From: couturie <couturie@carcariass.(none)> Date: Mon, 25 Feb 2013 10:23:12 +0000 (+0100) Subject: ch13 X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/commitdiff_plain/18cb85fd3967a4c46c9582e8515cef9af3448269 ch13 --- diff --git a/BookGPU/BookGPU.tex b/BookGPU/BookGPU.tex index 76dc8f6..f4f8323 100755 --- a/BookGPU/BookGPU.tex +++ b/BookGPU/BookGPU.tex @@ -148,6 +148,7 @@ \include{Chapters/chapter9/ch9} \include{Chapters/chapter11/ch11} \include{Chapters/chapter12/ch12} +\include{Chapters/chapter13/ch13} \include{Chapters/chapter14/ch14} \include{Chapters/chapter15/ch15} \include{Chapters/chapter16/ch16} diff --git a/BookGPU/Chapters/chapter13/biblio13.bib b/BookGPU/Chapters/chapter13/biblio13.bib new file mode 100644 index 0000000..b5ca7f0 --- /dev/null +++ b/BookGPU/Chapters/chapter13/biblio13.bib @@ -0,0 +1,155 @@ +@article{ref1, +title = {Asynchronous grid computation for {A}merican options derivatives}, +author = {Chau, Ming and Couturier, Rapha\"el and Bahi, Jacques M. and Spiteri, Pierre}, +journal = {Advances in Engineering Software}, +volume = {}, +number = {0}, +pages = {}, +note = {}, +year = {2012}, +} + +@article{ref2, +title = {Matrix iterative analysis}, +author = {Varga, Richard S.}, +journal = {Prentice Hall}, +volume = {}, +number = {}, +pages = {}, +note = {}, +year = {}, +} + +@article{ref3, + author = {Baudet, G\'erard M.}, + title = {Asynchronous iterative methods for multiprocessors}, + journal = {Journal Assoc. Comput. Mach.}, + volume = {25}, + number = {}, + pages = {226--244}, + year = {1978}, +} + +@article{ref4, + author = {Bertsekas, Dimitri P. and Tsitsiklis, John N.}, + title = {Parallel and distributed computation, numerical methods}, + journal = {Prentice Hall Englewood Cliffs N. J. (1989)}, + volume = {}, + number = {}, + pages = {}, + year = {1989}, +} + +@article{ref5, + author = {Bahi, Jacques M. and Contassot-Vivier, Sylvain and Couturier, Rapha\"el}, + title = {Parallel iterative algorithms: from sequential to grid computing}, + journal = {Chapman \& Hall/CRC, Numerical Analysis \& Scientific Computating 1 (2007)}, + volume = {}, + number = {}, + pages = {}, + year = {2007}, +} + +@article{ref6, + author = {Miellou, Jean-Claude and Spiteri, Pierre}, + title = {Two criteria for the convergence of asynchronous iterations}, + journal = {in Computers and computing, P. Chenin et al. ed., Wiley Masson}, + volume = {}, + number = {}, + pages = {91--95}, + year = {1985}, +} + +@article{ref7, + author = {Miellou, Jean-Claude}, + title = {Algorithmes de relaxation chaotique \`a retards}, + journal = {RAIRO Analyse num\'erique}, + volume = {R1}, + number = {}, + pages = {55--82}, + year = {1975}, +} + +@article{ref8, +title = {{CUDA} Toolkit 4.2 {CUBLAS} Library}, +author = {NVIDIA Corporation}, +journal = {}, +volume = {}, +number = {}, +pages = {}, +note = {\url{http://developer.download.nvidia.com/compute/DevZone/docs/html/CUDALibraries/doc/CUBLAS_Library.pdf}}, +year = {2012}, +} + +@article{ref9, + author = {Micikevicius, Paulius}, + title = {{3D} finite difference computation on {GPUs} using {CUDA}}, + journal = {Proceedings of 2nd Workshop on General Purpose Processing on Graphics Processing Units}, + volume = {}, + number = {}, + pages = {79--84}, + year = {2009}, +} + +@article{ref10, + author = {Leist, Aron and Playne, Daniel P. and Hawick, Ken A.}, + title = {Exploiting graphical processing units for data-parallel scientific applications}, + journal = {Concurrency and Computation: Practice and Experience}, + volume = {21}, + number = {18}, + pages = {2400--2437}, + year = {2009}, +} + +@article{ref11, + author = {Chau, Ming and Couturier, Rapha\"el and Bahi, Jacques M. and Spiteri, Pierre}, + title = {Parallel solution of the obstacle problem in grid environments}, + journal = {International Journal of High Performance Computing Applications}, + volume = {25}, + number = {4}, + pages = {488--495}, + year = {2011}, +} + +@article{ref12, + author = {Nvidia}, + title = {{NVIDIA} {CUDA} {C} {P}rogramming {G}uide}, + journal = {Version 4.2 (2012)}, + volume = {}, + number = {}, + pages = {}, + year = {2012}, +} + +@article{ref13, + author = {Evans, David J.