Speedup curve
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# 1
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# 8
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#16
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#32
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#64
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# TZ : AMD EPYC 7402 24-Core Processor
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#1
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#2
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#4
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#6
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#12
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#24
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#32 123.718727827072
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#48 121.613038063049
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# DZ: ARM Q80
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1 740.99828964984044 0.109375
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2 368.10031103389338 0.21875
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4 183.47006468195468 0.4375
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8 92.296218489762396 0.875
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16 47.492009534966201 1.75
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24 32.694960118737072 2.625
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32 25.408835886977613 3.5
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40 21.404467605985701 4.375
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48 19.549200831912458 5.25
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56 18.556575596332550 6.125 # Max
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64 18.981703117024153 7.0
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68 21.941064534708858 7.4375
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72 27.031634503975511 7.87
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76 32.015603828709573 8.3125
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80 39.939096361864358 8.75
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#!/usr/bin/gnuplot -persist
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#
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#
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# G N U P L O T
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# Version 5.4 patchlevel 2 last modified 2021-06-01
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#
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# Copyright (C) 1986-1993, 1998, 2004, 2007-2021
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# Thomas Williams, Colin Kelley and many others
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#
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# gnuplot home: http://www.gnuplot.info
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# faq, bugs, etc: type "help FAQ"
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# immediate help: type "help" (plot window: hit 'h')
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GNUTERM = "qt"
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I = {0.0, 1.0}
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VoxelDistance = 0.0
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## Last datafile plotted: "scaling.dat"
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plot 'scaling.dat' u 1:(740.99828964984044/$2) w lp notitle, x title "Ideal"
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# EOF
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@ -2,10 +2,11 @@
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%\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace}
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\usepackage{graphicx,xcolor,physics,siunitx,xspace}
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\usepackage[version=4]{mhchem}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{algorithm2e}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks,
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@ -71,23 +72,23 @@
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\Vienna}{Vienna}
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\begin{document}
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\begin{document}
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\title{Stochastically accelerated perturbative triples correction in coupled cluster calculations}
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% Alphabetic order
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\author{Yann \surname{Damour}}
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\affiliation{\LCPQ}
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\affiliation{\LCPQ}
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\author{Alejandro \surname{Gallo}}
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\affiliation{\Vienna}
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\affiliation{\Vienna}
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\author{Andreas \surname{Irmler}}
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\affiliation{\Vienna}
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\affiliation{\Vienna}
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\author{Andreas \surname{Gr\"uneis}}
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\affiliation{\Vienna}
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\affiliation{\Vienna}
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\author{Anthony \surname{Scemama}}
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\email{scemama@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\email{scemama@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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We introduce a novel algorithm that leverages stochastic sampling
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techniques to approximate perturbative triples correction in the
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@ -109,7 +110,7 @@ computations, enabling investigations of complex molecular systems
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that were previously computationally prohibitive.
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\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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@ -161,7 +162,7 @@ Each individual term is expressed as
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\end{equation}
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and depends on the canonical orbital energies $\epsilon$, and on the tensors $W$ and $V$:
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\begin{align}
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W_{ijk}^{abc} & = P_{ijk}^{abc} \qty( \sum_d^{\text{virt}} \qty(bd|ai) t_{kj}^{cd} -
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W_{ijk}^{abc} & = P_{ijk}^{abc} \qty( \sum_d^{\text{virt}} \qty(bd|ai) t_{kj}^{cd} -
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\sum_l^{\text{occ}} \qty(ck|jl) t_{ab}^{il}) \\
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V_{ijk}^{abc} & = W_{ijk}^{abc} + \qty(bj|ck) t_i^a + \qty(ai|ck) t_j^b + \qty(ai|bj) t_k^c
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\end{align}
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@ -204,7 +205,7 @@ In the algorithm proposed by Rendell\cite{rendell_1991}, for each given triplet
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\section{Implementation Details}
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\label{sec:implementation}
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The algorithm was implemented in the \textsc{Quantum Package} software.
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The algorithm presented in Algorithm~\cite{alg:stoch} was implemented in the \textsc{Quantum Package} software.
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\cite{garniron_2019}
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The stochastic algorithm is implemented using OpenMP tasks, where each task
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consists in the computation of a single component $E^{abc}$.
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@ -214,15 +215,79 @@ for printing or for exiting when the statistical error gets below a given thresh
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The number of samples $N^{abc}$ of each triplet $(abc)$ is initialized to $-1$, to identify
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the contributions that have not been already computed.
