. -Also, the Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_i|W|\varphi_j}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. +Also, the Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_i}{W}{\varphi_j}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. % \begin{table}[h] \centering \begin{tabular}{|l | r |} \hline - \multicolumn{1}{|C}{\textbf{Propagator}} & - \multicolumn{1}{|C|}{\textbf{Timing [s]}} \\ + \multicolumn{1}{|c}{\textbf{Propagator}} & + \multicolumn{1}{|c|}{\textbf{Timing [s]}} \\ \hline Hagedorn & 12.54 \\ Semiclassical & 12.51 \\ @@ -103,11 +101,11 @@ \subsubsection{Splitting Schemes} \centering \begin{tabular}{|l | r | r | r | r |} \hline - \multicolumn{1}{|C}{\textbf{Splitting}} & - \multicolumn{1}{|C}{\textbf{No. of coefs}} & - \multicolumn{1}{|C}{\textbf{Python timing [s]}} & - \multicolumn{1}{|C}{\textbf{C++ timing [s]}} & - \multicolumn{1}{|C|}{\textbf{Speedup}} \\ + \multicolumn{1}{|c}{\textbf{Splitting}} & + \multicolumn{1}{|c}{\textbf{No. of coefs}} & + \multicolumn{1}{|c}{\textbf{Python timing [s]}} & + \multicolumn{1}{|c}{\textbf{C++ timing [s]}} & + \multicolumn{1}{|c|}{\textbf{Speedup}} \\ \hline LT & 1 & 174.6 & 5.706 &\textbf{30.3} \\ S2 & 2 & 202.9 & 5.766 &\textbf{35.6} \\ diff --git a/LWB/propagators/uml.png b/LWB/propagators/uml.png new file mode 100644 index 0000000000000000000000000000000000000000..ef027926f0825f1e4e1fa634b102412347d932f5 GIT binary patch literal 99714 zcmc$`cUY8bmiAlbZA6ryf}pTPByMsB13}3-2PNkmB%^Mm$RZ#a6eK6f85BtpC1;Qf 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efficient implementation for time propagation mainly translates into finding an efficient way for calling existing functions with minimal overhead and by making use of compiler optimizations where possible. \par\medskip % -First, in subsection \ref{subsec:callback}, it will be discussed how the \proc{Evolve} method (see algorithm \ref{alg:evolve}) was implemented in order to provide a minimal working interface to the user while still preserving all the flexibility that may be required for intermediate measurements or other interaction. -Secondly, subsection \ref{subsec:poly} outlines how C++ features were used in order to achieve class inheritance and polymorphism witout any additional runtime cost. +First, subsection \ref{subsec:poly} outlines how C++ features were used in order to achieve class inheritance and polymorphism witout any additional runtime cost. +Secondly, in subsection \ref{subsec:callback}, it will be discussed how the \proc{Evolve} method (see algorithm \ref{alg:Evolve}) was implemented in order to provide a minimal working interface to the user while still preserving all the flexibility that may be required for intermediate measurements or other interaction. Finally, subsection \ref{subsec:intsplit} dives into the technical details of achieving inlined, highly optimized function calls for the \proc{IntSplit} method (see algorithm \ref{alg:intsplit}) that is used intensively in all the presented propagators. \par\medskip % Some details about the code were not found worthy to be mentioned here, -but more information may be found in the Doxygen code documentation. +but more information may be found in the Doxygen documentation. \par\medskip % The Hagedorn Propagator was already implemented in C++ as part of another project on the waveblocks library \cite{libwaveblocks}. -Many components of that code were re-used for the present implementation, but they were fundamentally re-organized into a new structure that has better support for many, diverse propagators. +Many components of that code were re-used for the present implementation, but they were fundamentally re-organized into a new structure. \subsection{Static Polymorphism through CRTP and enable\_if} @@ -47,14 +48,13 @@ \subsection{Static Polymorphism through CRTP and enable\_if} If the boolean is true, the function is enabled. Otherwise it is omitted. This handy metafunction from the C++ \proc{type\_traits} header was used in order to conditionally enable functions based on whether the number of levels $N$ was equal or greater to one or to distinguish between functions for homogeneous or inhomogeneous wave functions, which effectively provides a form of compile time polymorphism. % -\begin{figure}[h] - \begin{center} - \includegraphics[height=10cm]{figures/uml.png} - \end{center} - \label{fig:uml} +\begin{figure}[ht] + \centering + \includegraphics[height=10cm]{figures/uml.png} \caption{Simplified UML diagram for the propagator implementation. The derived propagator provides implementations for \proc{PrePropagate}, \proc{Propagate} and \proc{PostPropagate} which are called from the public \proc{Evolve} function. Some functions and function parameters are omitted for better readability of the graph.} + \label{fig:uml} \end{figure} @@ -62,10 +62,10 @@ \subsection{Static Polymorphism through CRTP and enable\_if} \subsection{Evolve and callback function} \label{subsec:callback} % -The simple \proc{Evolve} method as described in algorithm \ref{alg:evolve} in the section about propagator building blocks provides a handy wrapper for calling the \proc{PreProcessing}, \proc{Processing} and \proc{PostProcessing} methods in sequence and dividing the propagation time into smaller steps. \\ +The simple \proc{Evolve} method as described in algorithm \ref{alg:Evolve} in the section about propagator building blocks provides a handy wrapper for calling the \proc{PreProcessing}, \proc{Processing} and \proc{PostProcessing} methods in sequence and dividing the propagation time into smaller steps. \\ % However, one may still want to interact with the wave packet during the simulation, for example for writing the wave packet norm to file every 100 steps, throwing errors when energy conservation is violated or to carry out any other task that requires intermediate simulation results. -In order not to loose any flexibility in this regard, an optional callback-function argument was added to the \proc{Evolve} function. +In order not to lose any flexibility in this regard, an optional callback-function argument was added to the \proc{Evolve} function. The callback function is passed as a lambda function taking two arguments, the current simulation time \proc{t} and the index of the current timestep \proc{m}. If no callback function is passed, it defaults to an empty function. \par\medskip @@ -84,11 +84,11 @@ \subsection{Evolve and callback function} \subsection{IntSplit using alternating templates} -\label{subsec:instplit} +\label{subsec:intsplit} % The \proc{IntSplit} function is merely a helper function that alternatingly applies \proc{StepT} and \proc{StepU}. -Steps with the operators $\opT$ and $\opU$ are usually much cheaper than steps with $\opW$ (which requires numerical integration for approximating the inner product $\matrixel{\varphi_i}{W}{\varphi_j}$), but are in many cases found to be one of the main soures of error. -Therefore the idea of sub-dividing the steps into many smaller intervals - an approach introduced in the semiclassical splitting \ref{subsec:semiclassical} - is applied in many other propagators as well. +Steps with the operators $\opT$ and $\opU$ are usually much cheaper than steps with $\opW$ (which requires numerical integration for approximating the inner product $\matrixel{\varphi_k}{W}{\varphi_l}$), but are in many cases found to be one of the main soures of error. +Therefore the idea of sub-dividing the steps into many smaller intervals - an approach introduced in the semiclassical splitting \ref{sub:semiclassical_propagator} - is applied in many other propagators as well. Consequently, it is desirable to have a generic and efficient implementation available. \par\medskip % @@ -121,10 +121,9 @@ \subsection{IntSplit using alternating templates} In order to prevent the inappropriate use of the \proc{IntSplit} function through incompatible coefficient lists $w_T$ and $w_U$, their sizes are checked in the code at compile time via static assertions. \par\medskip % -Also, as seen in section \ref{sec:blub}, a splitting of the Hamiltonian into three operators $\opT + \opU + \opW$ was used, but $\opW$ is significantly more costly to compute. -This justifies the approach of using templates that are hard-coded to call functions \proc{StepT} and \proc{StepU}. +The fact that $\opW$ is significantly more costly to compute than its counterparts $\opT$ and $\opU$ also justifies the approach of using templates that are hard-coded to call functions \proc{StepT} and \proc{StepU}, as these are the only logical candidates for \proc{IntSplit}. -\begin{figure}[h] +\begin{figure}[ht] \begin{minipage}[c]{\textwidth} \begin{minipage}[l][10cm][t]{.4\textwidth} \begin{minipage}[t]{\textwidth} @@ -217,10 +216,8 @@ \subsection{IntSplit using alternating templates} \end{lstlisting} \end{minipage} \end{minipage} - - \end{minipage} - \label{fig:intsplit} - \caption{Simplified C++ snippets for outlining the template invocation sequence in the intsplit function. The template parameters \proc{S\_T} and \proc{S\_U} denote the total size of the coefficient arrays \proc{w\_T} and \proc{w\_U} while the \proc{N\_T} and \proc{N\_U} are the indices of the current coefficient to be applied. The \proc{TU::split} and \proc{UT::split} methods first apply their own step and then invoke each other by gradually incrementing the incices \proc{N\_T} and \proc{N\_U}. \\ + \caption{Simplified C++ snippets for outlining the template invocation sequence in the \proc{IntSplit} function. The template parameters \proc{S\_T} and \proc{S\_U} denote the total size of the coefficient arrays \proc{w\_T} and \proc{w\_U} while the \proc{N\_T} and \proc{N\_U} are the indices of the current coefficient to be applied. The \proc{TU::split} and \proc{UT::split} methods first apply their own step and then invoke each other by gradually incrementing the incices \proc{N\_T} and \proc{N\_U}. \\ When the conditions \proc{N\_T == S\_T} and \proc{N\_U == S\_U} are satisfied, the partially specialized templates become active and the recursion is terminated.} + \label{fig:intsplit} \end{figure} diff --git a/LWB/propagators/sections/introduction.tex b/LWB/propagators/sections/introduction.tex index c940998..033d0d3 100644 --- a/LWB/propagators/sections/introduction.tex +++ b/LWB/propagators/sections/introduction.tex @@ -9,8 +9,8 @@ \section*{Introduction} The entire WaveBlocks project, which is partially based on an existing Python project with the same name, can be found at \cite{libwaveblocks}. \par\medskip % -In this report, section \ref{sec:timeprop} repeats some of the most important mathematical ideas underlying the time propagation of wave packets and introduces the notations that will be used throughout the rest of the document. +In this report, section \ref{sec:operatorsplitting} repeats some of the most important mathematical ideas underlying the time propagation of wave packets and introduces the notations that will be used throughout the rest of the document. Based on these mathematical observations, section \ref{sec:buildingblocks} prepares a toolbox of essential building blocks that will help in the construction of propagators as well as for the final understanding of the source code. In section \ref{sec:propagators}, a series of numerical propagators for quantum wave packets are presented and expressed in terms of the components provided earlier. Some detailed insights into the most important technical ideas of the C++ implementation are given in section \ref{sec:implementation}. -Finally, section \ref{sec:benchmark} ... +Finally, section \ref{sec:results} presents the results of a series of numerical experiments to show the correctness of the implemented algorithms and analyse some performance aspects like the scaling with step size, dimensionality of the wave packet and order of the splitting coefficients. diff --git a/LWB/propagators/sections/operatorsplitting.tex b/LWB/propagators/sections/operatorsplitting.tex index ca4109b..50cc446 100644 --- a/LWB/propagators/sections/operatorsplitting.tex +++ b/LWB/propagators/sections/operatorsplitting.tex @@ -12,7 +12,7 @@ \section{Operator Splitting for Time Propagation} with wave function $\psi(\bvec{x},t)$ depending on the spatial variables $\bvec{x} = (x_1,\dots,x_D) \in \R^D$ and the time variable $t \in \R$. The parameter $\eps$, also known as the semiclassical parameter, determines the degree of influence of quantum dynamics, approaching classical dynamics in the limit $\eps \rightarrow 0$ and full quantum mechanics for $\eps = 1$. \\ Moreover, $\eps$ plays an important role in the stability of numerical schemes. -As underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\sim \eps^{-2})$ for Lie-Trotter splitting, Strang splitting and other related methods, meaning that smaller $\eps$ would lead to a larger error. +As underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\sim \eps^{-2}$ for Lie-Trotter splitting, Strang splitting and other related methods, meaning that smaller $\eps$ would lead to a larger error. \par\medskip % The solution to the above Schrödinger equation \ref{math:tdse} can be approximated as a finite linear combination of Hagedorn functions @@ -22,7 +22,7 @@ \section{Operator Splitting for Time Propagation} = e^{\im S(t)/\eps} \sum_{k \in \K} c_k(t) \varphi_k^\eps [ \bvec{\Pi} ] (\bvec{x}) \end{align} where $\K$ represents a finite multi-index set and the notation for the parameter pack $\bvec{\Pi} = (\bvec{q}(t),\bvec{p}(t),\bvec{Q}(t),\bvec{P}(t))$ is introduced for convenience. -A more detailed analysis of Hagedorn functions is given in \cite{FGL_semiclassical_dynamics}. +A more detailed analysis of Hagedorn functions is given in \cite{FGL_semiclassical_dynamics} and a comprehensive overview over the relevant mathematical details can be found in \cite{B_master_thesis}. \par\medskip % For the rest of this report, $\opU$ and $\opW$ will denote the quadratic and non-quadratic component of $\opV$ respectively and it will be used that diff --git a/LWB/propagators/sections/propagators.tex b/LWB/propagators/sections/propagators.tex index 44d800b..438138a 100644 --- a/LWB/propagators/sections/propagators.tex +++ b/LWB/propagators/sections/propagators.tex @@ -4,14 +4,14 @@ \section{Propagators} This section gives a short overview of the numerical properties of the propagators that were implemented in the context of this project. Where appropriate, references are given to find a more detailed mathematical analysis of the propagators. \par\medskip -The methods are called in the sequence determined by the \proc{Evolve} function (\ref{alg:evolve}) and generally are expressed in terms of the basic building blocks that were presented in the previous section. +The methods are called in the sequence determined by the \proc{Evolve} function (\ref{alg:Evolve}) and generally are expressed in terms of the basic building blocks that were presented in the previous section. If the methods \proc{PrePropagate} and \proc{PostPropagate} are omitted, it means that they are empty. \\ Whenever convenient, the direct succession of functions \proc{StepT} and \proc{StepU} was replaced by a call to \proc{IntSplit} (see explanation in \ref{subsec:tuintsplit}) for better accuracy. \subsection{Hagedorn Propagator} \label{sub:hagedorn_propagator} % -The Hagedorn propagator, shown in algorithm \ref{alg:hagedorn}, is one of the simplest propagators that can be built by exploiting the numerical aspects discussed above. +The Hagedorn propagator, shown in algorithm \ref{alg:hagedorn}, is one of the simplest propagators that can be built by exploiting the numerical properties of the Hamiltonian splitting $\opH = \opT + \opU + \opW$ discussed earlier. As pointed out in \cite{FGL_semiclassical_dynamics}, this simple time stepping scheme has many beneficial properties like the preservation of the $L^2$ norm of the wavepacket, time reversibility and stability in the classical limit $\eps \rightarrow 0$. Also, in the limit of $\K$ approaching the full basis set, the variational approximation used for the propagation with the non-quadratic part $\opW$ becomes exact. \begin{algorithm}[h] @@ -40,7 +40,7 @@ \subsection{Semiclassical Propagator} % The central idea of the semiclassical splitting, as introduced in \cite{GH_convsemiclassical}, is to split the propagation with operators $\opT$ and $\opU$ into many smaller, alternating steps, thereby reducing the dominating error\footnote{for small $\eps$, -the main source of error lies in the updating of $\Pi$ and $S$} in the update of $\bvec{\Pi}$ and $S$. +the main source of error lies in the updating of $\bvec{\Pi}$ and $S$} in the update of $\bvec{\Pi}$ and $S$. While there is some additional computational cost caused by a higher number of updates for the parameters $\bvec{\Pi}$ and $S$, the extra effort is usually negligible compared to the propagation with $\opW$ which requires numerical evaluation of multi-dimensional integrals. \par\medskip % @@ -111,7 +111,7 @@ \subsection{Magnus Propagator} Although the method is not a symplectic scheme in the usual sense, \cite{Iserles1999} has shown that in practical applications it conserves the Hamiltonian energy just as well as symplectic integrators. \par\medskip Moreover, the numerical stability and good performance of the Magnus propagator are not limited to problems where the solution evolves within a Lie Group, but also apply to various problems when this is not the case. -Algorithm \ref{alg:magnus} shows the \emph{MG4} method as presented in \cite{Iserles1999} and implemented in C++ in the scope of this work. +Algorithm \ref{alg:magnus} shows the \emph{MG4} method as presented in \cite{Iserles1999} and implemented in C++ in the scope of this project. \begin{algorithm}[h] \caption{Single timestep with Magnus propagator} @@ -147,7 +147,7 @@ \subsection{Magnus Propagator} \subsection{Processing Propagators} \label{sub:pre764_propagator} % -The idea of processing propagators is to carry out the time evolution with a Hamiltonian that is slightly perturbed from the initial one. +The idea of processing propagators is to carry out the time evolution with a Hamiltonian that is slightly perturbed from the original one. To achieve this, a pre and post processing step is applied. \\ More formally, it holds that % @@ -157,14 +157,14 @@ \subsection{Processing Propagators} % where the pre and post processing steps are applied via the multiplication with matrices $e^P$ and $e^{-P}$ (also referred to as \emph{processors}). While the processors only need to be applied once at the very beginning and end of the propagation (in order to return to the original Hamiltonian), the multiplication with the \emph{kernel} $e^{-\Dt K}$ is repeated $M$ times, once for every timestep. \\ -As a consequence, one wants to choose the processor $e^P$ in such a way that the evaluation of the kernel $e^{-\Dt K}$ is as simple as possible, requiring a minimum of expensive function evaluations. +As a consequence, one wants to choose the processor $e^P$ in such a way that the evaluation of the kernel $e^{-\Dt K}$ is as simple as possible in order to minimize the work associated with the computation of the kernel. \par\medskip % Unlike most commonly used integration schemes, the family of processing propagators are symplectic integrators. They are not suited for every kind of Hamiltonian, and the work of Blanes, Casas and Ros in \cite{Blanes1999} gives a method for finding the neccessary conditions that need to be satisfied in order for processing methods to be applicable. \par\medskip % -Algorithm \ref{alg:pre764} presents the \emph{Pre764} processing method which is a ?sixth? order processing method that was derived in \cite{Blanes1999}. +Algorithm \ref{alg:pre764} presents the \emph{Pre764} processing method as derived in \cite{Blanes1999}. % \begin{algorithm}[h] \caption{Single timestep with Pre764 propagator} @@ -218,7 +218,7 @@ \subsection{Processing Propagators} \subsection{McLachlan Propagators} \label{sub:mcl_propagator} % -McLachlan (see \cite{McLachlan1995}) has investigated various ?symplectic? schemes for computing the effect of operators of the form $\opX = \opA + \epsilon \opB$ where $\opA$ describes an exactly solvable system and $\opB$ is a perturbation. +McLachlan (see \cite{McLachlan1995}) has investigated various symplectic schemes for computing the effect of operators of the form $\opX = \opA + \epsilon \opB$ where $\opA$ describes an exactly solvable system and $\opB$ is a perturbation. The factor $\epsilon$ is a small parameter that indicates the limited influence of $\opB$, not to be confused with the semiclassical parameter $\eps$. \par\medskip The \emph{McL} integration schemes are characterised by pairs of weighing coefficients $a_i$ and $b_i$ that allow to represent the exponential $e^{\opX}$ as a product diff --git a/LWB/propagators/sections/results.tex b/LWB/propagators/sections/results.tex index 007002b..8cc3841 100644 --- a/LWB/propagators/sections/results.tex +++ b/LWB/propagators/sections/results.tex @@ -1,7 +1,11 @@ \section{Results} - +\label{sec:results} +% +This section discusses the outcomes of some numerical experiments that were carried out with the newly developed code. +Aspects that need to be checked are the correctness of the propagators from a quantum mechanical point of view (subsection \ref{subsec::physical_correctness}), convergence of the methods (subsection~\ref{subsec:convergence}) and benchmarking of the compute time (subsection \ref{subsec:benchmark}). \subsection{Physical Correctness} +\label{subsec:physical_correctness} % This short subsection analyzes the physical correctness of the implemented propagators. To do this, a two dimensional wave packet was propagated with each of the presented propagators for a time $\tau = 10$ with timestep $\Dt = 0.01$. @@ -22,7 +26,8 @@ \subsection{Physical Correctness} As is clearly shown by the graphs, the propagators conserve the total energy (apart from very small oscillations of the order $10^{-7}$ for the Hagedorn propagator and $10^{-12}$ for the remaining propagators that use \proc{IntSplit}). Another interesting fact is that the \emph{McL84} operator seems to have a much longer oscillation period than all other propagators. % -\begin{figure} +\begin{figure}[ht] + \centering \begin{minipage}[c]{\textwidth} \begin{center} \large Semiclassical Propagator \\[1mm] @@ -32,13 +37,35 @@ \subsection{Physical Correctness} \end{minipage} \includegraphics[width=.45\textwidth]{figures/energy_Semiclassical.png} \includegraphics[width=.45\textwidth]{figures/drift_Semiclassical.png} - \label{fig:energy_Semiclassical} \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Semiclassical Propagator with \emph{Y4} splitting} + \label{fig:energy_Semiclassical} +\end{figure} + + +\subsection{Effect of Stepsize} +\label{subsec:convergence} +% +A convergence analysis was carried out in which the error of each propagator was recorded while reducing the stepsize. +A Semiclassical propagator with \emph{KL10} splitting coefficients and step size 0.001 was used to propagate the wave packet to a final time $\tau = 10$ and create a reference solution. +\par\medskip +% +The error between two wave packets was computed by evaluating them on a grid with 1000 grid points and taking the $L_2$ norm of the differences. +Figure \ref{fig:error_analysis} shows the error for different step sizes $\Dt$. +It is quite surprising how the Hagedorn Propagator seems to have even better convergence order than all the more sophisticated contestants. +Also, the \emph{McL84} propagator does not seem to converge. This is consistent with the observation made earlier in this section that - although satisfying energy conservation - the \emph{McL84} propagator seemed to have a vastly different period of oscillation. + +\begin{figure}[ht] + \centering + \includegraphics[width=.8\textwidth]{figures/error_analysis.png} + \caption{Convergence of the wavefunctions towards the "correct" solution. The $L_2$ norm was measured by projecting the wave function on a grid with 1000 nodes in the range $[-1,1]$ and taking the differences of these values. According to this experiment, the Hagedorn propagator seems to have higher convergence order than its contestants, despite its simplicity. + This observation could be an interesting starting point for further experiments.} + \label{fig:error_analysis} \end{figure} \subsection{Benchmark} +\label{subsec:benchmark} % -In order to benchmark the scaling of the code, three timing experiments were carried out: a simple comparison of runtimes for different propagators, an investigation of the computational cost associated with the usage of high order splitting schemes, and an analysis of the compute time as a function of the dimension $D$. +In order to benchmark the code, three timing experiments were carried out: a simple comparison of runtimes for different propagators, an investigation of the computational cost associated with the usage of high order splitting schemes, and an analysis of the compute time as a function of the dimension $D$. In all cases, the harmonic potential introduced at the beginning of this section was used. \par\medskip % @@ -50,10 +77,10 @@ \subsection{Benchmark} \subsubsection{Comparison of propagators} % -A quick comparison of timings for the different propagators is shown in table \ref{tab:comparison}. +A quick comparison of timings for the different propagators is shown in table \ref{tab:speedup}. The same setup as for the energy analysis was used (2D wave packet, $\tau = 10$, step size $\Dt = 0.001$, \emph{Y4} time splitting coefficients). It comes as quite a surprise, that the Semiclassical propagator is faster at computing the result than the Hagedorn propagator, which at first glance seems to have a simpler structure and involve fewer steps. -Also, the Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_i}{W}{\varphi_j}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. +Also, the Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_k}{W}{\varphi_l}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. % \begin{table}[h] \centering @@ -70,8 +97,8 @@ \subsubsection{Comparison of propagators} McL84 & 23.52 \\ \hline \end{tabular} - \label{tab:comparison} \caption{Runtimes for propagating a 2D wave packet to a time $\tau = 10$ with timestep $\Dt = 0.001$ with different propagators. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $1\%$ for the Python timings and below $0.1\%$ for the C++ timings.} + \label{tab:speedup} \end{table} @@ -84,7 +111,7 @@ \subsubsection{Splitting Schemes} Therefore, a numerical experiment was carried out in order to analyze how much computational time is consumed for coefficient pairs of different sizes. To do this, the Semiclassical Propagator was used to propagate a two dimensional wave packet in a harmonic potential over a time of $\tau = 400$ and with stepsize $\Dt = 0.01$. The same simulation was carried out for coefficient pairs of various different sizes and were run in Python as well as C++. -The results are shown in table \ref{tab:benchmarksplit}. % figure \ref{fig:benchmarksplit}. +The results are listed in table \ref{tab:benchmarksplit_t} and plotted in figure \ref{fig:benchmarksplit_f}. \par\medskip % Looking at the Python timings only, the measurements suggest that in the case of the \emph{KL10} coefficient set, at least 80\% of the total runtime is spend in the \proc{IntSplit} function (since larger coefficient sets $\{ w_T, w_U \}$ only affect the number of steps with operators $\opT$ and $\opU$, but not with operator $\opW$). @@ -107,8 +134,8 @@ \subsubsection{Splitting Schemes} \multicolumn{1}{|c}{\textbf{C++ timing [s]}} & \multicolumn{1}{|c|}{\textbf{Speedup}} \\ \hline - LT & 1 & 174.6 & 5.706 &\textbf{30.3} \\ - S2 & 2 & 202.9 & 5.766 &\textbf{35.6} \\ + LT & 1 & 174.6 & 5.766 &\textbf{30.3} \\ + S2 & 2 & 202.9 & 5.706 &\textbf{35.6} \\ Y4 & 4 & 258.6 & 5.861 &\textbf{44.1} \\ PRKS6 & 7 & 341.0 & 5.995 &\textbf{56.9} \\ Y61 & 8 & 366.0 & 6.004 &\textbf{61.0} \\ @@ -118,12 +145,29 @@ \subsubsection{Splitting Schemes} KL10 & 34 & 1067.2 & 7.457 &\textbf{143.1} \\ \hline \end{tabular} - \label{tab:benchmarksplit} \caption{Table comparing the computation time of Python code vs. C++ code for different sizes of the splitting coefficients $\{ w_T, w_U \}$. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $1\%$ for the Python timings and below $0.1\%$ for the C++ timings.} + \label{tab:benchmarksplit_t} \end{table} +% +\begin{figure}[ht] + \centering + \begin{center} + \includegraphics[width=.8\textwidth]{figures/coefficient_analysis.png} + \end{center} + \caption{Comparison of computation times for different splitting coefficients $\{ w_T, w_U \}$. The absolute timings are shown in the top graph, the speedup on the bottom. It is remarkable that the initial speedup factor is about 30, but grows for larger coefficient sets. The speedup factor for the largest tested coefficient pair \emph{KL10} is over 140.} + \label{fig:benchmarksplit_f} +\end{figure} \subsubsection{Scaling with dimensionality D} % -The second code benchmark analyzed the C++ code only. The question that was addressed is how the compute time of the code scales with increasing dimension $D$ of the wave packet. - +The question that was addressed in this benchmark is how the compute time of the code scales with increasing dimension $D$ of the wave packet. +Figure \ref{fig:dimension_analysis} shows a clear exponential scaling with the dimension $D$. +With a computation time of under 20 minutes for 100 time steps with the Semiclassical Propagator, dimension $D=5$ is still quite feasible. +% +\begin{figure}[ht] + \centering + \includegraphics[width=.8\textwidth]{figures/dimension_analysis.png} + \caption{Scaling of the computation time with wave pack dimension $D$. Measurements done for a wave packet in a generalized harmonic potential, propagated for 100 steps with the Semiclassical Propagator.} + \label{fig:dimension_analysis} +\end{figure} diff --git a/LWB/serialization/data_layout.pdf b/LWB/serialization/data_layout.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2421d878c8ab036017ca49c3b481ed2e46177f92 GIT binary patch literal 12107 zcma)ibx@tpvMufs++pMH?he7-g3HF;-6goY2Mg}*Zo%E%HArxH0aGawQE-`az!x-dS(VTIP%7esuws`029E@$P$i^55Oo3v^8@!2e5rY%5VSx zfKkH2+8OBh`L#B528scV?M#4h0s?SO&W=Ds8#wo@Xdk(1x^~-*4qx$D@T`m3X3GpI ze-Iol94H*@Y#f+uZ)9p4J~!4nI$52X`l#Yc=u)*Gu35;h57i&9k70g1 zL?5qySn@}&r=@()j~VrRz6ZBfDX))9AGeP_ul<7L>q&=8755I&)xJf;Q~eH4xZHwW zL>AuBn}WaF{5CKjTMjiYu?(tDYNx3{fSq@3U6Zqf3frob_N9CA&r4}oioCRnQY+H| znQPCx-^Xn_yKZOZiAsHS%h}Ipm!6l=S|*UzB_PY%JYz_wH9wp;Wbg@S%GvM1OIR5t z5WYP2{1z5Ao(Aqt+TKrA>(_(n+T?AJPu-ulw&^WY3LvAxdHGa@f-fv2LgH~5$Kj7J zQP6RD5&t^s(L8Y7RwoA=6OAQl39KSRIqvs3vsL$ 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diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 2dc3a23..dc07f47 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -11,7 +11,7 @@ \clearpage -\listoffigures +% \listoffigures \listofalgorithms diff --git a/LWB/propagators/sections/results.tex b/LWB/propagators/sections/results.tex index 8cc3841..905362a 100644 --- a/LWB/propagators/sections/results.tex +++ b/LWB/propagators/sections/results.tex @@ -2,10 +2,10 @@ \section{Results} \label{sec:results} % This section discusses the outcomes of some numerical experiments that were carried out with the newly developed code. -Aspects that need to be checked are the correctness of the propagators from a quantum mechanical point of view (subsection \ref{subsec::physical_correctness}), convergence of the methods (subsection~\ref{subsec:convergence}) and benchmarking of the compute time (subsection \ref{subsec:benchmark}). +Aspects that need to be checked are the correctness of the propagators from a quantum mechanical point of view (subsection~\ref{subsec::physical}), convergence of the methods (subsection~\ref{subsec:convergence}) and benchmarking of the compute time (subsection~\ref{subsec:benchmark}). \subsection{Physical Correctness} -\label{subsec:physical_correctness} +\label{subsec:physical} % This short subsection analyzes the physical correctness of the implemented propagators. To do this, a two dimensional wave packet was propagated with each of the presented propagators for a time $\tau = 10$ with timestep $\Dt = 0.01$. From f59a705b78d6ae820f48af8c89311766bdd28309 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sat, 11 Feb 2017 01:11:24 +0100 Subject: [PATCH 19/28] fixed some typos and notation problems --- LWB/propagators/header.tex | 2 +- LWB/propagators/main.tex | 42 ++++++++++ LWB/propagators/sections/abstract.tex | 7 +- LWB/propagators/sections/appendix.tex | 10 +-- LWB/propagators/sections/buildingblocks.tex | 78 +++++++++---------- LWB/propagators/sections/implementation.tex | 4 +- LWB/propagators/sections/introduction.tex | 4 +- .../sections/operatorsplitting.tex | 37 +++++---- LWB/propagators/sections/propagators.tex | 35 +++++---- LWB/propagators/sections/results.tex | 12 +-- LWB/propagators/sections/title.tex | 2 +- 11 files changed, 139 insertions(+), 94 deletions(-) diff --git a/LWB/propagators/header.tex b/LWB/propagators/header.tex index b7fcf31..668fd13 100644 --- a/LWB/propagators/header.tex +++ b/LWB/propagators/header.tex @@ -174,7 +174,7 @@ % \newcommand*{\op}[1]{\hat{\mathcal{#1}}} \newcommand*{\C}{\ensuremath{\mathbb{C}}} \newcommand*{\Dt}{\ensuremath{\Delta t}} -\newcommand*{\K}{\ensuremath{\mathcal{K}}} +\newcommand*{\K}{\ensuremath{\mathfrak{K}}} \newcommand*{\R}{\ensuremath{\mathbb{R}}} \newcommand*{\bmat}[1]{\bm{#1}} \newcommand*{\bvec}[1]{\bm{#1}} diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index dc07f47..56e0414 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -26,6 +26,48 @@ \clearemptydoublepage \input{sections/appendix.tex} +\begin{comment} +\subsection{Henon-Heiles Potential} +% +\begin{align} + V(\bvec{x}) = + \frac{1}{2} \sum_{j=1}^N \sigma_j x_j^2 + + \sigma_* \sum_{j=1}^{N-1} \left( x_j x_{j+1}^2 - \frac{1}{3} x_j^3 + \frac{\sigma_*}{16} \left( x_j^2 + x_{j+1}^2 \right)^2 \right) +\end{align} +% +with harmonic part coefficients $\sigma_j = 1$ and mixing coefficient $\sigma_* = 0.2$. +Initial conditions for 2D: (GENERALIZE) +?Normed gaussian wave packet? +with $\bvec{q}=(1.8,0)^T$, $\bvec{p}=(0,1.2)^T$, $\bmat{Q} = \sqrt{2} \cdot \text{diag}(\sqrt{0.56},\sqrt{0.24})$, $\bmat{P} = \im \bmat{Q}^{-1}$ + +\paragraph{Jacobian} +\begin{align} + \del_k V(\bvec{x}) + = \; &\delta_{k,j} \sigma_j x_j \\ + + \; &\delta_{k,j} \sigma_* \left( x_{j+1}^2 - x_j^2 + \frac{\sigma_*}{4} x_j \left( x_j^2 + x_{j+1}^2 \right) \right) \\ + + \; &\delta_{k,j+1} \sigma_* \left( 2 x_j x_{j+1} + \frac{\sigma_*}{4} x_{j+1} \left( x_j^2 + x_{j+1}^2 \right) \right) \\ + = \; &\delta_{k,j} \left( \sigma_j x_j + \sigma_* \left( x_{j+1}^2 - x_j^2 + \frac{\sigma_*}{4} x_j \left( x_j^2 + x_{j+1}^2 \right) \right) \right) \\ + + \; &\delta_{k,j+1} \sigma_* \left( 2 x_j x_{j+1} + \frac{\sigma_*}{4} x_{j+1} \left( x_j^2 + x_{j+1}^2 \right) \right) +\end{align} + +\paragraph{Hessian} +\begin{align} + \del_k \del_l V(\bvec{x}) + = \; &\delta_{l,j} \left[ \delta_{k,j} \left( \sigma_j + \sigma_* \left( - 2 x_j + \frac{\sigma_*}{4} \left( x_j^2 + x_{j+1}^2 \right) + \frac{\sigma_*}{2} x_j^2 \right) \right) + + \; \delta_{k,j+1} \sigma_* \left( 2 x_{j+1} + \frac{\sigma_*}{2} x_{j+1} x_j \right) \right] \\ + + \; &\delta_{l,j+1} \left[ \delta_{k,j} \left( \sigma_* \left( 2 x_{j+1} + \frac{\sigma_*}{2} x_j x_{j+1} \right) \right) + + \; \delta_{k,j+1} \sigma_* \left( 2 x_j + \frac{\sigma_*}{4} \left( x_j^2 + x_{j+1}^2 \right) + \frac{\sigma_*}{2} x_{j+1}^2 \right) \right] \\ + = \; &\delta_{l,j} \left[ \delta_{k,j} \left( \sigma_j + \sigma_* \left( - 2 x_j + \frac{\sigma_*}{4} \left( 3 x_j^2 + x_{j+1}^2 \right) \right) \right) + + \; \delta_{k,j+1} \sigma_* \left( 2 x_{j+1} + \frac{\sigma_*}{2} x_{j+1} x_j \right) \right] \\ + \; % TODO: + + \; &\delta_{l,j+1} \left[ \delta_{k,j} \left( \sigma_* \left( 2 x_{j+1} + \frac{\sigma_*}{2} x_j x_{j+1} \right) \right) + + \; \delta_{k,j+1} \sigma_* \left( 2 x_j + \frac{\sigma_*}{4} \left( x_j^2 + x_{j+1}^2 \right) + \frac{\sigma_*}{2} x_{j+1}^2 \right) \right] +\end{align} +\end{comment} + + + + \clearemptydoublepage \bibliographystyle{plain} diff --git a/LWB/propagators/sections/abstract.tex b/LWB/propagators/sections/abstract.tex index d35e94f..315aade 100644 --- a/LWB/propagators/sections/abstract.tex +++ b/LWB/propagators/sections/abstract.tex @@ -1,6 +1,5 @@ \begin{abstract} - This article explains the central design decisions and most important code details related to the efficient implementation of time propagation for Hagedorn quantum wave packets in the WaveBlocks framework. - A short introduction to operator splitting and a summary of some of the most important time propagation schemes for quantum wave packets are also given. - The correctness of the implemented algorithms is confirmed by showing energy conservation plots. - Finally, some benchmarking results are presented which compare different propagators, splitting schemes, step sizes and dimensionalities. + This article explains the central design decisions and most important code details related to the efficient implementation of time propagation for Hagedorn wave packets in the WaveBlocks framework. + A short introduction to operator splitting and a summary of some of the most important time propagation schemes for Hagedorn wave packets are also given. + Finally, some benchmarking results are presented which analyze energy conservation and compare different propagators, splitting schemes, step sizes and dimensionalities. \end{abstract} diff --git a/LWB/propagators/sections/appendix.tex b/LWB/propagators/sections/appendix.tex index 6e6f711..f697c35 100644 --- a/LWB/propagators/sections/appendix.tex +++ b/LWB/propagators/sections/appendix.tex @@ -12,7 +12,7 @@ \section{Energy Evolution and Drift Plots} \end{minipage} \includegraphics[width=.45\textwidth]{figures/energy_Hagedorn.png} \includegraphics[width=.45\textwidth]{figures/drift_Hagedorn.