}, + title = {Parallel {S}.{O}.{R}. iterative methods}, + journal = {Parallel Computing}, + volume = {1}, + number = {1}, + pages = {3--18}, + year = {1984}, +} + +@article{ref14, + author = {Iwashita, Takeshi and Shimasaki, Masaaki}, + title = {Block red-black ordering method for parallel processing of {ICCG} solver}, + journal = {High Performance Computing}, + volume = {2327}, + number = {}, + pages = {}, + year = {2006}, +} + +@article{ref15, +title = {Iterative methods for sparse linear systems}, +author = {Saad, Yousef}, +journal = {Society for Industrial and Applied Mathematics, 2nd edition}, +volume = {}, +number = {}, +pages = {}, +year = {2003}, +} + + + diff --git a/BookGPU/Chapters/chapter13/bibunits.sty b/BookGPU/Chapters/chapter13/bibunits.sty new file mode 100644 index 0000000..e5e7bb9 --- /dev/null +++ b/BookGPU/Chapters/chapter13/bibunits.sty @@ -0,0 +1,314 @@ +%% +%% This is file `bibunits.sty', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% bibunits.dtx (with options: `package') +%% +%% IMPORTANT NOTICE: +%% +%% For the copyright see the source file. +%% +%% Any modified versions of this file must be renamed +%% with new filenames distinct from bibunits.sty. +%% +%% For distribution of the original source see the terms +%% for copying and modification in the file bibunits.dtx. +%% +%% This generated file may be distributed as long as the +%% original source files, as listed above, are part of the +%% same distribution. (The sources need not necessarily be +%% in the same archive or directory.) +%% Package `bibunits' to use with LaTeX2e. +%% Copyright (C) 1999, 2000 by Thorsten Hansen. All rights reserved. +%% +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{bibunits} + [2000/10/10 v2.2 Multiple bibliographies in one document.] +\newif\iflabelstoglobalaux \labelstoglobalauxfalse +\DeclareOption{labelstoglobalaux}{\labelstoglobalauxtrue} +\newif\ifglobalcitecopy +\globalcitecopyfalse +\DeclareOption{globalcitecopy}{\globalcitecopytrue} +\ProcessOptions +\newwrite\@bibunitaux +\newcount\@bibunitauxcnt \@bibunitauxcnt=0 +\def\@bibunitname{bu\the\@bibunitauxcnt} +\newif\if@starredversion +\let\std@cite\cite +\DeclareRobustCommand\bu@cite{% + \@ifstar + {\@starredversiontrue\std@cite}% + {\@starredversionfalse\std@cite}% +} +\AtBeginDocument{% + \@ifpackageloaded{natbib}% + {% + \NAT@set@cites \let\std@@citex\@citex + \def\bu@@citex[#1][#2]#3{% + \@startbibunitorrelax + \leavevmode + \begingroup\let\@auxout\@bibunitaux\std@@citex[#1][#2]{#3}\endgroup + \ifglobalcitecopy + \std@nocite{#3}% + \fi + }% + }% + {% + \@ifpackageloaded{overcite}% + {% + \let\std@@citew\@citew + \def\bu@@citew#1{% + \@startbibunitorrelax + \leavevmode + {\let\@auxout\@bibunitaux \std@@citew{#1}}% + \ifglobalcitecopy + \@nocite{#1}% + \else + \if@starredversion + \@nocite{#1}% + \fi + \fi + } + \let\std@@citex\@citex + \def\bu@@citex[#1]#2{% + \@startbibunitorrelax + \leavevmode + {\let\@auxout\@bibunitaux \std@@citex[#1]{#2}}% + \ifglobalcitecopy + \@nocite{#2}% + \else + \if@starredversion + \@nocite{#2}% + \fi + \fi + } + }% + {% + \@ifpackageloaded{jurabib}% + {% + \let\std@@citex\@citex + \def\bu@@citex[#1][#2]#3{% + \@startbibunitorrelax + \leavevmode + {\let\@auxout\@bibunitaux \std@@citex[#1][#2]{#3}}% + \ifglobalcitecopy + \std@nocite{#3}% + \else + \if@starredversion + \std@nocite{#3}% + \fi + \fi + }% + }% + {% + \let\std@@citex\@citex + \def\bu@@citex[#1]#2{% + \@startbibunitorrelax + \leavevmode + {\let\@auxout\@bibunitaux \std@@citex[#1]{#2}}% + \ifglobalcitecopy + \std@nocite{#2}% + \else + \if@starredversion + \std@nocite{#2}% + \fi + \fi + }% + }% + }% + }% +}% +\let\std@nocite\nocite +\def\bu@nocite{% + \@ifstar + {\@starredversiontrue\@bu@nocite}% + {\@starredversionfalse\@bu@nocite}% +} +\def\@bu@nocite#1{% + \@startbibunitorrelax + {\let\@auxout\@bibunitaux \std@nocite{#1}}% + \ifglobalcitecopy + \std@nocite{#1}% + \else + \if@starredversion + \std@nocite{#1}% + \fi + \fi +} +\def\bu@bibdata{} +\AtBeginDocument{% + \iflabelstoglobalaux + \else + \let\orig@bibliography\bibliography + \def\bibliography#1{% + \if@filesw + \immediate\openout\@bibunitaux bu.aux + \immediate\write\@mainaux{\string\@input{bu.aux}}% + \fi + \orig@bibliography{#1}% + \if@filesw + \immediate\closeout\@bibunitaux + \fi + }% + \fi + \let\std@bibliography\bibliography +} +\def\bu@bibliography{% + \@ifstar + {\@starredversiontrue\@bu@bibliography}% + {\@starredversionfalse\@bu@bibliography}% +} +\def\@bu@bibliography#1{% + \if@filesw + \immediate\write\@auxout{\string\gdef\string\bu@bibdata{#1}}% + \fi + \gdef\bu@bibdata{#1}% + \if@starredversion + \else + \std@bibliography{#1}% + \fi +} +\def\bu@bibstyle{} +\let\std@bibliographystyle\bibliographystyle +\def\bu@bibliographystyle{% + \@ifstar + {\@starredversiontrue\@bu@bibliographystyle}% + {\@starredversionfalse\@bu@bibliographystyle}% +} +\def\@bu@bibliographystyle#1{% + \if@filesw + \immediate\write\@auxout{\string\gdef\string\bu@bibstyle{#1}}% + \fi + \gdef\bu@bibstyle{#1}% + \if@starredversion + \else + \std@bibliographystyle{#1}% + \fi +} +\def\bibunit{% + \global\let\cite\bu@cite + \global\let\@citex\bu@@citex + \global\let\@citew\bu@@citew + \global\let\nocite\bu@nocite + \global\let\@startbibunitorrelax\@startbibunit + \global\let\@finishbibunit\relax + \@ifnextchar[{\@bibunitx}{\@bibunitx[\bu@bibstyle]}% +} +\def\@bibunitx[#1]{\gdef\@localbibstyle{#1}} +\def\endbibunit{% + \global\let\cite\std@cite + \global\let\@citex\std@@citex + \global\let\@citew\std@@citew + \global\let\nocite\std@nocite + \@finishbibunit + \@input{bu.aux}% +} +\def\@startbibunit{% + \global\let\@startbibunitorrelax\relax + \global\let\@finishbibunit\@finishstartedbibunit + \global\advance\@bibunitauxcnt 1 + \if@filesw + {\endlinechar-1 + \@input{\@bibunitname.aux}}% + \immediate\openout\@bibunitaux\@bibunitname.aux + \immediate\write\@bibunitaux{\string\bibstyle{\@localbibstyle}}% + \fi +} +\let\@finishbibunit\relax +\def\@finishstartedbibunit{% + \if@filesw + \immediate\closeout\@bibunitaux + \fi +} +\let\old@bibunit\@gobble +\def\@bibunit{\endbibunit\bibunit\old@bibunit} +\def\@endbibunit{} +\def\bibliographyunit{% + \@endbibunit + \@ifnextchar[{\@bibliographyunit}{% + \global\let\old@bibunit\@gobble + \global\let\bibliography\std@bibliography + \global\let\bibliographystyle\std@bibliographystyle + \endbibunit + \gdef\@endbibunit{}}% +} +\def\@bibliographyunit[#1]{% + \global\let\bibliography\bu@bibliography + \global\let\bibliographystyle\bu@bibliographystyle + \global\let\old@bibunit#1 + \global\let#1\@bibunit + \gdef\@endbibunit{\global\let#1\old@bibunit}% +} +\def\putbib{\@ifnextchar[{\@putbib}{\@putbib[\bu@bibdata]}} +\def\@putbib[#1]{% + \@startbibunitorrelax + \if@filesw + \immediate\write\@bibunitaux{\string\bibdata{#1}}% + \fi + \@input@{\@bibunitname.bbl}% +} +\AtBeginDocument{% + \iflabelstoglobalaux + \else + \let\std@bibitem\@bibitem + \let\std@lbibitem\@lbibitem + \def\@bibitem#1{% + \let\temp@auxout\@auxout + \let\@auxout\@bibunitaux + \std@bibitem{#1}% + \let\@auxout\temp@auxout + } + \def\@lbibitem[#1]#2{% + \let\temp@auxout\@auxout + \let\@auxout\@bibunitaux + \std@lbibitem[#1]{#2}% + \let\@auxout\temp@auxout + } + \fi +} +\def\remequivalent#1\from#2{% + \let\given=#1% + \ifx#2\empty + \else + \edef#2{\expandafter\plugh#2\plugh}% + \fi +} +\def\plugh\do#1#2{% + \ifx#1\given + \else + \noexpand\do\noexpand#1% + \fi + \ifx#2\plugh + \hgulp\fi\plugh#2% +} +\def\hgulp\fi\plugh\plugh{\fi} +\remequivalent\bibcite\from\@preamblecmds +\AtBeginDocument{% + \@ifpackageloaded{natbib}% + {\renewcommand\bibcite[2]{\global\@namedef{b@#1\@extra@binfo}{#2}}}% + {\renewcommand\bibcite[2]{\global\@namedef{b@#1}{#2}}}% +} +\AtBeginDocument{% + \@ifundefined{bbl@redefine}% + {}% + {% + \bbl@redefine\@input#1{% + \@safe@activestrue\org@@input{#1}\@safe@activesfalse}% + \@ifpackageloaded{natbib}% + {% + \bbl@redefine\std@@citex[#1][#2]#3{% + \@safe@activestrue\org@std@@citex[#1][#2]{#3}\@safe@activesfalse}% + \bbl@redefine\bu@@citex[#1][#2]#3{% + \@safe@activestrue\org@bu@@citex[#1][#2]{#3}\@safe@activesfalse}% + }% + {% natbib not loaded + \bbl@redefine\std@@citex[#1]#2{% + \@safe@activestrue\org@std@@citex[#1]{#2}\@safe@activesfalse}% + }% + }% +} +\endinput +%% +%% End of file `bibunits.sty'. diff --git a/BookGPU/Chapters/chapter13/ch12.tex~ b/BookGPU/Chapters/chapter13/ch12.tex~ new file mode 100755 index 0000000..370c6c7 --- /dev/null +++ b/BookGPU/Chapters/chapter13/ch12.tex~ @@ -0,0 +1,1080 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% CHAPTER 12 %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\chapterauthor{}{} +\chapter{Solving sparse linear systems with GMRES and CG methods on GPU clusters} + +%%--------------------------%% +%% SECTION 1 %% +%%--------------------------%% +\section{Introduction} +\label{sec:01} +The sparse linear systems are used to model many scientific and industrial problems, such as the environmental simulations or +the industrial processing of the complex or non-Newtonian fluids. Moreover, the resolution of these problems often involves the +solving of such linear systems which is considered as the most expensive process in terms of time execution and memory space. +Therefore, solving sparse linear systems must be as efficient as possible in order to deal with problems of ever increasing size. + +There are, in the jargon of numerical analysis, different methods of solving sparse linear systems that we can classify in two +classes: the direct and iterative methods. However, the iterative methods are often more suitable than their counterpart, direct +methods, for solving large sparse linear systems. Indeed, they are less memory consuming and easier to parallelize on parallel +computers than direct methods. Different computing platforms, sequential and parallel computers, are used for solving sparse +linear systems with iterative solutions. Nowadays, graphics processing units (GPUs) have become attractive for solving these +linear systems, due to their computing power and their ability to compute faster than traditional CPUs. + +In Section~\ref{sec:02}, we describe the general principle of two well-known iterative methods: the conjugate gradient method and +the generalized minimal residual method. In Section~\ref{sec:03}, we give the main key points of the parallel implementation of both +methods on a cluster of GPUs. Then, in Section~\ref{sec:04}, we present the experimental results obtained on a CPU cluster and on +a GPU cluster, for solving sparse linear systems associated to matrices of different structures. Finally, in Section~\ref{sec:05}, +we apply the hypergraph partitioning technique to reduce the total communication volume between the computing nodes and, thus, to +improve the execution times of the parallel algorithms of both iterative methods. + + +%%--------------------------%% +%% SECTION 2 %% +%%--------------------------%% +\section{Krylov iterative methods} +\label{sec:02} +Let us consider the following system of $n$ linear equations in $\mathbb{R}$: +\begin{equation} +Ax=b, +\label{eq:01} +\end{equation} +where $A\in\mathbb{R}^{n\times n}$ is a sparse nonsingular square matrix, $x\in\mathbb{R}^{n}$ is the solution vector, +$b\in\mathbb{R}^{n}$ is the right-hand side and $n\in\mathbb{N}$ is a large integer number. + +The iterative methods for solving