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An outer \emph{for} loop runs over the maximum number of iteration, equal to
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$N_{abc}$, the number of different triplets $(abc)$.
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the number of different triplets $N_{\text{triplets}}$.
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Within a loop iteration, the index of the first non-computed triplet $(abc)$ is identified, and the task associated with its computation is sent to the task queue.
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As this triplet has never been drawn, $N^{abc}$ is set to zero.
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Then, a triplet $(abc)$ is drawn randomly.
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If the $E^{abc}$ has not been computed (identified by $N^{abc}=-1$), the number of samples is set to zero and the task for the computation of this contribution is enqueued.
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If the $E^{abc}$ has not been computed (identified by $N^{abc}=-1$), the number of samples is set to zero and the task for the computation of this contribution is enqueued.
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In any case, $N^{abc}$ is then incremented.
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\begin{algorithm}[H]
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\caption{\label{alg:stoch} Pseudo-code for the computation of the perturbative triples correction implemented in Quantum Package. $i_\text{min}$ denotes the first non-computed triplet, $w_{\text{accu}}$ contains the cumulative probability density, $\text{Search}(A, x)$ searches for $x$ in array $A$, $\text{First}(i)$ and $\text{Last}(i)$ return the first last indices belonging to bucket $i$.}
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$i_{\text{min}} \leftarrow 1$ ;
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$N^{abc}[1,\dots,N_{\text{triplets}}] \leftarrow [-1, -1, \dots]$ \;
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$t_0 \leftarrow \text{WallClockTime}()$ \;
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\For {$i_{\text{iter}}=1,\ldots,N_{\text{triplets}}$}
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{
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\tcc{Deterministic computation}
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\While {$N^{abc}[i_{\text{min}}] > -1$ and $i_{\text{min}} \le N_{\text{triplets}}$}
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{
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$i_{\text{min}} \leftarrow i_{\text{min}}+1$ \;
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}
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\If{$i_{\text{min}} \le N_{\text{triplets}}$}
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{
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Send OpenMP task \{ {
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$E[i_{\text{min}}] \gets \text{Compute}(i_{\text{min}})$\;
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\} }\;
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}
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\tcc{Stochastic computation}
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$\eta \gets \text{RandomNumber}()$ \;
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\For {$i_\text{bucket} = 1,\ldots, N_{\text{buckets}}$}
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{
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\If{$i_{\text{min}} \le \text{Last}(i_\text{bucket})$}
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{
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$i_\eta \gets \text{Search}(w_{\text{accu}}, \frac{\eta + i_\text{bucket}-1}{N_{\text{buckets}}})+1$ \;
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\If{$N^{abc}[i_{\eta}] = -1$}
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{
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$N^{abc}[i_{\eta}] \gets 0$ \;
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Send OpenMP task \{ {
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$E[i_{\eta}] \gets \text{Compute}(i_{\eta})$\;
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\} }\;
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}
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$N^{abc}[i_{\eta}] \gets N^{abc}[i_\eta]+1$ \;
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}
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}
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\tcc{Compute the mean and error every second}
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$t_1 \gets \text{WallClockTime}()$ \;
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\If {$t_1 - t_0 > 1$ or $i_{\text{min}} \ge N_{\text{triplets}}$}
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{
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$i_\text{bucket} = 0$ \;
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\While {$i_{\text{bucket}} < N_\text{buckets}$ and
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$i_{\text{min}} > \text{Last}(i_\text{bucket}+1)$}
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{
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$i_\text{bucket} \gets i_\text{bucket} + 1$\;
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}
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$\mathcal{N} \gets \frac{ \sum_{i=\text{First}(i_\text{bucket}+1)}^{N_\text{triplets}} \max(N^{abc}[i],0)}
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{ 1 - \sum_{i=1}^{\text{Last}(i_\text{bucket})} P[i] }$ \;
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$E_{d} \gets \sum_{i=1}^{\text{Last}(i_\text{bucket})} E^{abc}[i]$ \;
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$E_{s} \gets 1/\mathcal{N}\, \sum_{i=\text{First}(i_\text{bucket}+1)}^{N_\text{triplets}}
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\max(N^{abc}[i],0)\, E^{abc}[i]/P[i]$ \;
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$E_{s^2} \gets 1/\mathcal{N}\, \sum_{i=\text{First}(i_\text{bucket}+1)}^{N_\text{triplets}}
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\max(N^{abc}[i],0)\, \qty(E^{abc}[i]/P[i])^2$ \;