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Hagedorn Propagator with \emph{Y4} splitting} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Hagedorn Propagator with \emph{Y4} splitting} \label{fig:energy_Hagedorn} \end{figure} @@ -26,7 +26,7 @@ \section{Energy Evolution and Drift Plots} \end{minipage} \includegraphics[width=.45\textwidth]{figures/energy_MG4.png} \includegraphics[width=.45\textwidth]{figures/drift_MG4.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the MG4 Propagator with \emph{Y4} splitting} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the MG4 Propagator with \emph{Y4} splitting} \label{fig:energy_MG4} \end{figure} @@ -40,7 +40,7 @@ \section{Energy Evolution and Drift Plots} \end{minipage} \includegraphics[width=.45\textwidth]{figures/energy_Pre764.png} \includegraphics[width=.45\textwidth]{figures/drift_Pre764.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Pre764 Propagator with \emph{Y4} splitting} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Pre764 Propagator with \emph{Y4} splitting} \label{fig:energy_Pre764} \end{figure} @@ -54,7 +54,7 @@ \section{Energy Evolution and Drift Plots} \end{minipage} \includegraphics[width=.45\textwidth]{figures/energy_McL42.png} \includegraphics[width=.45\textwidth]{figures/drift_McL42.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the McL42 Propagator with \emph{Y4} splitting} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the McL42 Propagator with \emph{Y4} splitting} \label{fig:energy_McL42} \end{figure} @@ -68,6 +68,6 @@ \section{Energy Evolution and Drift Plots} \end{minipage} \includegraphics[width=.45\textwidth]{figures/energy_McL84.png} \includegraphics[width=.45\textwidth]{figures/drift_McL84.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the McL84 Propagator with \emph{Y4} splitting} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the McL84 Propagator with \emph{Y4} splitting} \label{fig:energy_McL84} \end{figure} diff --git a/LWB/propagators/sections/buildingblocks.tex b/LWB/propagators/sections/buildingblocks.tex index 8c55d56..181b9a6 100644 --- a/LWB/propagators/sections/buildingblocks.tex +++ b/LWB/propagators/sections/buildingblocks.tex @@ -2,7 +2,7 @@ \section{Common building blocks for time evolution schemes} \label{sec:buildingblocks} % In order to be able to create an efficient, still flexible, implementation of various time propagators, it is helpful to break them down in common, optimized building blocks in terms of which all (or most) propagators can be expressed. -This section will briefly introduce the most important, generic building blocks that were found neccessary in order to build the propagators in section~\ref{sec:propagators}. +This section will briefly introduce the most important, generic building blocks that were found necessary in order to build the propagators in section~\ref{sec:propagators}. For each of them, it will be assumed that the following wave packet variables are implicitly accessible, without passing them as an argument to the procedure: the parameter pack $\bvec{\Pi} = (\bvec{q},\bvec{p},\bvec{Q},\bvec{P})$, the phase factor $S$, the coefficients $\{ c_k \}_{k \in \K}$, the potential function $V(\bvec{q})$ (including Jacobian and Hessian thereof) and the potential remainder $W(\bvec{q})$. \par\medskip % @@ -17,15 +17,15 @@ \subsection{Evolve} but will generally be a combination of the basic building blocks that follow in this section. % \begin{algorithm}[h] - \caption{Evolve the wave packet for a time period $\tau$} + \caption{Evolve the wave packet for a time period $T$} \label{alg:Evolve} \begin{algorithmic} \State - \Procedure{Evolve}{$\tau, \Dt$} + \Procedure{Evolve}{$T, \Dt$} \State - \State $M = \tau/\Dt$ + \State $M_{tot} = T/\Dt$ \State \Call{PrePropagate}{$\Dt$} - \For{$1 \le m \le M$} + \For{$m = 1, \dots, M_{tot}$} \State \Call{Propagate}{$\Dt$} \EndFor \State \Call{PostPropagate}{$\Dt$} @@ -40,7 +40,7 @@ \subsection{StepT, StepU and IntSplit} % %%% StepT & StepU The time stepping with operators $\opT$ and $\opU = U(\bvec{x})$ follows directly from the propositions in \cite{FGL_semiclassical_dynamics} and is outlined in the algorithms~\ref{alg:StepT} and~\ref{alg:StepU} respectively. -Both functions take a variable $\eta$ as an argument which describes the size of the time step. \\ +Both functions take a variable $\xi$ as an argument which describes the size of the time step. \\ In the inhomogeneous implementation of the algorithms, the subscript $n$ is used to denote the relative parameters at energy level $n$. \par\medskip % @@ -49,28 +49,28 @@ \subsection{StepT, StepU and IntSplit} \label{alg:StepT} \begin{algorithmic} \State - \Procedure{StepT[homogeneous]}{$\eta$} + \Procedure{StepT[homogeneous]}{$\xi$} \State - \State $\bvec{q} = \bvec{q} + \eta \bmat{M}^{-1} \bvec{p}$ + \State $\bvec{q} = \bvec{q} + \xi \bvec{p}$ \Comment update $\bvec{q}$ - \State $\bmat{Q} = \bmat{Q} + \eta \bmat{M}^{-1} \bmat{P}$ + \State $\bmat{Q} = \bmat{Q} + \xi \bmat{P}$ \Comment update $\bmat{Q}$ - \State $\bvec{S} = \bvec{S} + \frac{\eta}{2} \bvec{p}^T \bmat{M}^{-1} \bvec{p}$ - \Comment update $\bmat{S}$ + \State $S = S + \frac{\xi}{2} \bvec{p}^T \bvec{p}$ + \Comment update $S$ \State \EndProcedure \\\hrulefill \State - \Procedure{StepT[inhomogeneous]}{$\eta$} + \Procedure{StepT[inhomogeneous]}{$\xi$} \State \For{$n=1,...,N$} \Comment loop over all energy levels - \State $\bvec{q}_n = \bvec{q}_n + \eta \bmat{M}^{-1} \bvec{p}_n$ + \State $\bvec{q}_n = \bvec{q}_n + \xi \bvec{p}_n$ \Comment update $\bvec{q}$ for level $n$ - \State $\bmat{Q}_n = \bmat{Q}_n + \eta \bmat{M}^{-1} \bmat{P}_n$ + \State $\bmat{Q}_n = \bmat{Q}_n + \xi \bmat{P}_n$ \Comment update $\bmat{Q}$ for level $n$ - \State $\bvec{S}_n = \bvec{S}_n + \frac{\eta}{2} \bvec{p}_n^T \bmat{M}^{-1} \bvec{p}_n$ - \Comment update $\bmat{S}$ for level $n$ + \State $S_n = S_n + \frac{\xi}{2} \bvec{p}_n^T \bvec{p}_n$ + \Comment update $S$ for level $n$ \EndFor \State \EndProcedure @@ -82,28 +82,28 @@ \subsection{StepT, StepU and IntSplit} \label{alg:StepU} \begin{algorithmic} \State - \Procedure{StepU[homogeneous]}{$\eta$} + \Procedure{StepU[homogeneous]}{$\xi$} \State - \State $\bvec{p} = \bvec{p} - \eta \nabla V (\bvec{q})$ + \State $\bvec{p} = \bvec{p} - \xi \nabla V (\bvec{q})$ \Comment update $\bvec{p}$ - \State $\bvec{P} = \bvec{P} - \eta \nabla^2 V (\bvec{q}) \bvec{Q}$ + \State $\bvec{P} = \bvec{P} - \xi \nabla^2 V (\bvec{q}) \bvec{Q}$ \Comment update $\bvec{P}$ - \State $\bvec{S} = \bvec{S} - \eta V (\bvec{q})$ - \Comment update $\bvec{S}$ + \State $S = S - \xi V (\bvec{q})$ + \Comment update $S$ \State \EndProcedure \\\hrulefill \State - \Procedure{StepU[inhomogeneous]}{$\eta$} + \Procedure{StepU[inhomogeneous]}{$\xi$} \State \For{$n=1,...,N$} \Comment loop over all energy levels - \State $\bvec{p}_n = \bvec{p}_n - \eta \nabla V (\bvec{q}_n)$ + \State $\bvec{p}_n = \bvec{p}_n - \xi \nabla V (\bvec{q}_n)$ \Comment update $\bvec{p}$ for level $n$ - \State $\bvec{P}_n = \bvec{P}_n - \eta \nabla^2 V (\bvec{q}_n) \bvec{Q}_n$ + \State $\bvec{P}_n = \bvec{P}_n - \xi \nabla^2 V (\bvec{q}_n) \bvec{Q}_n$ \Comment update $\bvec{P}$ for level $n$ - \State $\bvec{S}_n = \bvec{S}_n - \eta V (\bvec{q}_n)$ - \Comment update $\bvec{S}$ for level $n$ + \State $S_n = S_n - \xi V (\bvec{q}_n)$ + \Comment update $S$ for level $n$ \EndFor \State \EndProcedure @@ -113,7 +113,7 @@ \subsection{StepT, StepU and IntSplit} %%% IntSplit In the practical implementations of the propagators that will be presented in the next section, propagation with $\opT$ and $\opU$ is usually replaced by a series of smaller, alternating steps with the two operators. In order to facilitate the use of such an alternating pattern, the helper function \proc{IntSplit} (algorithm~\ref{alg:intsplit}) is introduced. -This function splits the timestep $\Dt$ into $M$ smaller timesteps of size $\dt$ and uses the pair of weight lists $\{ w_T, w_U \}$ to alternatingly apply operators $\opT$ and $\opU$ on each of the sub-intervals. +This function splits the timestep $\Dt$ into $M_{split}$ smaller timesteps of size $\dt$ and uses the pair of weight lists $\{ w_T, w_U \}$ to alternatingly apply operators $\opT$ and $\opU$ on each of the sub-intervals. More detailed information on the individual parameters to this function and the practical implementation can be found in section \ref{sec:implementation} % \begin{algorithm}[h] @@ -121,16 +121,16 @@ \subsection{StepT, StepU and IntSplit} \label{alg:intsplit} \begin{algorithmic} \State - \Procedure{IntSplit}{$\Dt, M, \{w_T,w_U\}$} + \Procedure{IntSplit}{$\Dt, M_{split}, \{w_T,w_U\}$} \State - \State $\dt = \Dt/M$ + \State $\dt = \Dt/M_{split}$ \Comment step size for splitting - \For{$n = 1,\dots,M$} - \Comment split interval in M substeps - \For{$x_{w_T}$ in $w_T$ and $x_{w_U}$ in $w_U$} + \For{$n = 1,\dots,M_{split}$} + \Comment split interval in $M_{split}$ substeps + \For{$\alpha$ in $w_T$ and $\beta$ in $w_U$} \Comment alternatingly apply $\opT$ and $\opU$ - \State \Call{StepT}{$x_{w_T} \dt$} - \State \Call{StepU}{$x_{w_U} \dt$} + \State \Call{StepT}{$\alpha \cdot \dt$} + \State \Call{StepU}{$\beta \cdot \dt$} \EndFor \EndFor \State @@ -145,14 +145,14 @@ \subsection{StepW and building of the interaction matrix} The update rule is % \begin{align} - \bvec{c}(t) = \exp \left( - \frac{\im t}{\eps} \bmat{F} \right) \bvec{c}(0) + \bvec{c}(t + dt) = \exp \left( - \frac{\im t}{\eps^2} \bmat{F} \right) \bvec{c}(t) \end{align} % -where the matrix $\bmat{F} = \{ f_{k,l} \}_{k,l \in \K}$ has entries +where the matrix $\bmat{F} = \left[ f_{k,l} \right]_{k,l \in \K}$ has entries % \begin{align} f_{k,l} = \matrixel{\varphi_k}{W}{\varphi_l} - = \int_{\R^N} \conj{\varphi_k(\bvec{x})} W(\bvec{x}) \varphi_l(\bvec{x}) \; \dif \bvec{x} + = \int_{\R^D} \conj{\varphi_k(\bvec{x})} W(\bvec{x}) \varphi_l(\bvec{x}) \; \dif \bvec{x} \; . \end{align} % The calculation of such interaction matrices $\bmat{F}$ makes heavy use of the work on inner products that has been done in previous student projects for the libwaveblocks library \cite{libwaveblocks}. @@ -163,8 +163,8 @@ \subsection{StepW and building of the interaction matrix} \label{alg:StepW} \begin{algorithmic} \State - \Procedure{StepW}{$\upic,\Dt$} - \State $\bm{\Sigma} = - \im \frac{\Dt}{\eps^2} \cdot$ \Call{BuildF}{$\Pi$} + \Procedure{StepW}{$\xi$} + \State $\bm{\Sigma} = - \im \frac{\xi}{\eps^2} \cdot$ \Call{BuildF}{$\Pi$} \State $\bm{c} = \exp{\bm{\Sigma}} \bm{c}$ \EndProcedure \State diff --git a/LWB/propagators/sections/implementation.tex b/LWB/propagators/sections/implementation.tex index fedd88e..72bfb91 100644 --- a/LWB/propagators/sections/implementation.tex +++ b/LWB/propagators/sections/implementation.tex @@ -42,7 +42,7 @@ \subsection{Static Polymorphism through CRTP and enable\_if} \par\medskip % While CRTP is an excellent candidate for providing static polymorphism for derived classes, there was also the need for some polymorphism in the Propagator base class itself, since some of the building blocks need to have different implementations based on the dimensionality or homogeneity of the wave packet. -Providing a partially specialized class for every possible combination of parameters is often tedious and unneccessarily bloats the code. +Providing a partially specialized class for every possible combination of parameters is often tedious and unnecessarily bloats the code. Therefore, the C++11 \proc{enable\_if} keyword was used. \proc{enable\_if} allows to define multiple versions of a function and (de)activate them based on a compile-time boolean value. If the boolean is true, the function is enabled. Otherwise it is omitted. @@ -93,7 +93,7 @@ \subsection{IntSplit using alternating templates} \par\medskip % The current \proc{IntSplit} algorithm described in \ref{alg:intsplit} takes three arguments: -the size of the time interval to be covered $\Dt$, the number $M$ of subintervals of size $\dt = \Dt/M$ in which the initial interval should be divided, and a pair of coefficient lists $\{ w_T, w_U \}$ denoting the weights with which the single steps of size $\dt$ should be scaled. +the size of the time interval to be covered $\Dt$, the number $M_{split}$ of subintervals of size $\dt = \Dt/M_{split}$ in which the initial interval should be divided, and a pair of coefficient lists $\{ w_T, w_U \}$ denoting the weights with which the single steps of size $\dt$ should be scaled. In the simplest case, the lists $w_T$ and $w_U$ are both only of length one (Lie-Trotter), but higher order splitting strategies can have much larger coefficient sets. In general, it must hold that $\text{length}(w_T) = \text{length}(w_U)$ (for TU..TU schemes) or $\text{length}(w_T) = \text{length}(w_U)$ (for TU..UT schemes). \par\medskip diff --git a/LWB/propagators/sections/introduction.tex b/LWB/propagators/sections/introduction.tex index 033d0d3..179a3ef 100644 --- a/LWB/propagators/sections/introduction.tex +++ b/LWB/propagators/sections/introduction.tex @@ -1,6 +1,6 @@ \section*{Introduction} % -With the growing understanding for quantum mechanics and quantum dynamics, one can attempt to solve increasingly complex systems and computational problems. +With the growing understanding for quantum mechanics and dynamics, one can attempt to solve increasingly complex systems and computational problems. Without doubt, time propagation is at the core of simulating quantum systems and getting a better understanding for them. \par\medskip % @@ -13,4 +13,4 @@ \section*{Introduction} Based on these mathematical observations, section \ref{sec:buildingblocks} prepares a toolbox of essential building blocks that will help in the construction of propagators as well as for the final understanding of the source code. In section \ref{sec:propagators}, a series of numerical propagators for quantum wave packets are presented and expressed in terms of the components provided earlier. Some detailed insights into the most important technical ideas of the C++ implementation are given in section \ref{sec:implementation}. -Finally, section \ref{sec:results} presents the results of a series of numerical experiments to show the correctness of the implemented algorithms and analyse some performance aspects like the scaling with step size, dimensionality of the wave packet and order of the splitting coefficients. +Finally, section \ref{sec:results} presents the results of a series of numerical experiments to show the correctness of the implemented algorithms and analyze some performance aspects like the scaling with step size, dimensionality of the wave packet and order of the splitting coefficients. diff --git a/LWB/propagators/sections/operatorsplitting.tex b/LWB/propagators/sections/operatorsplitting.tex index 50cc446..e96b4d2 100644 --- a/LWB/propagators/sections/operatorsplitting.tex +++ b/LWB/propagators/sections/operatorsplitting.tex @@ -9,23 +9,32 @@ \section{Operator Splitting for Time Propagation} \im \eps \; \del_t \psi (\bvec{x},t) = \opH (\eps) \; \psi (\bvec{x},t) \end{align} % -with wave function $\psi(\bvec{x},t)$ depending on the spatial variables $\bvec{x} = (x_1,\dots,x_D) \in \R^D$ and the time variable $t \in \R$. +with Hamiltonian operator $\opH$, wave function $\psi(\bvec{x},t)$ depending on the spatial variables $\bvec{x} = (x_1,\dots,x_D) \in \R^D$ and the time variable $t \in \R$. The parameter $\eps$, also known as the semiclassical parameter, determines the degree