the large sparse linear system~(\ref{eq:01}) proceed by successive iterations of a same +block of elementary operations, during which an infinite number of approximate solutions $\{x_k\}_{k\geq 0}$ are computed. +Indeed, from an initial guess $x_0$, an iterative method determines at each iteration $k>0$ an approximate solution $x_k$ +which, gradually, converges to the exact solution $x^{*}$ as follows: +\begin{equation} +x^{*}=\lim\limits_{k\to\infty}x_{k}=A^{-1}b. +\label{eq:02} +\end{equation} +The number of iterations necessary to reach the exact solution $x^{*}$ is not known beforehand and can be infinite. In +practice, an iterative method often finds an approximate solution $\tilde{x}$ after a fixed number of iterations and/or +when a given convergence criterion is satisfied as follows: +\begin{equation} +\|b-A\tilde{x}\| < \varepsilon, +\label{eq:03} +\end{equation} +where $\varepsilon<1$ is the required convergence tolerance threshold. + +Some of the most iterative methods that have proven their efficiency for solving large sparse linear systems are those +called \textit{Krylov sub-space methods}~\cite{ref1}. In the present chapter, we describe two Krylov methods which are +widely used: the conjugate gradient method (CG) and the generalized minimal residual method (GMRES). In practice, the +Krylov sub-space methods are usually used with preconditioners that allow to improve their convergence. So, in what +follows, the CG and GMRES methods are used for solving the left-preconditioned sparse linear system: +\begin{equation} +M^{-1}Ax=M^{-1}b, +\label{eq:11} +\end{equation} +where $M$ is the preconditioning matrix. + +%%****************%% +%%****************%% +\subsection{CG method} +\label{sec:02.01} +The conjugate gradient method is initially developed by Hestenes and Stiefel in 1952~\cite{ref2}. It is one of the well +known iterative method for solving large sparse linear systems. In addition, it can be adapted for solving nonlinear +equations and optimization problems. However, it can only be applied to problems with positive definite symmetric matrices. + +The main idea of the CG method is the computation of a sequence of approximate solutions $\{x_k\}_{k\geq 0}$ in a Krylov +sub-space of order $k$ as follows: +\begin{equation} +x_k \in x_0 + \mathcal{K}_k(A,r_0), +\label{eq:04} +\end{equation} +such that the Galerkin condition must be satisfied: +\begin{equation} +r_k \bot \mathcal{K}_k(A,r_0), +\label{eq:05} +\end{equation} +where $x_0$ is the initial guess, $r_k=b-Ax_k$ is the residual of the computed solution $x_k$ and $\mathcal{K}_k$ the Krylov +sub-space of order $k$: \[\mathcal{K}_k(A,r_0) \equiv\text{span}\{r_0, Ar_0, A^2r_0,\ldots, A^{k-1}r_0\}.\] +In fact, CG is based on the construction of a sequence $\{p_k\}_{k\in\mathbb{N}}$ of direction vectors in $\mathcal{K}_k$ +which are pairwise $A$-conjugate ($A$-orthogonal): +\begin{equation} +\begin{array}{ll} +p_i^T A p_j = 0, & i\neq j. +\end{array} +\label{eq:06} +\end{equation} +At each iteration $k$, an approximate solution $x_k$ is computed by recurrence as follows: +\begin{equation} +\begin{array}{ll} +x_k = x_{k-1} + \alpha_k p_k, & \alpha_k\in\mathbb{R}. +\end{array} +\label{eq:07} +\end{equation} +Consequently, the residuals $r_k$ are computed in the same way: +\begin{equation} +r_k = r_{k-1} - \alpha_k A p_k. +\label{eq:08} +\end{equation} +In the case where all residuals are nonzero, the direction vectors $p_k$ can be determined so that the following recurrence +holds: +\begin{equation} +\begin{array}{lll} +p_0=r_0, & p_k=r_k+\beta_k p_{k-1}, & \beta_k\in\mathbb{R}. +\end{array} +\label{eq:09} +\end{equation} +Moreover, the scalars $\{\alpha_k\}_{k>0}$ are chosen so as to minimize the $A$-norm error $\|x^{*}-x_k\|_A$ over the Krylov +sub-space $\mathcal{K}_{k}$ and the scalars $\{\beta_k\}_{k>0}$ are chosen so as to ensure that the