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$E \gets E_{d} + E_{s}$ \;
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$\Delta E \gets \sqrt{ \qty(E_{s^2} - {E_{s}}^2) / \qty(\mathcal{N}-1) }$ \;
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\If{$\Delta E < \epsilon$}
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{
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Exit outermost loop \;
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}
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}
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}
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\end{algorithm}
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%a. Description of the computational framework and software used
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%b. Discussion of any specific optimizations or parallelization techniques employed
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@ -252,19 +317,19 @@ The calculations were performed on an Intel Xeon Gold 6130 dual socket server (3
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\begin{figure}
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\includegraphics[width=\columnwidth]{benzene_tz.pdf}
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\includegraphics[width=\columnwidth]{benzene_qz.pdf}
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\caption{\label{fig:benzene} Energy convergence of benzene plotted against the program execution time, showing comparisons between the cc-pVTZ (upper curve) and cc-pVQZ (lower curve) basis sets. The blue lines indicate the exact CCSD(T) energies.}
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\caption{\label{fig:benzene} Energy convergence of benzene plotted against the program execution time, showing comparisons between the cc-pVTZ (upper curve) and cc-pVQZ (lower curve) basis sets. The blue lines indicate the exact CCSD(T) energies.}
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\end{figure}
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\begin{figure}
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\includegraphics[width=\columnwidth]{benzene_err.pdf}
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\caption{\label{fig:benzene_err} Convergence of the statistical error of the perturbative triples contribution in benzene as a function of the percentage of computed contributions, for both cc-pVTZ and cc-pVQZ basis sets.}
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\caption{\label{fig:benzene_err} Convergence of the statistical error of the perturbative triples contribution in benzene as a function of the percentage of computed contributions, for both cc-pVTZ and cc-pVQZ basis sets.}
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\end{figure}
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Figure~\ref{fig:benzene} shows the convergence of the CCSD(T) energy as a function of the program execution time using the two basis sets.
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Notably, the exact CCSD(T) energy always falls within $2\sigma$, affirming the reliability of the statistical error.
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Figure~\ref{fig:benzene_err} displays the statistical error as a function of the percentage of computed contributions.
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Noteworthy in both figures are the curve discontinuities, attributable to readjustments in the separation between the deterministic and stochastic components of the calculation.
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These updates lead to revised estimates and a diminution in statistical error.
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These updates lead to revised estimates and a diminution in statistical error.
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Achieving chemical accuracy, defined as \SI{1.6}{\milli\hartree}, necessitates less than 1\% of the total contributions in both basis sets.
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Attaining a \SI{0.1}{\milli\hartree} precision level requires computation of 32\% and 15\% of the contributions for cc-pVTZ and cc-pVQZ, respectively.
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@ -303,6 +368,9 @@ The vibrational frequency and equilibrium distance estimated using this data, $\
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Figure \ref{fig:cucl} illustrates the potential energy surface of \ce{CuCl}, displaying both the exact CCSD(T) energies and those estimated via the semi-stochastic method.
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\subsection{Parallel efficiency}
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\subsection{Performance analysis}
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The primary bottleneck of our proposed algorithm lies in the generation of the sub-tensor $W^{abc}$ for each $(a,b,c)$ triplet, as discussed in Section~\ref{sec:theory}.
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|
@ -318,10 +386,10 @@ For this, we utilized the ArmPL library for BLAS operations.
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\begin{tabular}{lcccccc}
|
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CPU & $N_{\text{cores}}$ & $V$ & $F$ & Memory Bandwidth & Peak DP & Measured performance \\
|
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& & & (GHz) & (GB/s) & (GFlop/s) & (GFlop/s) \\
|
||||
\hline
|
||||
\hline
|
||||
EPYC 7513 & 64 & 4 & 2.6 & 409.6 & 2~662 & 1~576 \\
|
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Xeon Gold 6130 & 32 & 8 & 2.1 & 256.0 & 2~150 & 667 \\ % 239.891
|
||||
ARM Q80 & 80 & 2 & 2.8 & 204.8 & 1~792 & 547 \\ % 292.492
|
||||
ARM Q80 & 80 & 2 & 2.8 & 204.8 & 1~792 & 547 \\ % 292.492
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
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\caption{\label{tab:flops} Average performance of the code measured as the number of double precision (DP) floating-point operations per second (Flop/s) on different machines.}
|
||||
|
|
@ -336,8 +404,12 @@ where $F$ represents the frequency, $V$ the number of double precision elements
|
|||
|
||||
|
||||
The relatively modest performance, at around 30\% efficiency, is attributed to the small dimensions of the matrices involved.