of influence of quantum dynamics, approaching classical dynamics in the limit $\eps \rightarrow 0$ and full quantum mechanics for $\eps = 1$. \\ Moreover, $\eps$ plays an important role in the stability of numerical schemes. -As underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\sim \eps^{-2}$ for Lie-Trotter splitting, Strang splitting and other related methods, meaning that smaller $\eps$ would lead to a larger error. +As underlined in \cite{GH_convsemiclassical}, a small parameter $\eps$ can often impose severe constraints on the step size of splitting methods and significantly increase the error. For example, the error was shown to be proportional to $\eps^{-2}$ for Lie-Trotter splitting, Strang splitting and other related methods, meaning that smaller $\eps$ would lead to a larger error. \par\medskip % The solution to the above Schrödinger equation \ref{math:tdse} can be approximated as a finite linear combination of Hagedorn functions \begin{align} \label{math:hagedornwp} \psi(\bvec{x},t) \approx u(\bvec{x},t) - = e^{\im S(t)/\eps} \sum_{k \in \K} c_k(t) \varphi_k^\eps [ \bvec{\Pi} ] (\bvec{x}) + = e^{\im S(t)/\eps^2} \sum_{k \in \K} c_k(t) \varphi_k^\eps [ \bvec{\Pi} (t) ] (\bvec{x}) \end{align} -where $\K$ represents a finite multi-index set and the notation for the parameter pack $\bvec{\Pi} = (\bvec{q}(t),\bvec{p}(t),\bvec{Q}(t),\bvec{P}(t))$ is introduced for convenience. +where $\K$ represents a finite multi-index set, $c_k \in \C$ are coefficients that depend on time only, $S(t)$ is a phase factor and $\bvec{q},\bvec{p} \in \R^D$ and $\bmat{Q},\bmat{P} \in \C^{D \times D}$ are some parameters. +For convenience, the notation for the parameter pack $\bvec{\Pi} (t) = (\bvec{q}(t),\bvec{p}(t),\bvec{Q}(t),\bvec{P}(t))$ is introduced. +It is worth noting that $\varphi_k^\eps [ \bvec{\Pi} (t) ]$ depends on time $t$ only implicitly through the time dependency of $\bvec{q}(t),\bvec{p}(t),\bvec{Q}(t),\bvec{P}(t)$. A more detailed analysis of Hagedorn functions is given in \cite{FGL_semiclassical_dynamics} and a comprehensive overview over the relevant mathematical details can be found in \cite{B_master_thesis}. \par\medskip % -For the rest of this report, $\opU$ and $\opW$ will denote the quadratic and non-quadratic component of $\opV$ respectively and it will be used that +The Schrödinger equation \ref{math:tdse} can be solved using the ansatz +\begin{align} + \psi (\bvec{x},t+dt) = \exp \left( - \frac{\im}{\eps^2} \opH t \right) \; \psi (\bvec{x},t) \; . +\end{align} +% +This matrix exponential is very expensive to compute. +Instead, in the search for a numerical solution, one can exploit the structure of the Hamiltonian operator $\opH$, by splitting it into its kinetic part $\opT$ and potential part $\opV$. +Furthermore, for the rest of this report we will denote by $\opU$ and $\opW$ the quadratic and non-quadratic component of $\opV$ respectively and it will be used that % \begin{align} \opV &= V(\bvec{x}) \\ @@ -37,9 +46,9 @@ \section{Operator Splitting for Time Propagation} In other words, $\opU = U(\bvec{q},\bvec{x})$ is the second order Taylor expansion of the potential energy $\opV = V(\bvec{x})$ around $\bvec{q}$ (the coordinate $\bvec{q}$ will be omitted for simplicity of notation) and $\opW = W(\bvec{q},\bvec{x})$ is the corresponding remainder. \par\medskip % -By further recalling the form of the kinetic operator $\opT$ +By further recalling the form of the kinetic operator $\opT$ (for a particle of mass $m=1$) \begin{align} - \opT &= - \sum_{j=1}^D \frac{\eps^2}{2m_j} \frac{\del^2}{\del x_j^2} + \opT &= - \sum_{j=1}^D \frac{\eps^4}{2} \frac{\del^2}{\del x_j^2} \end{align} % one can then split the Hamiltonian operator $\opH$ into the components @@ -48,20 +57,14 @@ \section{Operator Splitting for Time Propagation} \opH = \opT + \opV = \opT + \opU + \opW \end{align} % -This splitting is so central to the construction of the time propagators in the following sections that it is worthwile repeating the most important findings from \cite{FGL_semiclassical_dynamics}: +This splitting is so central to the construction of the time propagators in the following sections that it is worthwhile repeating the most important findings from \cite{FGL_semiclassical_dynamics}: \begin{itemize} - \item The free linear Schrödinger equation ($\opV=0$) can be solved exactly and the wavepacket remains in Hagedorn wavepacket form (equation~\ref{math:hagedornwp}). + \item The free particle Schrödinger equation ($\opV=0$) can be solved exactly and the wave packet remains in Hagedorn wave packet form (equation~\ref{math:hagedornwp}). For time propagation, only the parameters $\bvec{\Pi},S$ need to be updated, the coefficients $\{c_k\}_{k \in \K}$ remain unchanged. \item The Schrödinger equation~\ref{math:tdse} can be solved exactly in a pure quadratic potential, i.e. in a potential $\opV=U(\bvec{x})$ with $\opT=0$. Again, time propagation only affects the parameters $\bvec{\Pi},S$, not the coefficients $\{c_k\}_{k \in \K}$. - \item In the case of an arbitrary potential $\opV = W(\bvec{x})$ that is not quadratic, a set of Galerkin functions can be propagated by adapting the coefficients $\{c_k\}_{k \in \K}$ without changing the parameters $\bvec{\Pi},S$. + \item In the case of an arbitrary potential $\opV = W(\bvec{x})$ that is not quadratic, a variational approach can be used. + A set of Galerkin functions can be propagated by adapting the coefficients $\{c_k\}_{k \in \K}$ without changing the parameters $\bvec{\Pi},S$. \end{itemize} % The following sections will heavily make use of these properties in order to build composition methods for time integration of quantum wave packets. - -% TODO: describe how exact propagation works -% TODO: describe idea of operator splitting in a nutshell -% All based on composition methods -% methods that compose elementary flows/operators to build an approximation of high order -% TODO: operators don't commute -% TODO?: $q^n, p^n \in V_N, \; Q^n, P^n \in M_{N \times N}, \; S^n \in \mathbb{R}, \bm{c}^n \in (c_k^n)_{k \in \K}$ diff --git a/LWB/propagators/sections/propagators.tex b/LWB/propagators/sections/propagators.tex index 438138a..528a674 100644 --- a/LWB/propagators/sections/propagators.tex +++ b/LWB/propagators/sections/propagators.tex @@ -2,17 +2,18 @@ \section{Propagators} \label{sec:propagators} This section gives a short overview of the numerical properties of the propagators that were implemented in the context of this project. -Where appropriate, references are given to find a more detailed mathematical analysis of the propagators. +Where appropriate, references are given to a more detailed mathematical analysis of the propagators. \par\medskip The methods are called in the sequence determined by the \proc{Evolve} function (\ref{alg:Evolve}) and generally are expressed in terms of the basic building blocks that were presented in the previous section. +In order to specify to which propagator each of the implementations belongs, the C++ inspired notation \proc{PropagatorName.Propagate} is used for the pseudo algorithms in this section. If the methods \proc{PrePropagate} and \proc{PostPropagate} are omitted, it means that they are empty. \\ -Whenever convenient, the direct succession of functions \proc{StepT} and \proc{StepU} was replaced by a call to \proc{IntSplit} (see explanation in \ref{subsec:tuintsplit}) for better accuracy. +Whenever convenient, the direct succession of functions \proc{StepT} and \proc{StepU} was replaced by a call to \proc{IntSplit} which effectively corresponds to a series of $M_{split}$ alternating calls to \proc{StepT} and \proc{StepU} (see explanation in \ref{subsec:tuintsplit}) for better accuracy. \subsection{Hagedorn Propagator} \label{sub:hagedorn_propagator} % The Hagedorn propagator, shown in algorithm \ref{alg:hagedorn}, is one of the simplest propagators that can be built by exploiting the numerical properties of the Hamiltonian splitting $\opH = \opT + \opU + \opW$ discussed earlier. -As pointed out in \cite{FGL_semiclassical_dynamics}, this simple time stepping scheme has many beneficial properties like the preservation of the $L^2$ norm of the wavepacket, time reversibility and stability in the classical limit $\eps \rightarrow 0$. +As pointed out in \cite{FGL_semiclassical_dynamics}, this simple time stepping scheme has many beneficial properties like the preservation of the $L^2$ norm of the wave packet, time reversibility and stability in the classical limit $\eps \rightarrow 0$. Also, in the limit of $\K$ approaching the full basis set, the variational approximation used for the propagation with the non-quadratic part $\opW$ becomes exact. \begin{algorithm}[h] \caption{Single timestep with Hagedorn propagator} @@ -67,15 +68,15 @@ \subsection{Semiclassical Propagator} \State \Procedure{Semiclassical.Propagate}{$\Dt$} \State - \State $M := \lceil 1 + \frac{\sqrt{\Dt}}{\eps^{3/4}} \rceil$ + \State $M_{split} := \lceil 1 + \frac{\sqrt{\Dt}}{\eps^{3/4}} \rceil$ \Comment{Divide $\Dt$ into smaller steps} \State - \State \Call{intSplit}{$\frac{\Dt}{2}, \frac{M}{2}, \{ w_T, w_U \}$} - \Comment{$M/2$ split steps with $T+U$} + \State \Call{intSplit}{$\frac{\Dt}{2}, M_{split}, \{ w_T, w_U \}$} + \Comment{$M_{split}$ split steps with $\opT+\opU$} \State \Call{stepW}{$\Dt$} - \Comment{Single step with $W$} - \State \Call{intSplit}{$\frac{\Dt}{2}, \frac{M}{2}, \{ w_T, w_U \}$} - \Comment{$M/2$ split steps with $T+U$} + \Comment{Single step with $\opW$} + \State \Call{intSplit}{$\frac{\Dt}{2}, M_{split}, \{ w_T, w_U \}$} + \Comment{$M_{split}$ split steps with $\opT+\opU$} \State \EndProcedure \end{algorithmic} @@ -156,12 +157,12 @@ \subsection{Processing Propagators} \end{align} % where the pre and post processing steps are applied via the multiplication with matrices $e^P$ and $e^{-P}$ (also referred to as \emph{processors}). -While the processors only need to be applied once at the very beginning and end of the propagation (in order to return to the original Hamiltonian), the multiplication with the \emph{kernel} $e^{-\Dt K}$ is repeated $M$ times, once for every timestep. \\ +While the processors only need to be applied once at the very beginning and end of the propagation (in order to return to the original Hamiltonian), the multiplication with the \emph{kernel} $e^{-\Dt K}$ is repeated $M_{tot}$ times, once for every timestep. \\ As a consequence, one wants to choose the processor $e^P$ in such a way that the evaluation of the kernel $e^{-\Dt K}$ is as simple as possible in order to minimize the work associated with the computation of the kernel. \par\medskip % Unlike most commonly used integration schemes, the family of processing propagators are symplectic integrators. -They are not suited for every kind of Hamiltonian, and the work of Blanes, Casas and Ros in \cite{Blanes1999} gives a method for finding the neccessary conditions that need to be satisfied in order for processing methods to be applicable. +They are not suited for every kind of Hamiltonian, and the work of Blanes, Casas and Ros in \cite{Blanes1999} gives a method for finding the necessary conditions that need to be satisfied in order for processing methods to be applicable. \par\medskip % Algorithm \ref{alg:pre764} presents the \emph{Pre764} processing method as derived in \cite{Blanes1999}. @@ -173,11 +174,11 @@ \subsection{Processing Propagators} \State \Procedure{Pre764.PrePropagate}{$\Dt$} \State - \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \State $M_{pre} = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ \Comment compute number of time steps \For{$j=0,...,v-1$} \Comment $v=6$ for Pre(7,6,4) - \State \Call{intSplit}{$-Z_j \Dt, M, \{w_T, w_U\}$} - \Comment $M$ alternating steps with $\opT$ and $\opU$ + \State \Call{intSplit}{$-Z_j \Dt, M_{pre}, \{w_T, w_U\}$} + \Comment $M_{pre}$ alternating steps with $\opT$ and $\opU$ \State \Call{stepW}{$\upic, -Y_j \Dt$} \Comment single step with $\opW$ \EndFor @@ -201,13 +202,13 @@ \subsection{Processing Propagators} \State \Procedure{Pre764.PostPropagate}{$\Dt$} \State - \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ + \State $M_{post} = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ \Comment compute number of time steps \For{$j=v-1,...,0$} \Comment $v=6$ for Pre(7,6,4) \State \Call{stepW}{$Y_j \Dt$} \Comment single step with $\opW$ - \State \Call{intSplit}{$Z_j \Dt, M, \{w_T, w_U\}$} - \Comment $M$ alternating steps with $\opT$ and $\opU$ + \State \Call{intSplit}{$Z_j \Dt, M_{post}, \{w_T, w_U\}$} + \Comment $M_{post}$ alternating steps with $\opT$ and $\opU$ \EndFor \State \EndProcedure diff --git a/LWB/propagators/sections/results.tex b/LWB/propagators/sections/results.tex index 905362a..d261d1c 100644 --- a/LWB/propagators/sections/results.tex +++ b/LWB/propagators/sections/results.tex @@ -8,7 +8,7 @@ \subsection{Physical Correctness} \label{subsec:physical} % This short subsection analyzes the physical correctness of the implemented propagators. -To do this, a two dimensional wave packet was propagated with each of the presented propagators for a time $\tau = 10$ with timestep $\Dt = 0.01$. +To do this, a two dimensional wave packet was propagated with each of the presented propagators for a time $T = 10$ with timestep $\Dt = 0.01$. All propagators used the \emph{Y4} splitting coefficients for the \proc{IntSplit} method. \par\medskip % @@ -37,7 +37,7 @@ \subsection{Physical Correctness} \end{minipage} \includegraphics[width=.45\textwidth]{figures/energy_Semiclassical.png} \includegraphics[width=.45\textwidth]{figures/drift_Semiclassical.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $\tau = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Semiclassical Propagator with \emph{Y4} splitting} + \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Semiclassical Propagator with \emph{Y4} splitting} \label{fig:energy_Semiclassical} \end{figure} @@ -46,7 +46,7 @@ \subsection{Effect of Stepsize} \label{subsec:convergence} % A convergence analysis was carried out in which the error of each propagator was recorded while reducing the stepsize. -A Semiclassical propagator with \emph{KL10} splitting coefficients and step size 0.001 was used to propagate the wave packet to a final time $\tau = 10$ and create a reference solution. +A Semiclassical propagator with \emph{KL10} splitting coefficients and step size 0.001 was used to propagate the wave packet to a final time $T = 10$ and create a reference solution. \par\medskip % The error between two wave packets was computed by evaluating them on a grid with 1000 grid points and taking the $L_2$ norm of the differences. @@ -78,7 +78,7 @@ \subsection{Benchmark} \subsubsection{Comparison of propagators} % A quick comparison of timings for the different propagators is shown in table \ref{tab:speedup}. -The same setup as for the energy analysis was used (2D wave packet, $\tau = 10$, step size $\Dt = 0.001$, \emph{Y4} time splitting coefficients). +The same setup as for the energy analysis was used (2D wave packet, $T = 10$, step size $\Dt = 0.001$, \emph{Y4} time splitting coefficients). It comes as quite a surprise, that the Semiclassical propagator is faster at computing the result than the Hagedorn propagator, which at first glance seems to have a simpler structure and involve fewer steps. Also, the Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_k}{W}{\varphi_l}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. % @@ -97,7 +97,7 @@ \subsubsection{Comparison of propagators} McL84 & 23.52 \\ \hline \end{tabular} - \caption{Runtimes for propagating a 2D wave packet to a time $\tau = 10$ with timestep $\Dt = 0.001$ with different propagators. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $1\%$ for the Python timings and below $0.1\%$ for the C++ timings.} + \caption{Runtimes for propagating a 2D wave packet to a time $T = 10$ with timestep $\Dt = 0.001$ with different propagators. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $1\%$ for the Python timings and below $0.1\%$ for the C++ timings.