direction vectors are +pairwise $A$-conjugate. So, the assumption that matrix $A$ is symmetric and the recurrences~(\ref{eq:08}) and~(\ref{eq:09}) +allow to deduce that: +\begin{equation} +\begin{array}{ll} +\alpha_{k}=\frac{r^{T}_{k-1}r_{k-1}}{p_{k}^{T}Ap_{k}}, & \beta_{k}=\frac{r_{k}^{T}r_{k}}{r_{k-1}^{T}r_{k-1}}. +\end{array} +\label{eq:10} +\end{equation} + +Algorithm~\ref{alg:01} shows the main key points of the preconditioned CG method. It allows to solve the left-preconditioned +sparse linear system~(\ref{eq:11}). In this algorithm, $\varepsilon$ is the convergence tolerance threshold, $maxiter$ is the maximum +number of iterations and $(\cdot,\cdot)$ defines the dot product between two vectors in $\mathbb{R}^{n}$. At every iteration, a direction +vector $p_k$ is determined, so that it is orthogonal to the preconditioned residual $z_k$ and to the direction vectors $\{p_i\}_{i<k}$ +previously determined (from line~$8$ to line~$13$). Then, at lines~$16$ and~$17$ , the iterate $x_k$ and the residual $r_k$ are computed +using formulas~(\ref{eq:07}) and~(\ref{eq:08}), respectively. The CG method converges after, at most, $n$ iterations. In practice, the CG +algorithm stops when the tolerance threshold $\varepsilon$ and/or the maximum number of iterations $maxiter$ are reached. + +\begin{algorithm} + \SetLine + \linesnumbered + Choose an initial guess $x_0$\; + $r_{0} = b - A x_{0}$\; + $convergence$ = false\; + $k = 1$\; + \Repeat{convergence}{ + $z_{k} = M^{-1} r_{k-1}$\; + $\rho_{k} = (r_{k-1},z_{k})$\; + \eIf{$k = 1$}{ + $p_{k} = z_{k}$\; + }{ + $\beta_{k} = \rho_{k} / \rho_{k-1}$\; + $p_{k} = z_{k} + \beta_{k} \times p_{k-1}$\; + } + $q_{k} = A \times p_{k}$\; + $\alpha_{k} = \rho_{k} / (p_{k},q_{k})$\; + $x_{k} = x_{k-1} + \alpha_{k} \times p_{k}$\; + $r_{k} = r_{k-1} - \alpha_{k} \times q_{k}$\; + \eIf{$(\rho_{k} < \varepsilon)$ {\bf or} $(k \geq maxiter)$}{ + $convergence$ = true\; + }{ + $k = k + 1$\; + } + } +\caption{Left-preconditioned CG method} +\label{alg:01} +\end{algorithm} + +%%****************%% +%%****************%% +\subsection{GMRES method} +\label{sec:02.02} +The iterative method GMRES is developed by Saad and Schultz in 1986~\cite{ref3} as a generalization of the minimum residual method +MINRES~\cite{ref4}. Indeed, GMRES can be applied for solving symmetric or asymmetric linear systems. + +The main principle of the GMRES method is to find an approximation minimizing at best the residual norm. In fact, GMRES +computes a sequence of approximate solutions $\{x_k\}_{k>0}$ in a Krylov sub-space $\mathcal{K}_k$ as follows: +\begin{equation} +\begin{array}{ll} +x_k \in x_0 + \mathcal{K}_k(A, v_1),& v_1=\frac{r_0}{\|r_0\|_2}, +\end{array} +\label{eq:12} +\end{equation} +so that the Petrov-Galerkin condition is satisfied: +\begin{equation} +\begin{array}{ll} +r_k \bot A \mathcal{K}_k(A, v_1). +\end{array} +\label{eq:13} +\end{equation} +GMRES uses the Arnoldi process~\cite{ref5} to construct an orthonormal basis $V_k$ for the Krylov sub-space $\mathcal{K}_k$ +and an upper Hessenberg matrix $\bar{H}_k$ of order $(k+1)\times k$: +\begin{equation} +\begin{array}{ll} +V_k = \{v_1, v_2,\ldots,v_k\}, & \forall k>1, v_k=A^{k-1}v_1, +\end{array} +\label{eq:14} +\end{equation} +and +\begin{equation} +V_k A = V_{k+1} \bar{H}_k. +\label{eq:15} +\end{equation} + +Then, at each iteration $k$, an approximate solution $x_k$ is computed in the Krylov sub-space $\mathcal{K}_k$ spanned by $V_k$ +as follows: +\begin{equation} +\begin{array}{ll} +x_k = x_0 + V_k y, & y\in\mathbb{R}^{k}. +\end{array} +\label{eq:16} +\end{equation} +From both formulas~(\ref{eq:15}) and~(\ref{eq:16}) and $r_k=b-Ax_k$, we can deduce that: +\begin{equation} +\begin{array}{lll} + r_{k} & = & b - A (x_{0} + V_{k}y) \\ + & = & r_{0} - AV_{k}y \\ + & = & \beta v_{1} - V_{k+1}\bar{H}_{k}y \\ + & = & V_{k+1}(\beta