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The largest matrix multiplications in the computational task entail a matrix of size $N_\text{o}^2 \times N_\text{v}$ and another of size $N_\text{v} \times N_\text{o}$ to yield an $N_\text{o}^2 \times N_\text{o}$ matrix.
|
||||
These multiplications exhibit an arithmetic intensity below $N_\text{o} / 4$ flops/byte, which is usually relatively low.
|
||||
The largest matrix multiplications in the computational task entail a matrix of size ${N_\text{o}}^2 \times N_\text{v}$ and another of size $N_\text{v} \times N_\text{o}$ to yield an ${N_\text{o}}^2 \times N_\text{o}$ matrix.
|
||||
These multiplications exhibit an arithmetic intensity of
|
||||
\begin{equation}
|
||||
I = \frac{2\, {N_\text{o}}^3\, N_\text{v}}{8\, \qty({N_\text{o}}^3 + {N_\text{o}}^2 N_\text{v} + {N_\text{o}} N_\text{v})}
|
||||
\end{equation}
|
||||
which can be approximated by $N_\text{o} / 4$ flops/byte as an upper bound, which is usually relatively low.
|
||||
For instance, in the case of benzene with a triple-zeta basis set, the arithmetic intensity is calculated to be 3.52 flops/byte, falling short of the threshold required to attain peak performance on any of the CPUs.
|
||||
By leveraging memory bandwidth and double precision throughput peak, we determined the critical arithmetic intensity necessary to achieve peak performance. On the Xeon and ARM CPUs, this critical value stands at approximately 8.4 and 8.8 flops/byte, respectively. Meanwhile, the EPYC CPU exhibits a value of 6.5 flops/byte, thanks to its superior memory bandwidth.
|
||||
|
||||
|
|
@ -362,7 +434,7 @@ By leveraging memory bandwidth and double precision throughput peak, we determin
|
|||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\acknowledgements{
|
||||
This work was supported by the European Centre of Excellence in Exascale Computing TREX --- Targeting Real Chemical Accuracy at the Exascale.
|
||||
This work was supported by the European Centre of Excellence in Exascale Computing TREX --- Targeting Real Chemical Accuracy at the Exascale.
|
||||
This project has received funding from the European Union's Horizon 2020 — Research and Innovation program --- under grant agreement No.~952165.
|
||||
This work was performed using HPC resourced from CALMIP (Toulouse) under allocations p18005 and p22001.}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
|
|
|||
19
triples.org
19
triples.org
|
|
@ -7706,16 +7706,16 @@ plot E(x), data using ($1*a0):2:3 w err
|
|||
|
||||
|
||||
#+begin_example
|
||||
a = 0.84615 +/- 0.03216 (3.8%)
|
||||
re = 3.92539 +/- 0.01058 (0.2696%)
|
||||
De = 0.101589 +/- 0.00932 (9.174%)
|
||||
E0 = -2099.77 +/- 0.0008014 (3.817e-05%)
|
||||
a = 0.849036 +/- 0.02921 (3.44%)
|
||||
re = 3.92202 +/- 0.009502 (0.2423%)
|
||||
De = 0.101614 +/- 0.008801 (8.661%)
|
||||
E0 = -2099.77 +/- 0.0007748 (3.69e-05%)
|
||||
#+end_example
|
||||
|
||||
#+CALL:freq(0.84615,0.101589)
|
||||
#+CALL:freq(0.849036,0.101614)
|
||||
|
||||
#+RESULTS:
|
||||
: 413.5302408975902
|
||||
: 414.99173933985907
|
||||
|
||||
** CCSD(T) exact
|
||||
|
||||
|
|
@ -7805,6 +7805,13 @@ plot data2 using ($1*a0-0.002):2 pointtype 2 lt 3 title "Full", data using ($1*a
|
|||
#+RESULTS:
|
||||
[[file:cucl_ccsdt2.png]]
|
||||
|
||||
|
||||
* Arithmetic intensity
|
||||
|
||||
No^3 x Nv / ( No^2 x Nv + No x Nv + No^3)
|
||||
|
||||
No^2 / ( No + 1 + No^2/Nv)
|
||||
|
||||
* Export :noexport:
|
||||
#+BEGIN_SRC elisp :output none
|
||||
(setq org-latex-listings 'minted
|
||||
|
|
|
|||
Loading…
Reference in New Issue