} \label{tab:speedup} \end{table} @@ -109,7 +109,7 @@ \subsubsection{Splitting Schemes} \par\medskip % Therefore, a numerical experiment was carried out in order to analyze how much computational time is consumed for coefficient pairs of different sizes. -To do this, the Semiclassical Propagator was used to propagate a two dimensional wave packet in a harmonic potential over a time of $\tau = 400$ and with stepsize $\Dt = 0.01$. +To do this, the Semiclassical Propagator was used to propagate a two dimensional wave packet in a harmonic potential over a time of $T = 400$ and with stepsize $\Dt = 0.01$. The same simulation was carried out for coefficient pairs of various different sizes and were run in Python as well as C++. The results are listed in table \ref{tab:benchmarksplit_t} and plotted in figure \ref{fig:benchmarksplit_f}. \par\medskip diff --git a/LWB/propagators/sections/title.tex b/LWB/propagators/sections/title.tex index 942b34a..88506ac 100644 --- a/LWB/propagators/sections/title.tex +++ b/LWB/propagators/sections/title.tex @@ -5,7 +5,7 @@ \vspace{2cm} - {\Large \textsc{Time Propagation\\ of Hagedorn Quantum Wave Packets\\[5pt]}} + {\Large \textsc{Time Propagation\\ of Hagedorn Wave Packets\\[5pt]}} {\large \textsc{an efficient implementation in C++}} \vspace{1cm} From d1289a4eb961b498f94b30da7b6079eefb721000 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sat, 11 Feb 2017 01:40:59 +0100 Subject: [PATCH 20/28] some more small fixes --- LWB/propagators/sections/abstract.tex | 6 + LWB/propagators/sections/conclusion.tex | 7 - LWB/propagators/sections/implementation.tex | 7 +- LWB/propagators/sections/propagators.tex | 148 ++++++++++++-------- LWB/propagators/sections/results.tex | 10 +- 5 files changed, 105 insertions(+), 73 deletions(-) diff --git a/LWB/propagators/sections/abstract.tex b/LWB/propagators/sections/abstract.tex index 315aade..bc4112a 100644 --- a/LWB/propagators/sections/abstract.tex +++ b/LWB/propagators/sections/abstract.tex @@ -3,3 +3,9 @@ A short introduction to operator splitting and a summary of some of the most important time propagation schemes for Hagedorn wave packets are also given. Finally, some benchmarking results are presented which analyze energy conservation and compare different propagators, splitting schemes, step sizes and dimensionalities. \end{abstract} +% +\vspace{5mm} +% +\begin{center} + The code to the project can be found on Github at \cite{libwaveblocks}. +\end{center} diff --git a/LWB/propagators/sections/conclusion.tex b/LWB/propagators/sections/conclusion.tex index ba91d94..dfb47ef 100644 --- a/LWB/propagators/sections/conclusion.tex +++ b/LWB/propagators/sections/conclusion.tex @@ -5,10 +5,3 @@ \section{Conclusion} At the same time, as shown by the benchmarking results, the performance of the resulting code is by orders of magnitude better than that of the corresponding Python code. In particular, the additional cost for using higher-order splitting coefficients is very contained thanks to a templated implementation of the \proc{IntSplit} method which is resolved at compile time and avoids the usage of any loops. Given these performance improvements, more complex quantum mechanical simulations are now possible with this new framework. - - -\vspace{5cm} - -\begin{center} - The code to the project can be found on Github at \cite{libwaveblocks}. -\end{center} diff --git a/LWB/propagators/sections/implementation.tex b/LWB/propagators/sections/implementation.tex index 72bfb91..509817a 100644 --- a/LWB/propagators/sections/implementation.tex +++ b/LWB/propagators/sections/implementation.tex @@ -95,7 +95,7 @@ \subsection{IntSplit using alternating templates} The current \proc{IntSplit} algorithm described in \ref{alg:intsplit} takes three arguments: the size of the time interval to be covered $\Dt$, the number $M_{split}$ of subintervals of size $\dt = \Dt/M_{split}$ in which the initial interval should be divided, and a pair of coefficient lists $\{ w_T, w_U \}$ denoting the weights with which the single steps of size $\dt$ should be scaled. In the simplest case, the lists $w_T$ and $w_U$ are both only of length one (Lie-Trotter), but higher order splitting strategies can have much larger coefficient sets. -In general, it must hold that $\text{length}(w_T) = \text{length}(w_U)$ (for TU..TU schemes) or $\text{length}(w_T) = \text{length}(w_U)$ (for TU..UT schemes). +In general, it must hold that $\text{length}(w_T) = \text{length}(w_U)$ (for TU..TU schemes) or $\text{length}(w_T) = \text{length}(w_U)+1$ (for TU..UT schemes). \par\medskip % The obvious implementation of using a dynamic container (like a C++ vector) for the coefficients and a simple \emph{for loop} over all its entries brings a two (minimal) drawbacks: First, the overhead of the loop itself and second, the fact that the number of coefficients is not known at compile time and thus no optimizations can be done. @@ -111,12 +111,11 @@ \subsection{IntSplit using alternating templates} % One technical subtlety of this approach is, that the C++ language does not allow partially specialized function templates. This can, however, be overcome quite easily by declaring \proc{TU} and \proc{UT} as classes with a static function. In contrast to function templates, class templates can be partially specialized. -By using static member functions, the actual class objects do not need to be created and all the function calls will be resolved at compile time. -\footnote{One limitation is the maximal template recursion depth. The default value on gcc compiler is 900 and can be increased if required. Typical template nesting depths for coefficient pairs should lie significantly below 100.} +By using static member functions, the actual class objects do not need to be created and all the function calls will be resolved at compile time\footnote{One limitation is the maximal template recursion depth. The default value on gcc compiler is 900 and can be increased if required. Typical template nesting depths for coefficient pairs should lie significantly below 100.}. All templates, no loops, and the resulting sequential code can then be further optimized by the compiler. \par\medskip % -Some minimal code snippets explaining the working principle of \proc{IntSplit} are shown in figure~\ref{fig:intsplit} +Some minimal code snippets explaining the working principle of \proc{IntSplit} are shown in Figure~\ref{fig:intsplit}. % In order to prevent the inappropriate use of the \proc{IntSplit} function through incompatible coefficient lists $w_T$ and $w_U$, their sizes are checked in the code at compile time via static assertions. \par\medskip diff --git a/LWB/propagators/sections/propagators.tex b/LWB/propagators/sections/propagators.tex index 528a674..0dc4df2 100644 --- a/LWB/propagators/sections/propagators.tex +++ b/LWB/propagators/sections/propagators.tex @@ -47,13 +47,13 @@ \subsection{Semiclassical Propagator} % In addition, due to the numerical properties of the semiclassical splitting, it even allows to take larger timesteps $\Dt$ than conventional -splitting methods like the YL-splitting, while maintaining the same error. +splitting methods like the YL-splitting (see \cite{GH_convsemiclassical}), while maintaining the same error. \par\medskip % Finally and most importantly, the error is no longer proportional to $1/\eps^2$ but instead scales linearly in the semiclassical parameter $\eps$, meaning that a smaller $\eps$ will now reduce the error instead of increasing it. -The error scales with $\eps (\Delta t)^2$ for the semiclassical splitting using the Y-splitting, +The error scales with $\eps (\Delta t)^2$ for the semiclassical splitting using the Y-splitting (see \cite{GH_convsemiclassical}), but the dependency on the timestep $\Delta t$ can be improved even further by using different splittings which effectively corresponds to higher order coefficient pairs $w_T$ and $w_U$. \par\medskip @@ -86,13 +86,13 @@ \subsection{Semiclassical Propagator} \subsection{Magnus Propagator} \label{sub:magnus_propagator} % -As was noted by Magnus in \cite{Magnus1954}, the solution to a differential equation of the form +As was noted by Magnus in \cite{Magnus1954}, the solution to a differential equation of the form % \begin{align} y'(t) = a(t) y(t) \qquad t \ge 0 \end{align} % -can be written as +with initial value $y(0) = y_0$ can be written as % \begin{align} \label{math:magnussolution} @@ -121,9 +121,9 @@ \subsection{Magnus Propagator} \State \Procedure{Magnus.Propagate}{$\Dt$} \State - \State $h_1 = \frac{3-\sqrt{3}}{6} \Dt$, $h_2 = \frac{2\sqrt{3}}{6} \Dt$ + \State $h_1 = \frac{3-\sqrt{3}}{6} \Dt$ $h_2 = \frac{2\sqrt{3}}{6} \Dt$ \Comment Gauss-Legendre coefficients on $[0,\Dt]$ - \State $M_{k} = 1+\sqrt{h_{k} \eps^{-3/8}}, \quad k=1,2$ + \State $M_{k} = 1 + \lfloor \sqrt{h_{k} \eps^{-3/8}} \rfloor, \quad k=1,2$ \Comment number of timesteps for splitting \State \State \Call{intSplit}{$h_1, M_1, \{w_T, w_U\}$} @@ -145,11 +145,94 @@ \subsection{Magnus Propagator} \end{algorithmic} \end{algorithm} +\subsection{McLachlan Propagators} +\label{sub:mcl_propagator} +% +McLachlan (see \cite{McLachlan1995}) has investigated various symplectic schemes for computing the effect of operators of the form $\opX = \opA + \epsilon \opB$ where $\opA$ describes an exactly solvable system and $\opB$ is a perturbation. +The factor $\epsilon$ is a small parameter that indicates the limited influence of $\opB$, not to be confused with the semiclassical parameter $\eps$. +\par\medskip +The \emph{McL} integration schemes are characterised by pairs of weighing coefficients $a_i$ and $b_i$ that allow to represent the exponential $e^{\opX}$ as a product +% +\begin{align} + e^\opX = \prod_i e^{b_i t \epsilon \opB} e^{a_i t \opA} \; . +\end{align} +% +McLachlan also goes through the process of deriving the coefficients for the following schemes +\begin{align*} + &\text{\emph{McL(2s,2)}} &&\text{with error of order } \epsilon (\Dt)^{2s} + \epsilon^2 (\Dt)^2 \\ + &\text{\emph{McL(6,4)}} &&\text{with error of order } \epsilon (\Dt)^{6} + \epsilon^2 (\Dt)^4 \\ + &\text{\emph{McL(8,4)}} &&\text{with error of order } \epsilon (\Dt)^{8} + \epsilon^2 (\Dt)^4 \; . +\end{align*} +% +Note that the error depends on two small parameters, the timestep $\Dt$ and the parameter $\epsilon$ that quantifies the influence of $\opB$. +\par\medskip +% +In our case, $\opX = \opH = \opT + \opU + \opW$, the computationally expensive operator $\opW$ takes the role of $\epsilon \opB$, while the remaining Hamiltonian $\opT + \opU$ corresponds to $\opA$. +Our preferred method will therefore be of the form $\opA \opB \opA$ because this minimizes the number of steps with $\opB = \opW$. +\par\medskip +% +The source code of this project contains the C++ implementation of the propagators \emph{McL(4,2)} and \emph{McL(8,4)} that require only two respectively five evaluations of $\opW$ per step. + +\begin{algorithm}[h] + \caption{Single timestep with McL42 propagator} + \label{alg:mcl} + \begin{algorithmic} + \State + \Procedure{McL42.Propagate}{$\Dt$} + \State + \State $M = \lceil \left( \Dt^4 / \eps^3 \right)^{\frac{1}{8}} \rceil$ + \Comment compute number of substeps + \State \Call{intSplit}{$A_0 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_0 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_1 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_1 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_2 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State + \EndProcedure + \\\hrulefill + \State + \Procedure{McL84.Propagate}{$\Dt$} + \State + \State $M = \lceil \left( \Dt^4 / \eps^3 \right)^{\frac{1}{8}} \rceil$ + \Comment compute number of substeps + \State \Call{intSplit}{$A_0 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_0 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_1 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_1 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_2 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_2 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_3 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_3 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_4 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State \Call{stepW}{$B_4 \Dt$} + \Comment $\opW = B$ + \State \Call{intSplit}{$A_5 \Dt, M, \{w_T, w_U\}$} + \Comment $\opT + \opU = A$ + \State + \EndProcedure + \end{algorithmic} +\end{algorithm} + + \subsection{Processing Propagators} \label{sub:pre764_propagator} % The idea of processing propagators is to carry out the time evolution with a Hamiltonian that is slightly perturbed from the original one. -To achieve this, a pre and post processing step is applied. \\ +To achieve this, a pre and post processing step is applied. More formally, it holds that % \begin{align} @@ -216,54 +299,3 @@ \subsection{Processing Propagators} \end{algorithm} -\subsection{McLachlan Propagators} -\label{sub:mcl_propagator} -% -McLachlan (see \cite{McLachlan1995}) has investigated various symplectic schemes for computing the effect of operators of the form $\opX = \opA + \epsilon \opB$ where $\opA$ describes an exactly solvable system and $\opB$ is a perturbation. -The factor $\epsilon$ is a small parameter that indicates the limited influence of $\opB$, not to be confused with the semiclassical parameter $\eps$. -\par\medskip -The \emph{McL} integration schemes are characterised by pairs of weighing coefficients $a_i$ and $b_i$ that allow to represent the exponential $e^{\opX}$ as a product -% -\begin{align} - e^\opX = \prod_i e^{b_i t \epsilon \opB} e^{a_i t \opA} -\end{align} -% -McLachlan also goes through the process of deriving the coefficients for the following schemes -\begin{align*} - &\text{\emph{McL(2s,2)}} &&\text{with error of order } \epsilon (\Dt)^{2s} + \epsilon^2 (\Dt)^2 \\ - &\text{\emph{McL(6,4)}} &&\text{with error of order } \epsilon (\Dt)^{6} + \epsilon^2 (\Dt)^4 \\ - &\text{\emph{McL(8,4)}} &&\text{with error of order } \epsilon (\Dt)^{8} + \epsilon^2 (\Dt)^4 -\end{align*} -% -Note that the error depends on two small parameters, the timestep $\Dt$ and the parameter $\epsilon$ that quantifies the influence of $\opB$. -\par\medskip -% -In our case, $\opX = \opH = \opT + \opV + \opW$, the computationally expensive operator $\opW$ takes the role of $\epsilon \opB$, while the remaining Hamiltonian $\opT + \opU$ corresponds to $\opA$. -Our preferred method will therefore be of the form $ABA$ because this minimizes the number of steps with $\opB = \opW$. -\par\medskip -% -The source code of this project contains the C++ implementation of the propagators \emph{McL(4,2)} (asymmetric) and \emph{McL(8,4)} (symmetric) that require only two respectively three evaluations of $\opW$ per step. - -\begin{algorithm}[h] - \caption{Single timestep with McL42 propagator} - \label{alg:mcl} - \begin{algorithmic} - \State - \Procedure{McL42.Propagate}{$\Dt$} - \State - \State $M = \lceil \left( \Dt^4 / \eps^3 \right)^{\frac{1}{8}} \rceil$ - \Comment compute number of substeps - \State \Call{intSplit}{$A_0 \Dt, M, \{w_T, w_U\}$} - \Comment $\opT + \opU = A$ - \State \Call{stepW}{$B_0 \Dt$} - \Comment $\opW = B$ - \State \Call{intSplit}{$A_1 \Dt, M, \{w_T, w_U\}$} - \Comment $\opT + \opU = A$ - \State \Call{stepW}{$B_1 \Dt$} - \Comment $\opW = B$ - \State \Call{intSplit}{$A_2 \Dt, M, \{w_T, w_U\}$} - \Comment $\opT + \opU = A$ - \State - \EndProcedure - \end{algorithmic} -\end{algorithm} diff --git a/LWB/propagators/sections/results.tex b/LWB/propagators/sections/results.tex index d261d1c..390b6ee 100644 --- a/LWB/propagators/sections/results.tex +++ b/LWB/propagators/sections/results.tex @@ -2,10 +2,11 @@ \section{Results} \label{sec:results} % This section discusses the outcomes of some numerical experiments that were carried out with the newly developed code. -Aspects that need to be checked are the correctness of the propagators from a quantum mechanical point of view (subsection~\ref{subsec::physical}), convergence of the methods (subsection~\ref{subsec:convergence}) and benchmarking of the compute time (subsection~\ref{subsec:benchmark}). +Aspects that need to be checked are the correctness of the propagators from a quantum mechanical point of view (subsection~\ref{subsec:physical}), convergence of the methods (subsection~\ref{subsec:convergence}) and benchmarking of the compute time (subsection~\ref{subsec:benchmark}). \subsection{Physical Correctness} \label{subsec:physical} +{\Huge TODO: Review this part} % This short subsection analyzes the physical correctness of the implemented propagators. To do this, a two dimensional wave packet was propagated with each of the presented propagators for a time $T = 10$ with timestep $\Dt = 0.01$. @@ -44,6 +45,7 @@ \subsection{Physical Correctness} \subsection{Effect of Stepsize} \label{subsec:convergence} +{\Huge TODO: Review this part} % A convergence analysis was carried out in which the error of each propagator was recorded while reducing the stepsize. A Semiclassical propagator with \emph{KL10} splitting coefficients and step size 0.001 was used to propagate the wave packet to a final time $T = 10$ and create a reference solution. @@ -57,7 +59,7 @@ \subsection{Effect of Stepsize} \begin{figure}[ht] \centering \includegraphics[width=.8\textwidth]{figures/error_analysis.png} - \caption{Convergence of the wavefunctions towards the "correct" solution. The $L_2$ norm was measured by projecting the wave function on a grid with 1000 nodes in the range $[-1,1]$ and taking the differences of these values. According to this experiment, the Hagedorn propagator seems to have higher convergence order than its contestants, despite its simplicity. + \caption{Convergence of the wavefunctions towards the reference solution. The $L_2$ norm was measured by projecting the wave function on a grid with 1000 nodes in the range $[-1,1]$ and taking the differences of these values. According to this experiment, the Hagedorn propagator seems to have higher convergence order than its contestants, despite its simplicity. This observation could be an interesting starting point for further experiments.} \label{fig:error_analysis} \end{figure} @@ -97,7 +99,7 @@ \subsubsection{Comparison of propagators} McL84 & 23.52 \\ \hline \end{tabular} - \caption{Runtimes for propagating a 2D wave packet to a time $T = 10$ with timestep $\Dt = 0.001$ with different propagators. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $1\%$ for the Python timings and below $0.1\%$ for the C++ timings.} + \caption{Runtimes for propagating a 2D wave packet to a time $T = 10$ with timestep $\Dt = 0.001$ with different propagators in C++. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $0.1\%$.} \label{tab:speedup} \end{table} @@ -168,6 +170,6 @@ \subsubsection{Scaling with dimensionality D} \begin{figure}[ht] \centering \includegraphics[width=.8\textwidth]{figures/dimension_analysis.png} - \caption{Scaling of the computation time with wave pack dimension $D$. Measurements done for a wave packet in a generalized harmonic potential, propagated for 100 steps with the Semiclassical Propagator.} + \caption{Scaling of the computation time with wave packet dimension $D$. Measurements done for a wave packet in a generalized harmonic potential, propagated for 100 steps with the Semiclassical Propagator.