e_{1} - \bar{H}_{k}y), +\end{array} +\label{eq:17} +\end{equation} +such that $\beta=\|r_0\|_2$ and $e_1=(1,0,\cdots,0)$ is the first vector of the canonical basis of $\mathbb{R}^k$. So, +the vector $y$ is chosen in $\mathbb{R}^k$ so as to minimize at best the Euclidean norm of the residual $r_k$. Consequently, +a linear least-squares problem of size $k$ is solved: +\begin{equation} +\underset{y\in\mathbb{R}^{k}}{min}\|r_{k}\|_{2}=\underset{y\in\mathbb{R}^{k}}{min}\|\beta e_{1}-\bar{H}_{k}y\|_{2}. +\label{eq:18} +\end{equation} +The QR factorization of matrix $\bar{H}_k$ is used to compute the solution of this problem by using Givens rotations~\cite{ref1,ref3}, +such that: +\begin{equation} +\begin{array}{lll} +\bar{H}_{k}=Q_{k}R_{k}, & Q_{k}\in\mathbb{R}^{(k+1)\times (k+1)}, & R_{k}\in\mathbb{R}^{(k+1)\times k}, +\end{array} +\label{eq:19} +\end{equation} +where $Q_kQ_k^T=I_k$ and $R_k$ is an upper triangular matrix. + +The GMRES method computes an approximate solution with a sufficient precision after, at most, $n$ iterations ($n$ is the size of the +sparse linear system to be solved). However, the GMRES algorithm must construct and store in the memory an orthonormal basis $V_k$ whose +size is proportional to the number of iterations required to achieve the convergence. Then, to avoid a huge memory storage, the GMRES +method must be restarted at each $m$ iterations, such that $m$ is very small ($m\ll n$), and with $x_m$ as the initial guess to the +next iteration. This allows to limit the size of the basis $V$ to $m$ orthogonal vectors. + +Algorithm~\ref{alg:02} shows the main key points of the GMRES method with restarts. It solves the left-preconditioned sparse linear +system~(\ref{eq:11}), such that $M$ is the preconditioning matrix. At each iteration $k$, GMRES uses the Arnoldi process (defined +from line~$7$ to line~$17$) to construct a basis $V_m$ of $m$ orthogonal vectors and an upper Hessenberg matrix $\bar{H}_m$ of size +$(m+1)\times m$. Then, it solves the linear least-squares problem of size $m$ to find the vector $y\in\mathbb{R}^{m}$ which minimizes +at best the residual norm (line~$18$). Finally, it computes an approximate solution $x_m$ in the Krylov sub-space spanned by $V_m$ +(line~$19$). The GMRES algorithm is stopped when the residual norm is sufficiently small ($\|r_m\|_2<\varepsilon$) and/or the maximum +number of iterations ($maxiter$) is reached. + +\begin{algorithm} + \SetLine + \linesnumbered + Choose an initial guess $x_0$\; + $convergence$ = false\; + $k = 1$\; + $r_{0} = M^{-1}(b-Ax_{0})$\; + $\beta = \|r_{0}\|_{2}$\; + \While{$\neg convergence$}{ + $v_{1} = r_{0}/\beta$\; + \For{$j=1$ \KwTo $m$}{ + $w_{j} = M^{-1}Av_{j}$\; + \For{$i=1$ \KwTo $j$}{ + $h_{i,j} = (w_{j},v_{i})$\; + $w_{j} = w_{j}-h_{i,j}v_{i}$\; + } + $h_{j+1,j} = \|w_{j}\|_{2}$\; + $v_{j+1} = w_{j}/h_{j+1,j}$\; + } + Set $V_{m}=\{v_{j}\}_{1\leq j \leq m}$ and $\bar{H}_{m}=(h_{i,j})$ a $(m+1)\times m$ upper Hessenberg matrix\; + Solve a least-squares problem of size $m$: $min_{y\in\mathrm{I\!R}^{m}}\|\beta e_{1}-\bar{H}_{m}y\|_{2}$\; + $x_{m} = x_{0}+V_{m}y_{m}$\; + $r_{m} = M^{-1}(b-Ax_{m})$\; + $\beta = \|r_{m}\|_{2}$\; + \eIf{ $(\beta<\varepsilon)$ {\bf or} $(k\geq maxiter)$}{ + $convergence$ = true\; + }{ + $x_{0} = x_{m}$\; + $r_{0} = r_{m}$\; + $k = k + 1$\; + } + } +\caption{Left-preconditioned GMRES method with restarts} +\label{alg:02} +\end{algorithm} + + +%%--------------------------%% +%% SECTION 3 %% +%%--------------------------%% +\section{Parallel implementation on a GPU cluster} +\label{sec:03} +In this section, we present the parallel algorithms of both iterative CG and GMRES methods for GPU clusters. +The implementation is performed on a GPU cluster composed of different computing nodes, such that each node +is a CPU core managed by a MPI process and equipped with a GPU card. The parallelization of these algorithms +is carried