} \label{fig:dimension_analysis} \end{figure} From ed361c5865b7a8e854ca606b04fd9cdb9d82c8d6 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Sat, 11 Feb 2017 21:50:46 +0100 Subject: [PATCH 21/28] Some more minor fixes in writeup --- LWB/propagators/sections/buildingblocks.tex | 20 +++++----- LWB/propagators/sections/conclusion.tex | 2 +- LWB/propagators/sections/implementation.tex | 15 ++++---- LWB/propagators/sections/introduction.tex | 4 +- .../sections/operatorsplitting.tex | 9 +++-- LWB/propagators/sections/propagators.tex | 38 +++++++++---------- LWB/propagators/sections/results.tex | 19 +++++----- references/propagators.bib | 38 ++++++++++++++++--- 8 files changed, 87 insertions(+), 58 deletions(-) diff --git a/LWB/propagators/sections/buildingblocks.tex b/LWB/propagators/sections/buildingblocks.tex index 181b9a6..b6455b1 100644 --- a/LWB/propagators/sections/buildingblocks.tex +++ b/LWB/propagators/sections/buildingblocks.tex @@ -2,7 +2,7 @@ \section{Common building blocks for time evolution schemes} \label{sec:buildingblocks} % In order to be able to create an efficient, still flexible, implementation of various time propagators, it is helpful to break them down in common, optimized building blocks in terms of which all (or most) propagators can be expressed. -This section will briefly introduce the most important, generic building blocks that were found necessary in order to build the propagators in section~\ref{sec:propagators}. +The present section will briefly introduce the most important, generic building blocks that were found necessary in order to build the propagators in section~\ref{sec:propagators}. For each of them, it will be assumed that the following wave packet variables are implicitly accessible, without passing them as an argument to the procedure: the parameter pack $\bvec{\Pi} = (\bvec{q},\bvec{p},\bvec{Q},\bvec{P})$, the phase factor $S$, the coefficients $\{ c_k \}_{k \in \K}$, the potential function $V(\bvec{q})$ (including Jacobian and Hessian thereof) and the potential remainder $W(\bvec{q})$. \par\medskip % @@ -16,7 +16,7 @@ \subsection{Evolve} The concrete implementation of the methods \proc{PrePropagate}, \proc{Propagate} and \proc{PostPropagate} is delegated to the derived classes (see section~\ref{sec:implementation} for more details), but will generally be a combination of the basic building blocks that follow in this section. % -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Evolve the wave packet for a time period $T$} \label{alg:Evolve} \begin{algorithmic} @@ -44,7 +44,7 @@ \subsection{StepT, StepU and IntSplit} In the inhomogeneous implementation of the algorithms, the subscript $n$ is used to denote the relative parameters at energy level $n$. \par\medskip % -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Propagate with Kinetic Energy Operator $\opT$} \label{alg:StepT} \begin{algorithmic} @@ -77,7 +77,7 @@ \subsection{StepT, StepU and IntSplit} \end{algorithmic} \end{algorithm} % -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Propagate with (Quadratic) Potential Energy Operator $\opU$} \label{alg:StepU} \begin{algorithmic} @@ -112,11 +112,11 @@ \subsection{StepT, StepU and IntSplit} % %%% IntSplit In the practical implementations of the propagators that will be presented in the next section, propagation with $\opT$ and $\opU$ is usually replaced by a series of smaller, alternating steps with the two operators. -In order to facilitate the use of such an alternating pattern, the helper function \proc{IntSplit} (algorithm~\ref{alg:intsplit}) is introduced. -This function splits the timestep $\Dt$ into $M_{split}$ smaller timesteps of size $\dt$ and uses the pair of weight lists $\{ w_T, w_U \}$ to alternatingly apply operators $\opT$ and $\opU$ on each of the sub-intervals. -More detailed information on the individual parameters to this function and the practical implementation can be found in section \ref{sec:implementation} +In order to facilitate the use of such an alternating pattern, the helper function \proc{IntSplit} (algorithm~\ref{alg:intsplit}) is introduced: +it splits the timestep $\Dt$ into $M_{split}$ smaller timesteps of size $\dt$ and uses the pair of weight lists $\{ w_T, w_U \}$ to alternatingly apply operators $\opT$ and $\opU$ on each of the sub-intervals. +More detailed information on the individual parameters to the \proc{IntSplit} function and the practical implementation can be found in section \ref{sec:implementation} % -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Split a time interval and alternatingly apply $\opT$ and $\opU$} \label{alg:intsplit} \begin{algorithmic} @@ -155,10 +155,10 @@ \subsection{StepW and building of the interaction matrix} = \int_{\R^D} \conj{\varphi_k(\bvec{x})} W(\bvec{x}) \varphi_l(\bvec{x}) \; \dif \bvec{x} \; . \end{align} % -The calculation of such interaction matrices $\bmat{F}$ makes heavy use of the work on inner products that has been done in previous student projects for the libwaveblocks library \cite{libwaveblocks}. +The calculation of such interaction matrices $\bmat{F}$ makes heavy use of the work on inner products that has been presented in \cite{LWB_innerproducts}, another student project in the WaveBlocks framework. For the further treatment in this report, it is assumed that an efficient implementation of \proc{BuildF} is readily available. % -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Propagate with (Non-Quadratic) Potential Energy Operator $\opW$} \label{alg:StepW} \begin{algorithmic} diff --git a/LWB/propagators/sections/conclusion.tex b/LWB/propagators/sections/conclusion.tex index dfb47ef..1a2d307 100644 --- a/LWB/propagators/sections/conclusion.tex +++ b/LWB/propagators/sections/conclusion.tex @@ -4,4 +4,4 @@ \section{Conclusion} The C++ code, which is the main outcome of the conduced work, is based on a clean software design that makes it very easy to add new Propagators and integrate them into the existing framework. At the same time, as shown by the benchmarking results, the performance of the resulting code is by orders of magnitude better than that of the corresponding Python code. In particular, the additional cost for using higher-order splitting coefficients is very contained thanks to a templated implementation of the \proc{IntSplit} method which is resolved at compile time and avoids the usage of any loops. -Given these performance improvements, more complex quantum mechanical simulations are now possible with this new framework. +Given these performance improvements, more complex quantum mechanical simulations are now possible with the WaveBlocks framework. diff --git a/LWB/propagators/sections/implementation.tex b/LWB/propagators/sections/implementation.tex index 509817a..ef5e790 100644 --- a/LWB/propagators/sections/implementation.tex +++ b/LWB/propagators/sections/implementation.tex @@ -12,7 +12,7 @@ \section{Implementation in C++} In fact, the current implementation of the \proc{build\_matrix} routine for computing the inner product requires to always compute a full $N \times N$ sub-matrix, even if just one single entry is needed, which is a very severe inefficiency if the number of levels $N>1$. \par\medskip % -However, the aim of this project was to provide a fast implementation of the propagators by re-using existing functions, therefore possible performance enhancements through changing of the existing code was out of the scope of this project and could be targeted in future work. +However, the aim of the work presented here was to provide a fast implementation of the propagators by re-using existing functions, therefore possible performance enhancements through changing of the existing code was out of the scope of this project and could be targeted in future work. Consequently, providing an efficient implementation for time propagation mainly translates into finding an efficient way for calling existing functions with minimal overhead and by making use of compiler optimizations where possible. \par\medskip % @@ -38,15 +38,16 @@ \subsection{Static Polymorphism through CRTP and enable\_if} This idea is essentially an application of the Strategy Pattern as introduced by Gamma et al. in \cite{Gamma1995}, one of the most important references for software design patterns. \par\medskip % -A popular C++ design pattern for achieving static polymorphism is the \emph{Curiously Recurring Template Pattern}, \emph{CRTP} for short, which is described in more detail in \cite{C_CRTP} and in earlier work on the WaveBlocks project \cite{libwaveblocks}. +A popular C++ design pattern for achieving static polymorphism is the \emph{Curiously Recurring Template Pattern}, \emph{CRTP} for short, which is described in more detail in \cite{C_CRTP} and in earlier work on the WaveBlocks project \cite{LWB_wavepackets}. \par\medskip % While CRTP is an excellent candidate for providing static polymorphism for derived classes, there was also the need for some polymorphism in the Propagator base class itself, since some of the building blocks need to have different implementations based on the dimensionality or homogeneity of the wave packet. Providing a partially specialized class for every possible combination of parameters is often tedious and unnecessarily bloats the code. -Therefore, the C++11 \proc{enable\_if} keyword was used. -\proc{enable\_if} allows to define multiple versions of a function and (de)activate them based on a compile-time boolean value. +Therefore, the C++11 \proc{enable\_if} keyword was used, which is a handy C++ metafunction from the \proc{type\_traits} header. +It allows to define multiple versions of a function and conditionally (de)activate them based on a compile-time boolean value. If the boolean is true, the function is enabled. Otherwise it is omitted. -This handy metafunction from the C++ \proc{type\_traits} header was used in order to conditionally enable functions based on whether the number of levels $N$ was equal or greater to one or to distinguish between functions for homogeneous or inhomogeneous wave functions, which effectively provides a form of compile time polymorphism. +In the code, such boolean values are indicating whether the number of levels $N$ is equal or greater to one or to distinguish between functions for homogeneous or inhomogeneous wave functions. +The presented use of \proc{enable\_if} effectively provides a form of compile time polymorphism. % \begin{figure}[ht] \centering @@ -71,7 +72,7 @@ \subsection{Evolve and callback function} \par\medskip % Furthermore, since the wave packet is passed to the \proc{Propagator} class \emph{by reference}, it is of course available in the callback function as well and does not need to be passed as an argument explicitly (provided that it is captured in the angular brackets \proc{[]} of the lambda function). \\ -\emph{Disclaimer} if the value of the wave packet is (intentionally or unintentionally) changed in the callback function, this will of course also affect the further results of the simulation. +\emph{Disclaimer} if the value of the wave packet is (intentionally or unintentionally) changed in the callback function, the further results of the simulation will of course also be affected. Also, care must be taken when measuring energies of packets that are being propagated with propagators such as \emph{Pre764} which apply a pre-processor to the packet before starting the simulation. In order to get the ``physically correct'' wave packet, the corresponding post-processor needs to be applied before making any meaningful measurement. \par\medskip @@ -109,7 +110,7 @@ \subsection{IntSplit using alternating templates} Secondly, and with the information about container size readily available at compile time, one can then implement two sets of templates \proc{TU} and \proc{UT} that execute a step with operator $\opT$ or $\opU$ respectively and then mutually invoke each other, until both arrays of coefficients are used up. As a termination condition, when all coefficients have been used, a partially specialized instance of the templates will be called and interrupt the mutual invocation. \\ % -One technical subtlety of this approach is, that the C++ language does not allow partially specialized function templates. This can, however, be overcome quite easily by declaring \proc{TU} and \proc{UT} as classes with a static function. +One technical subtlety of the presented approach is, that the C++ language does not allow partially specialized function templates. This can, however, be overcome quite easily by declaring \proc{TU} and \proc{UT} as classes with a static function. In contrast to function templates, class templates can be partially specialized. By using static member functions, the actual class objects do not need to be created and all the function calls will be resolved at compile time\footnote{One limitation is the maximal template recursion depth. The default value on gcc compiler is 900 and can be increased if required. Typical template nesting depths for coefficient pairs should lie significantly below 100.}. All templates, no loops, and the resulting sequential code can then be further optimized by the compiler. diff --git a/LWB/propagators/sections/introduction.tex b/LWB/propagators/sections/introduction.tex index 179a3ef..7531dfb 100644 --- a/LWB/propagators/sections/introduction.tex +++ b/LWB/propagators/sections/introduction.tex @@ -5,11 +5,11 @@ \section*{Introduction} \par\medskip % The aim of this project was to give a summary of some state of the art quantum time propagators and provide an efficient C++ implementation for evolving Hagedorn wave packets in time. -The produced code is an addition to the C++ WaveBlocks Framework whose functionality has already been extended in the context of preceding student projects. +The produced code is an addition to the C++ WaveBlocks framework whose functionality has already been extended in the context of preceding student projects. The entire WaveBlocks project, which is partially based on an existing Python project with the same name, can be found at \cite{libwaveblocks}. \par\medskip % -In this report, section \ref{sec:operatorsplitting} repeats some of the most important mathematical ideas underlying the time propagation of wave packets and introduces the notations that will be used throughout the rest of the document. +Section \ref{sec:operatorsplitting} repeats some of the most important mathematical ideas underlying the time propagation of wave packets and introduces the notations that will be used throughout the rest of the document. Based on these mathematical observations, section \ref{sec:buildingblocks} prepares a toolbox of essential building blocks that will help in the construction of propagators as well as for the final understanding of the source code. In section \ref{sec:propagators}, a series of numerical propagators for quantum wave packets are presented and expressed in terms of the components provided earlier. Some detailed insights into the most important technical ideas of the C++ implementation are given in section \ref{sec:implementation}. diff --git a/LWB/propagators/sections/operatorsplitting.tex b/LWB/propagators/sections/operatorsplitting.tex index e96b4d2..d262305 100644 --- a/LWB/propagators/sections/operatorsplitting.tex +++ b/LWB/propagators/sections/operatorsplitting.tex @@ -32,15 +32,15 @@ \section{Operator Splitting for Time Propagation} \psi (\bvec{x},t+dt) = \exp \left( - \frac{\im}{\eps^2} \opH t \right) \; \psi (\bvec{x},t) \; . \end{align} % -This matrix exponential is very expensive to compute. +However, the matrix exponential is very expensive to compute. Instead, in the search for a numerical solution, one can exploit the structure of the Hamiltonian operator $\opH$, by splitting it into its kinetic part $\opT$ and potential part $\opV$. -Furthermore, for the rest of this report we will denote by $\opU$ and $\opW$ the quadratic and non-quadratic component of $\opV$ respectively and it will be used that +Furthermore, for the rest of the report we will denote by $\opU$ and $\opW$ the quadratic and non-quadratic component of $\opV$ respectively and it will be used that % \begin{align} \opV &= V(\bvec{x}) \\ \opU &= U(\bvec{q},\bvec{x}) := V(\bvec{q}) + \nabla V(\bvec{q}) (\bvec{x}-\bvec{q}) + \frac{1}{2} (\bvec{x}-\bvec{q})^T \nabla^2 V(\bvec{q}) (\bvec{x}-\bvec{q}) \\ - \opW &= W(\bvec{q},\bvec{x}) := V(\bvec{x}) - U(\bvec{q},\bvec{x}) + \opW &= W(\bvec{q},\bvec{x}) := V(\bvec{x}) - U(\bvec{q},\bvec{x}) \; . \end{align} % In other words, $\opU = U(\bvec{q},\bvec{x})$ is the second order Taylor expansion of the potential energy $\opV = V(\bvec{x})$ around $\bvec{q}$ (the coordinate $\bvec{q}$ will be omitted for simplicity of notation) and $\opW = W(\bvec{q},\bvec{x})$ is the corresponding remainder. @@ -54,7 +54,7 @@ \section{Operator Splitting for Time Propagation} one can then split the Hamiltonian operator $\opH$ into the components % \begin{align} - \opH = \opT + \opV = \opT + \opU + \opW + \opH = \opT + \opV = \opT + \opU + \opW \; . \end{align} % This splitting is so central to the construction of the time propagators in the following sections that it is worthwhile repeating the most important findings from \cite{FGL_semiclassical_dynamics}: @@ -68,3 +68,4 @@ \section{Operator Splitting for Time Propagation} \end{itemize} % The following sections will heavily make use of these properties in order to build composition methods for time integration of quantum wave packets. +A more in depth analysis of composition methods can be found in \cite{GeoNumInt}. diff --git a/LWB/propagators/sections/propagators.tex b/LWB/propagators/sections/propagators.tex index 0dc4df2..d533d8e 100644 --- a/LWB/propagators/sections/propagators.tex +++ b/LWB/propagators/sections/propagators.tex @@ -1,11 +1,11 @@ \section{Propagators} \label{sec:propagators} -This section gives a short overview of the numerical properties of the propagators that were implemented in the context of this project. +This section gives a short overview of the numerical properties of the propagators that were implemented in the context of time propagation. Where appropriate, references are given to a more detailed mathematical analysis of the propagators. \par\medskip The methods are called in the sequence determined by the \proc{Evolve} function (\ref{alg:Evolve}) and generally are expressed in terms of the basic building blocks that were presented in the previous section. -In order to specify to which propagator each of the implementations belongs, the C++ inspired notation \proc{PropagatorName.Propagate} is used for the pseudo algorithms in this section. +In order to specify to which propagator each of the implementations belongs, the C++ inspired notation \proc{PropagatorName.Propagate} is adapted also for pseudo algorithms. If the methods \proc{PrePropagate} and \proc{PostPropagate} are omitted, it means that they are empty. \\ Whenever convenient, the direct succession of functions \proc{StepT} and \proc{StepU} was replaced by a call to \proc{IntSplit} which effectively corresponds to a series of $M_{split}$ alternating calls to \proc{StepT} and \proc{StepU} (see explanation in \ref{subsec:tuintsplit}) for better accuracy. @@ -15,7 +15,7 @@ \subsection{Hagedorn Propagator} The Hagedorn propagator, shown in algorithm \ref{alg:hagedorn}, is one of the simplest propagators that can be built by exploiting the numerical properties of the Hamiltonian splitting $\opH = \opT + \opU + \opW$ discussed earlier. As pointed out in \cite{FGL_semiclassical_dynamics}, this simple time stepping scheme has many beneficial properties like the preservation of the $L^2$ norm of the wave packet, time reversibility and stability in the classical limit $\eps \rightarrow 0$. Also, in the limit of $\K$ approaching the full basis set, the variational approximation used for the propagation with the non-quadratic part $\opW$ becomes exact. -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Single timestep with Hagedorn propagator} \label{alg:hagedorn} \begin{algorithmic} @@ -61,7 +61,7 @@ \subsection{Semiclassical Propagator} The steps for the semiclassical propagator are shown in algorithm \ref{alg:semiclassical} and further details can be found in the original paper \cite{GH_convsemiclassical}. % -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Single timestep with Semiclassical propagator} \label{alg:semiclassical} \begin{algorithmic} @@ -103,18 +103,18 @@ \subsection{Magnus Propagator} The idea behind the Magnus approximation was extensively studied using the Baker-Campbell-Hausdorff (BCH) formula and rooted trees techniques, more details can be found in \cite{Blanes2006}, \cite{Blanes2000}, \cite{Iserles1999}. \par\medskip % -In order to approximate the solution from equation \ref{math:magnussolution}, one can take only a finite number of terms from this series whereby a truncation error is committed -(additional error sources in this method are the discretization of integrals and the approximation of matrix exponentials). +In order to approximate the solution from equation \ref{math:magnussolution}, one can take only a finite number of terms from the infinite series whereby a truncation error is committed +(additional error sources in the Magnus method are the discretization of integrals and the approximation of matrix exponentials). % -This method of the Magnus Propagator has several beneficial numerical properties that are described in \cite{Iserles1999}. +The Magnus Propagator has several beneficial numerical properties that are described in \cite{Iserles1999}. In particular, for solutions that evolve within a Lie group, the same holds for the approximate numerical solution calculated through a truncated Magnus series. It was also shown in the same paper that the Magnus series can compete with - and in fact may even outplay - classical schemes like Runge-Kutta or Gauss-Legendre. Although the method is not a symplectic scheme in the usual sense, \cite{Iserles1999} has shown that in practical applications it conserves the Hamiltonian energy just as well as symplectic integrators. \par\medskip -Moreover, the numerical stability and good performance of the Magnus propagator are not limited to problems where the solution evolves within a Lie Group, but also apply to various problems when this is not the case. +Moreover, the numerical stability and good performance of the Magnus propagator are not limited to problems where the solution evolves within a Lie Group, but also apply to various problems where this is not the case. Algorithm \ref{alg:magnus} shows the \emph{MG4} method as presented in \cite{Iserles1999} and implemented in C++ in the scope of this project. -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Single timestep with Magnus propagator} \label{alg:magnus} \begin{algorithmic} @@ -168,12 +168,12 @@ \subsection{McLachlan Propagators} \par\medskip % In our case, $\opX = \opH = \opT + \opU + \opW$, the computationally expensive operator $\opW$ takes the role of $\epsilon \opB$, while the remaining Hamiltonian $\opT + \opU$ corresponds to $\opA$. -Our preferred method will therefore be of the form $\opA \opB \opA$ because this minimizes the number of steps with $\opB = \opW$. +Our preferred method will therefore be of the form $\opA \opB \opA$ because such a scheme minimizes the number of steps with $\opB = \opW$. \par\medskip % -The source code of this project contains the C++ implementation of the propagators \emph{McL(4,2)} and \emph{McL(8,4)} that require only two respectively five evaluations of $\opW$ per step. +The source code of the project contains the C++ implementation of the propagators \emph{McL(4,2)} and \emph{McL(8,4)} that require only two respectively five evaluations of $\opW$ per step. -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Single timestep with McL42 propagator} \label{alg:mcl} \begin{algorithmic} @@ -231,9 +231,8 @@ \subsection{McLachlan Propagators} \subsection{Processing Propagators} \label{sub:pre764_propagator} % -The idea of processing propagators is to carry out the time evolution with a Hamiltonian that is slightly perturbed from the original one. -To achieve this, a pre and post processing step is applied. -More formally, it holds that +The idea of processing propagators is to carry out the time evolution with a Hamiltonian that is slightly perturbed from the original one, which can be achieved by applying a pre and post processing step. +The exponential of the Hamiltonian can be re-formulated as % \begin{align} e^{- \Dt \opH (\Dt)} = e^P e^{- \Dt K} e^{-P} @@ -249,8 +248,9 @@ \subsection{Processing Propagators} \par\medskip % Algorithm \ref{alg:pre764} presents the \emph{Pre764} processing method as derived in \cite{Blanes1999}. +The coefficient arrays $\alpha$ and $\beta$ (both of length $k=4$) are used for propagating the wave packet with \proc{StepW} and \proc{IntSplit} in the \proc{Propagate} method, while the coefficient arrays $Y$ and $Z$ (both of length $v=6$) are used in \proc{PrePropagate} and \proc{PostPropagate} to transform the wave packet with the processor. % -\begin{algorithm}[h] +\begin{algorithm}[ht] \caption{Single timestep with Pre764 propagator} \label{alg:pre764} \begin{algorithmic} @@ -259,7 +259,7 @@ \subsection{Processing Propagators} \State \State $M_{pre} = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ \Comment compute number of time steps - \For{$j=0,...,v-1$} \Comment $v=6$ for Pre(7,6,4) + \For{$j=1,...,v$} \Comment $v=6$ for Pre(7,6,4) \State \Call{intSplit}{$-Z_j \Dt, M_{pre}, \{w_T, w_U\}$} \Comment $M_{pre}$ alternating steps with $\opT$ and $\opU$ \State \Call{stepW}{$\upic, -Y_j \Dt$} @@ -273,7 +273,7 @@ \subsection{Processing Propagators} \State \State $M = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ \Comment compute number of time steps - \For{$j=0,...,k-1$} \Comment $k=4$ for Pre(7,6,4) + \For{$j=1,...,k$} \Comment $k=4$ for Pre(7,6,4) \State \Call{stepW}{$\alpha_j \Dt$} \Comment single step with $\opW$ \State \Call{intSplit}{$\beta_j \Dt, M, \{w_T, w_U\}$} @@ -287,7 +287,7 @@ \subsection{Processing Propagators} \State \State $M_{post} = 1+ \left\lfloor \sqrt{\Dt \eps^{-\frac{3}{4}}} \right\rfloor$ \Comment compute number of time steps - \For{$j=v-1,...,0$} \Comment $v=6$ for Pre(7,6,4) + \For{$j=v,...,1$} \Comment $v=6$ for Pre(7,6,4) \State \Call{stepW}{$Y_j \Dt$} \Comment single step with $\opW$ \State \Call{intSplit}{$Z_j \Dt, M_{post}, \{w_T, w_U\}$} diff --git a/LWB/propagators/sections/results.tex b/LWB/propagators/sections/results.tex index 390b6ee..4f713b7 100644 --- a/LWB/propagators/sections/results.tex +++ b/LWB/propagators/sections/results.tex @@ -1,19 +1,18 @@ \section{Results} \label{sec:results} % -This section discusses the outcomes of some numerical experiments that were carried out with the newly developed code. +The newly developed code was used to carry out some numerical experiments which are presented in this section. Aspects that need to be checked are the correctness of the propagators from a quantum mechanical point of view (subsection~\ref{subsec:physical}), convergence of the methods (subsection~\ref{subsec:convergence}) and benchmarking of the compute time (subsection~\ref{subsec:benchmark}). \subsection{Physical Correctness} \label{subsec:physical} {\Huge TODO: Review this part} % -This short subsection analyzes the physical correctness of the implemented propagators. -To do this, a two dimensional wave packet was propagated with each of the presented propagators for a time $T = 10$ with timestep $\Dt = 0.01$. +In order to analyze the physical correctness of the implemented propagators, a two dimensional wave packet was propagated with each of the presented propagators for a time $T = 10$ with timestep $\Dt = 0.01$. All propagators used the \emph{Y4} splitting coefficients for the \proc{IntSplit} method. \par\medskip % -The harmonic potential was used as an example potential as this is easy to verify. +The harmonic potential was used as an example potential as it is easy to verify. The generalized harmonic potential for $D$ dimensions is \begin{align} V(\bvec{x}) = @@ -23,7 +22,7 @@ \subsection{Physical Correctness} where $x_i, \; 1 \le i \le D$ denote the single entries of the vector $\bvec{x}$. \par\medskip % -Figure \ref{fig:energy_Semiclassical} shows the time evolution of energy and the energy drift for the semiclassical propagator. The corresponding energy plots for the remaining propagators can be found in the appendix to this report. \\ +Figure \ref{fig:energy_Semiclassical} shows the time evolution of energy and the energy drift for the semiclassical propagator. The corresponding energy plots for the remaining propagators can be found in the appendix to the report. \\ As is clearly shown by the graphs, the propagators conserve the total energy (apart from very small oscillations of the order $10^{-7}$ for the Hagedorn propagator and $10^{-12}$ for the remaining propagators that use \proc{IntSplit}). Another interesting fact is that the \emph{McL84} operator seems to have a much longer oscillation period than all other propagators. % @@ -55,7 +54,9 @@ \subsection{Effect of Stepsize} Figure \ref{fig:error_analysis} shows the error for different step sizes $\Dt$. It is quite surprising how the Hagedorn Propagator seems to have even better convergence order than all the more sophisticated contestants. Also, the \emph{McL84} propagator does not seem to converge. This is consistent with the observation made earlier in this section that - although satisfying energy conservation - the \emph{McL84} propagator seemed to have a vastly different period of oscillation. - +% +{\Huge TODO: Review this part} +% \begin{figure}[ht] \centering \includegraphics[width=.8\textwidth]{figures/error_analysis.png} @@ -84,7 +85,7 @@ \subsubsection{Comparison of propagators} It comes as quite a surprise, that the Semiclassical propagator is faster at computing the result than the Hagedorn propagator, which at first glance seems to have a simpler structure and involve fewer steps. Also, the Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_k}{W}{\varphi_l}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. % -\begin{table}[h] +\begin{table}[ht] \centering \begin{tabular}{|l | r |} \hline @@ -111,7 +112,7 @@ \subsubsection{Splitting Schemes} \par\medskip % Therefore, a numerical experiment was carried out in order to analyze how much computational time is consumed for coefficient pairs of different sizes. -To do this, the Semiclassical Propagator was used to propagate a two dimensional wave packet in a harmonic potential over a time of $T = 400$ and with stepsize $\Dt = 0.01$. +In order to create a reference solution, the Semiclassical Propagator was used to propagate a two dimensional wave packet in a harmonic potential over a time of $T = 400$ and with stepsize $\Dt = 0.01$. The same simulation was carried out for coefficient pairs of various different sizes and were run in Python as well as C++. The results are listed in table \ref{tab:benchmarksplit_t} and plotted in figure \ref{fig:benchmarksplit_f}. \par\medskip @@ -126,7 +127,7 @@ \subsubsection{Splitting Schemes} This is a very encouraging result as it means that high order splitting coefficients come at almost no extra cost. \par\medskip % -\begin{table}[h] +\begin{table}[ht] \centering \begin{tabular}{|l | r | r | r | r |} \hline diff --git a/references/propagators.bib b/references/propagators.bib index 3cc3547..89f71d8 100644 --- a/references/propagators.bib +++ b/references/propagators.bib @@ -1,7 +1,5 @@ @article{Iserles1999, - author="Iserles, A. - and Marthinsen, A. - and N{\o}rsett, S. P.", + author="A. Iserles and A. Marthinsen and S. P. N{\o}rsett", title="On the Implementation of the Method of Magnus Series for Linear Differential Equations", journal="BIT Numerical Mathematics", year="1999", @@ -14,7 +12,7 @@ @article{Iserles1999 } @article{Magnus1954, - author = {Magnus, Wilhelm}, + author = {Wilhelm Magnus}, title = {On the exponential solution of differential equations for a linear operator}, journal = {Communications on Pure and Applied Mathematics}, volume = {7}, @@ -46,7 +44,7 @@ @article{Blanes2006 } @Article{Blanes2000, - author="S. Blanes, F. Casas and J. Ros", + author="S. Blanes and F. Casas and J. Ros", title="Improved High Order Integrators Based on the Magnus Expansion", journal="BIT Numerical Mathematics", year="2000", @@ -86,10 +84,38 @@ @Article{McLachlan1995 @book{Gamma1995, - author = {Gamma, Erich and Helm, Richard and Johnson, Ralph and Vlissides, John}, + author = {Erich Gamma and Richard Helm and Ralph Johnson and John Vlissides}, title = {Design Patterns: Elements of Reusable Object-oriented Software}, year = {1995}, isbn = {0-201-63361-2}, publisher = {Addison-Wesley Longman Publishing Co., Inc.}, address = {Boston, MA, USA}, } + +@Article{LWB_wavepackets, + author="Lionel Miserez", + title="Porting WaveBlocksND Matrix Potential Functionality to C++", + journal="Libwaveblocks", + year="2015", + url="https://github.com/raoulbq/TeX" +} + + +@Article{LWB_innerproducts, + author="Benedek Vartok", + title="Implementation of WaveBlocksND’s Quadrature and Inner Products in C++", + journal="Libwaveblocks", + year="2015", + url="https://github.com/raoulbq/TeX" +} + + +@book{GeoNumInt, + author = {Ernst Hairer and Christian Lubich and Gerhard Wanner}, + title = {Geometric Numerical Integration}, + year = {2002}, + isbn = {3-540-30663-3}, + publisher = {Springer}, + address = {Berlin, Germany}, +} + From ed973d7b808b714b8eb754ff90783b77bd095d0a Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Mon, 13 Feb 2017 05:20:58 +0100 Subject: [PATCH 22/28] Corrections in Result section --- .../figures/coefficient_analysis.png | Bin 39896 -> 42838 bytes .../figures/dimension_analysis.png | Bin 40220 -> 37326 bytes LWB/propagators/figures/error_analysis.png | Bin 68026 -> 43654 bytes LWB/propagators/figures/error_analysis_lt.png | Bin 0 -> 88378 bytes LWB/propagators/figures/error_analysis_y4.png | Bin 0 -> 92563 bytes .../figures/harmonic_drift_Hagedorn.png | Bin 0 -> 50210 bytes .../figures/harmonic_drift_MG4.png | Bin 0 -> 56800 bytes .../figures/harmonic_drift_McL42.png | Bin 0 -> 55639 bytes .../figures/harmonic_drift_McL84.png | Bin 0 -> 58208 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a/LWB/propagators/figures/energy_McL84.png b/LWB/propagators/figures/rubbish/energy_McL84.png similarity index 100% rename from LWB/propagators/figures/energy_McL84.png rename to LWB/propagators/figures/rubbish/energy_McL84.png diff --git a/LWB/propagators/figures/energy_Pre764.png b/LWB/propagators/figures/rubbish/energy_Pre764.png similarity index 100% rename from LWB/propagators/figures/energy_Pre764.png rename to LWB/propagators/figures/rubbish/energy_Pre764.png diff --git a/LWB/propagators/figures/energy_Semiclassical.png b/LWB/propagators/figures/rubbish/energy_Semiclassical.png similarity index 100% rename from LWB/propagators/figures/energy_Semiclassical.png rename to LWB/propagators/figures/rubbish/energy_Semiclassical.png diff --git a/LWB/propagators/main.tex b/LWB/propagators/main.tex index 56e0414..dc07f47 100644 --- a/LWB/propagators/main.tex +++ b/LWB/propagators/main.tex @@ -26,48 +26,6 @@ \clearemptydoublepage \input{sections/appendix.tex} -\begin{comment} -\subsection{Henon-Heiles Potential} -% -\begin{align} - V(\bvec{x}) = - \frac{1}{2} \sum_{j=1}^N \sigma_j x_j^2 - + \sigma_* \sum_{j=1}^{N-1} \left( x_j x_{j+1}^2 - \frac{1}{3} x_j^3 + \frac{\sigma_*}{16} \left( x_j^2 + x_{j+1}^2 \right)^2 \right) -\end{align} -% -with harmonic part coefficients $\sigma_j = 1$ and mixing coefficient $\sigma_* = 0.2$. -Initial conditions for 2D: (GENERALIZE) -?Normed gaussian wave packet? -with $\bvec{q}=(1.8,0)^T$, $\bvec{p}=(0,1.2)^T$, $\bmat{Q} = \sqrt{2} \cdot \text{diag}(\sqrt{0.56},\sqrt{0.24})$, $\bmat{P} = \im \bmat{Q}^{-1}$ - -\paragraph{Jacobian} -\begin{align} - \del_k V(\bvec{x}) - = \; &\delta_{k,j} \sigma_j x_j \\ - + \; &\delta_{k,j} \sigma_* \left( x_{j+1}^2 - x_j^2 + \frac{\sigma_*}{4} x_j \left( x_j^2 + x_{j+1}^2 \right) \right) \\ - + \; &\delta_{k,j+1} \sigma_* \left( 2 x_j x_{j+1} + \frac{\sigma_*}{4} x_{j+1} \left( x_j^2 + x_{j+1}^2 \right) \right) \\ - = \; &\delta_{k,j} \left( \sigma_j x_j + \sigma_* \left( x_{j+1}^2 - x_j^2 + \frac{\sigma_*}{4} x_j \left( x_j^2 + x_{j+1}^2 \right) \right) \right) \\ - + \; &\delta_{k,j+1} \sigma_* \left( 2 x_j x_{j+1} + \frac{\sigma_*}{4} x_{j+1} \left( x_j^2 + x_{j+1}^2 \right) \right) -\end{align} - -\paragraph{Hessian} -\begin{align} - \del_k \del_l V(\bvec{x}) - = \; &\delta_{l,j} \left[ \delta_{k,j} \left( \sigma_j + \sigma_* \left( - 2 x_j + \frac{\sigma_*}{4} \left( x_j^2 + x_{j+1}^2 \right) + \frac{\sigma_*}{2} x_j^2 \right) \right) - + \; \delta_{k,j+1} \sigma_* \left( 2 x_{j+1} + \frac{\sigma_*}{2} x_{j+1} x_j \right) \right] \\ - + \; &\delta_{l,j+1} \left[ \delta_{k,j} \left( \sigma_* \left( 2 x_{j+1} + \frac{\sigma_*}{2} x_j x_{j+1} \right) \right) - + \; \delta_{k,j+1} \sigma_* \left( 2 x_j + \frac{\sigma_*}{4} \left( x_j^2 + x_{j+1}^2 \right) + \frac{\sigma_*}{2} x_{j+1}^2 \right) \right] \\ - = \; &\delta_{l,j} \left[ \delta_{k,j} \left( \sigma_j + \sigma_* \left( - 2 x_j + \frac{\sigma_*}{4} \left( 3 x_j^2 + x_{j+1}^2 \right) \right) \right) - + \; \delta_{k,j+1} \sigma_* \left( 2 x_{j+1} + \frac{\sigma_*}{2} x_{j+1} x_j \right) \right] \\ - \; % TODO: - + \; &\delta_{l,j+1} \left[ \delta_{k,j} \left( \sigma_* \left( 2 x_{j+1} + \frac{\sigma_*}{2} x_j x_{j+1} \right) \right) - + \; \delta_{k,j+1} \sigma_* \left( 2 x_j + \frac{\sigma_*}{4} \left( x_j^2 + x_{j+1}^2 \right) + \frac{\sigma_*}{2} x_{j+1}^2 \right) \right] -\end{align} -\end{comment} - - - - \clearemptydoublepage \bibliographystyle{plain} diff --git a/LWB/propagators/sections/appendix.tex b/LWB/propagators/sections/appendix.tex index f697c35..92849ef 100644 --- a/LWB/propagators/sections/appendix.tex +++ b/LWB/propagators/sections/appendix.tex @@ -2,7 +2,9 @@ \section{Energy Evolution and Drift Plots} + \begin{figure}[ht] + \centering \begin{minipage}[c]{\textwidth} \begin{center} \large Hagedorn Propagator \\[1mm] @@ -10,13 +12,17 @@ \section{Energy Evolution and Drift Plots} \vspace{4mm} \end{center} \end{minipage} - \includegraphics[width=.45\textwidth]{figures/energy_Hagedorn.png} - \includegraphics[width=.45\textwidth]{figures/drift_Hagedorn.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Hagedorn Propagator with \emph{Y4} splitting} + \includegraphics[width=.45\textwidth]{figures/harmonic_energies_Hagedorn.png} + \includegraphics[width=.45\textwidth]{figures/harmonic_drift_Hagedorn.png} \\ + \includegraphics[width=.45\textwidth]{figures/henon_energies_Hagedorn.png} + \includegraphics[width=.45\textwidth]{figures/henon_drift_Hagedorn.png} + \caption{Energy evolution and drift for a 2D wave packet propagated with the Hagedorn propagator in a harmonic potential (top) and a Henon-Heiles potential (bottom). + ($T = 10$, $\Dt = 0.01$, $| \K | = 4$, \emph{LT} splitting)} \label{fig:energy_Hagedorn} \end{figure} \begin{figure}[ht] + \centering \begin{minipage}[c]{\textwidth} \begin{center} \large MG4 Propagator \\[1mm] @@ -24,50 +30,65 @@ \section{Energy Evolution and Drift Plots} \vspace{4mm} \end{center} \end{minipage} - \includegraphics[width=.45\textwidth]{figures/energy_MG4.png} - \includegraphics[width=.45\textwidth]{figures/drift_MG4.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the MG4 Propagator with \emph{Y4} splitting} + \includegraphics[width=.45\textwidth]{figures/harmonic_energies_MG4.png} + \includegraphics[width=.45\textwidth]{figures/harmonic_drift_MG4.png} \\ + \includegraphics[width=.45\textwidth]{figures/henon_energies_MG4.png} + \includegraphics[width=.45\textwidth]{figures/henon_drift_MG4.png} + \caption{Energy evolution and drift for a 2D wave packet propagated with the MG4 propagator in a harmonic potential (top) and a Henon-Heiles potential (bottom). + ($T = 10$, $\Dt = 0.01$, $| \K | = 4$, \emph{LT} splitting)} \label{fig:energy_MG4} \end{figure} \begin{figure}[ht] + \centering \begin{minipage}[c]{\textwidth} \begin{center} - \large Pre764 Propagator \\[1mm] + \large McL42 Propagator \\[1mm] \normalsize Energy Evolution and Drift \vspace{4mm} \end{center} \end{minipage} - \includegraphics[width=.45\textwidth]{figures/energy_Pre764.png} - \includegraphics[width=.45\textwidth]{figures/drift_Pre764.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Pre764 Propagator with \emph{Y4} splitting} - \label{fig:energy_Pre764} + \includegraphics[width=.45\textwidth]{figures/harmonic_energies_McL42.png} + \includegraphics[width=.45\textwidth]{figures/harmonic_drift_McL42.png} \\ + \includegraphics[width=.45\textwidth]{figures/henon_energies_McL42.png} + \includegraphics[width=.45\textwidth]{figures/henon_drift_McL42.png} + \caption{Energy evolution and drift for a 2D wave packet propagated with the McL42 propagator in a harmonic potential (top) and a Henon-Heiles potential (bottom). + ($T = 10$, $\Dt = 0.01$, $| \K | = 4$, \emph{LT} splitting)} + \label{fig:energy_McL42} \end{figure} \begin{figure}[ht] + \centering \begin{minipage}[c]{\textwidth} \begin{center} - \large McL42 Propagator \\[1mm] + \large McL84 Propagator \\[1mm] \normalsize Energy Evolution and Drift \vspace{4mm} \end{center} \end{minipage} - \includegraphics[width=.45\textwidth]{figures/energy_McL42.png} - \includegraphics[width=.45\textwidth]{figures/drift_McL42.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the McL42 Propagator with \emph{Y4} splitting} - \label{fig:energy_McL42} + \includegraphics[width=.45\textwidth]{figures/harmonic_energies_McL84.png} + \includegraphics[width=.45\textwidth]{figures/harmonic_drift_McL84.png} \\ + \includegraphics[width=.45\textwidth]{figures/henon_energies_McL84.png} + \includegraphics[width=.45\textwidth]{figures/henon_drift_McL84.png} + \caption{Energy evolution and drift for a 2D wave packet propagated with the McL84 propagator in a harmonic potential (top) and a Henon-Heiles potential (bottom). + ($T = 10$, $\Dt = 0.01$, $| \K | = 4$, \emph{LT} splitting)} + \label{fig:energy_McL84} \end{figure} \begin{figure}[ht] + \centering \begin{minipage}[c]{\textwidth} \begin{center} - \large McL84 Propagator \\[1mm] + \large Pre764 Propagator \\[1mm] \normalsize Energy Evolution and Drift \vspace{4mm} \end{center} \end{minipage} - \includegraphics[width=.45\textwidth]{figures/energy_McL84.png} - \includegraphics[width=.45\textwidth]{figures/drift_McL84.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the McL84 Propagator with \emph{Y4} splitting} - \label{fig:energy_McL84} + \includegraphics[width=.45\textwidth]{figures/harmonic_energies_Pre764.png} + \includegraphics[width=.45\textwidth]{figures/harmonic_drift_Pre764.png} \\ + \includegraphics[width=.45\textwidth]{figures/henon_energies_Pre764.png} + \includegraphics[width=.45\textwidth]{figures/henon_drift_Pre764.png} + \caption{Energy evolution and drift for a 2D wave packet propagated with the Pre764 propagator in a harmonic potential (top) and a Henon-Heiles potential (bottom). + ($T = 10$, $\Dt = 0.01$, $| \K | = 4$, \emph{LT} splitting)} + \label{fig:energy_Pre764} \end{figure} diff --git a/LWB/propagators/sections/results.tex b/LWB/propagators/sections/results.tex index 4f713b7..1bbcefd 100644 --- a/LWB/propagators/sections/results.tex +++ b/LWB/propagators/sections/results.tex @@ -3,28 +3,43 @@ \section{Results} % The newly developed code was used to carry out some numerical experiments which are presented in this section. Aspects that need to be checked are the correctness of the propagators from a quantum mechanical point of view (subsection~\ref{subsec:physical}), convergence of the methods (subsection~\ref{subsec:convergence}) and benchmarking of the compute time (subsection~\ref{subsec:benchmark}). - -\subsection{Physical Correctness} -\label{subsec:physical} -{\Huge TODO: Review this part} +\par\medskip % -In order to analyze the physical correctness of the implemented propagators, a two dimensional wave packet was propagated with each of the presented propagators for a time $T = 10$ with timestep $\Dt = 0.01$. -All propagators used the \emph{Y4} splitting coefficients for the \proc{IntSplit} method. +The potentials used for the numerical experiments were the simple harmonic potential and the Henon-Heiles potential. +In both cases, the basis size $| \K | = 4$ was used. \par\medskip % -The harmonic potential was used as an example potential as it is easy to verify. -The generalized harmonic potential for $D$ dimensions is +The generalized harmonic potential \begin{align} - V(\bvec{x}) = - \frac{1}{2} \sum_{i=1}^D x_i^2 + \label{math:harmonic} + V(\bvec{x}) = \frac{1}{2} \sum_{i=1}^D x_i^2 \end{align} % -where $x_i, \; 1 \le i \le D$ denote the single entries of the vector $\bvec{x}$. +was used with initial values $\bvec{q} = (1,\dots,1)^T$ and $\bvec{q} = (1,\dots,1,-1,\dots,-1)$ that were normalized to a $L_2$ norm of $0.5$ and $0.8$ respectively. +$\bmat{Q}$ was the identity matrix and $\bmat{P} = \im \bmat{Q}^{-1}$. \\ \par\medskip % +The two dimensional Henon-Heiles potential is given by +% +\begin{align} + \label{math:henonheiles} + V(\bvec{x}) = \frac{1}{2} \left( x_1^2 + x_2^2 \right) + \sigma_* \left( x_1^2 x_2 - \frac{1}{3} x_2^3 \right) +\end{align} +% +where $\sigma_*$ is a mixing coefficient that was chosen to be equal to $0.2$. +The initial values for this potential were $\bvec{q}=(1.8,0)^T$, $ \bvec{p}=(0,1.2)^T$, $\bmat{Q} = \sqrt{2} \cdot \text{diag}( \sqrt{0.56}, \sqrt{0.24} )$ and $\bmat{P} = \im \bmat{Q}^{-1}$) + + +\subsection{Physical Correctness} +\label{subsec:physical} +% +In order to analyze the physical correctness of the implemented propagators, a two dimensional wave packet was propagated in the Harmonic potential and in the Henon-Heiles potential presented above with each of the presented propagators for a time $T = 10$ with timestep $\Dt = 0.01$. +All propagators used the \emph{LT} splitting coefficients for the \proc{IntSplit} method. +\par\medskip +% +% Figure \ref{fig:energy_Semiclassical} shows the time evolution of energy and the energy drift for the semiclassical propagator. The corresponding energy plots for the remaining propagators can be found in the appendix to the report. \\ -As is clearly shown by the graphs, the propagators conserve the total energy (apart from very small oscillations of the order $10^{-7}$ for the Hagedorn propagator and $10^{-12}$ for the remaining propagators that use \proc{IntSplit}). -Another interesting fact is that the \emph{McL84} operator seems to have a much longer oscillation period than all other propagators. +As is clearly shown by the graphs, the propagators conserve the total energy (apart from very small oscillations). % \begin{figure}[ht] \centering @@ -35,33 +50,33 @@ \subsection{Physical Correctness} \vspace{4mm} \end{center} \end{minipage} - \includegraphics[width=.45\textwidth]{figures/energy_Semiclassical.png} - \includegraphics[width=.45\textwidth]{figures/drift_Semiclassical.png} - \caption{Energy Evolution and Drift for a 2D Wave Packet evolved for $T = 10$ with step $\Dt = 0.001$ in a harmonic potential using the Semiclassical Propagator with \emph{Y4} splitting} + \includegraphics[width=.45\textwidth]{figures/harmonic_energies_Semiclassical.png} + \includegraphics[width=.45\textwidth]{figures/harmonic_drift_Semiclassical.png} \\ + \includegraphics[width=.45\textwidth]{figures/henon_energies_Semiclassical.png} + \includegraphics[width=.45\textwidth]{figures/henon_drift_Semiclassical.png} + \caption{Energy evolution and drift for a 2D wave packet propagated with the Semiclassical propagator in a harmonic potential (top) and a Henon-Heiles potential (bottom). + ($T = 10$, $\Dt = 0.01$, $| \K | = 4$, \emph{LT} splitting)} \label{fig:energy_Semiclassical} \end{figure} \subsection{Effect of Stepsize} \label{subsec:convergence} -{\Huge TODO: Review this part} % -A convergence analysis was carried out in which the error of each propagator was recorded while reducing the stepsize. -A Semiclassical propagator with \emph{KL10} splitting coefficients and step size 0.001 was used to propagate the wave packet to a final time $T = 10$ and create a reference solution. +A convergence analysis was carried out in which the error of each propagator was recorded while reducing the stepsize $\Dt$. +A Semiclassical propagator with \emph{KL10} splitting coefficients and step size $0.001$ was used to propagate the wave packet to a final time $T = 10$ and create a reference solution. \par\medskip % The error between two wave packets was computed by evaluating them on a grid with 1000 grid points and taking the $L_2$ norm of the differences. Figure \ref{fig:error_analysis} shows the error for different step sizes $\Dt$. -It is quite surprising how the Hagedorn Propagator seems to have even better convergence order than all the more sophisticated contestants. -Also, the \emph{McL84} propagator does not seem to converge. This is consistent with the observation made earlier in this section that - although satisfying energy conservation - the \emph{McL84} propagator seemed to have a vastly different period of oscillation. -% -{\Huge TODO: Review this part} +It is quite surprising how the Hagedorn Propagator has higher precision than the other schemes in the case of \emph{LT} splitting. +Only for higher order splitting coefficients like \emph{Y4} the advantages of \proc{IntSplit} start paying off and the other propagators show their advantages over the simplistic Hagedorn Propagator. % \begin{figure}[ht] \centering - \includegraphics[width=.8\textwidth]{figures/error_analysis.png} - \caption{Convergence of the wavefunctions towards the reference solution. The $L_2$ norm was measured by projecting the wave function on a grid with 1000 nodes in the range $[-1,1]$ and taking the differences of these values. According to this experiment, the Hagedorn propagator seems to have higher convergence order than its contestants, despite its simplicity. - This observation could be an interesting starting point for further experiments.} + \includegraphics[width=.8\textwidth]{figures/error_analysis_lt.png} \\ + \includegraphics[width=.8\textwidth]{figures/error_analysis_y4.png} + \caption{Convergence of the various propagators towards the reference solution for \emph{LT} splitting (top) and \emph{Y4} splitting (bottom). The $L_2$ norm was measured by projecting the wave function on a grid with $1000$ nodes in the range $[-1,1]$ and taking the differences of these values.} \label{fig:error_analysis} \end{figure} @@ -80,10 +95,9 @@ \subsection{Benchmark} \subsubsection{Comparison of propagators} % -A quick comparison of timings for the different propagators is shown in table \ref{tab:speedup}. -The same setup as for the energy analysis was used (2D wave packet, $T = 10$, step size $\Dt = 0.001$, \emph{Y4} time splitting coefficients). -It comes as quite a surprise, that the Semiclassical propagator is faster at computing the result than the Hagedorn propagator, which at first glance seems to have a simpler structure and involve fewer steps. -Also, the Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_k}{W}{\varphi_l}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. +A quick comparison of timings for the different propagators is shown in table \ref{tab:speedup}, the same setup as for the energy analysis was used (2D wave packet, $T = 10$, step size $\Dt = 0.01$, \emph{LT} time splitting coefficients). +The Magnus Propagator \emph{MG4} is cheaper than expected, as it involves two evaluations of $\matrixel{\varphi_k}{W}{\varphi_l}$ per time step but is only slightly slower than the Semiclassical operator with one evaluation of the inner product. +The slowest propagator is \emph{McL84} which involves five propagations with $\opW$ per step. % \begin{table}[ht] \centering @@ -92,15 +106,15 @@ \subsubsection{Comparison of propagators} \multicolumn{1}{|c}{\textbf{Propagator}} & \multicolumn{1}{|c|}{\textbf{Timing [s]}} \\ \hline - Hagedorn & 12.54 \\ - Semiclassical & 12.51 \\ - MG4 & 13.82 \\ - Pre764 & 17.10 \\ - McL42 & 22.82 \\ - McL84 & 23.52 \\ + Hagedorn & 1.06 \\ + Semiclassical & 1.14 \\ + MG4 & 1.23 \\ + McL42 & 1.39 \\ + McL84 & 2.08 \\ + Pre764 & 1.63 \\ \hline \end{tabular} - \caption{Runtimes for propagating a 2D wave packet to a time $T = 10$ with timestep $\Dt = 0.001$ with different propagators in C++. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $0.1\%$.} + \caption{Runtimes for propagating a 2D wave packet to a time $T = 10$ with timestep $\Dt = 0.01$ with different propagators in C++. All the timings in the table were measured by taking the average of 10 independent runs. The standard deviation of these measurements was below $1\%$.} \label{tab:speedup} \end{table} From ae21fd3132effc62c5831129c5e2bf3c75aa35d3 Mon Sep 17 00:00:00 2001 From: Matthias Untergassmair Date: Mon, 20 Feb 2017 03:25:40 +0100 Subject: [PATCH 23/28] intermediate commit --- .../BA_WaveBlocks_Propagators_v2.pdf | Bin 0 -> 1919882 bytes .../BA_WaveBlocks_Propagators_v3.pdf | Bin 0 -> 1920072 bytes LWB/propagators/declaration-orig.pdf | Bin 0 -> 416132 bytes .../figures/coefficient_analysis.png | Bin 42838 -> 43563 bytes LWB/propagators/figures/coefs.png | Bin 0 -> 41948 bytes .../figures/convergence_harmonic_1D_lt.png | Bin 0 -> 66632 bytes .../figures/convergence_harmonic_1D_y4.png | Bin 0 -> 66877 bytes .../figures/convergence_legend.png | Bin 0 -> 18737 bytes .../figures/convergence_morse_1D_lt.png | Bin 0 -> 60280 bytes .../figures/convergence_morse_1D_y4.png | Bin 0 -> 67060 bytes .../figures/convergence_torsional_1D_lt.png | Bin 0 -> 65856 bytes 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