out by using the MPI communication routines between the GPU computing nodes and the CUDA programming +environment inside each node. In what follows, the algorithms of the iterative methods are called iterative +solvers. + +%%****************%% +%%****************%% +\subsection{Data partitioning} +\label{sec:03.01} +The parallel solving of the large sparse linear system~(\ref{eq:11}) requires a data partitioning between the computing +nodes of the GPU cluster. Let $p$ denotes the number of the computing nodes on the GPU cluster. The partitioning operation +consists in the decomposition of the vectors and matrices, involved in the iterative solver, in $p$ portions. Indeed, this +operation allows to assign to each computing node $i$: +\begin{itemize*} +\item a portion of size $\frac{n}{p}$ elements of each vector, +\item a sparse rectangular sub-matrix $A_i$ of size $(n,\frac{n}{p})$ and, +\item a square preconditioning sub-matrix $M_i$ of size $(\frac{n}{p},\frac{n}{p})$, +\end{itemize*} +where $n$ is the size of the sparse linear system to be solved. In the first instance, we perform a naive row-wise partitioning +(decomposition row-by-row) on the data of the sparse linear systems to be solved. Figure~\ref{fig:01} shows an example of a row-wise +data partitioning between four computing nodes of a sparse linear system (sparse matrix $A$, solution vector $x$ and right-hand +side $b$) of size $16$ unknown values. + +\begin{figure} +\centerline{\includegraphics[scale=0.35]{Chapters/chapter12/figures/partition}} +\caption{A data partitioning of the sparse matrix $A$, the solution vector $x$ and the right-hand side $b$ into four portions.} +\label{fig:01} +\end{figure} + +%%****************%% +%%****************%% +\subsection{GPU computing} +\label{sec:03.02} +After the partitioning operation, all the data involved from this operation must be transferred from the CPU memories to the GPU +memories, in order to be processed by GPUs. We use two functions of the CUBLAS library (CUDA Basic Linear Algebra Subroutines), +developed by Nvidia~\cite{ref6}: \verb+cublasAlloc()+ for the memory allocations on GPUs and \verb+cublasSetVector()+ for the +memory copies from the CPUs to the GPUs. + +An efficient implementation of CG and GMRES solvers on a GPU cluster requires to determine all parts of their codes that can be +executed in parallel and, thus, take advantage of the GPU acceleration. As many Krylov sub-space methods, the CG and GMRES methods +are mainly based on arithmetic operations dealing with vectors or matrices: sparse matrix-vector multiplications, scalar-vector +multiplications, dot products, Euclidean norms, AXPY operations ($y\leftarrow ax+y$ where $x$ and $y$ are vectors and $a$ is a +scalar) and so on. These vector operations are often easy to parallelize and they are more efficient on parallel computers when +they work on large vectors. Therefore, all the vector operations used in CG and GMRES solvers must be executed by the GPUs as kernels. + +We use the kernels of the CUBLAS library to compute some vector operations of CG and GMRES solvers. The following kernels of CUBLAS +(dealing with double floating point) are used: \verb+cublasDdot()+ for the dot products, \verb+cublasDnrm2()+ for the Euclidean +norms and \verb+cublasDaxpy()+ for the AXPY operations. For the rest of the data-parallel operations, we code their kernels in CUDA. +In the CG solver, we develop a kernel for the XPAY operation ($y\leftarrow x+ay$) used at line~$12$ in Algorithm~\ref{alg:01}. In the +GMRES solver, we program a kernel for the scalar-vector multiplication (lines~$7$ and~$15$ in Algorithm~\ref{alg:02}), a kernel for +solving the least-squares problem and a kernel for the elements updates of the solution vector $x$. + +The least-squares problem in the GMRES method is solved by performing a QR factorization on the Hessenberg matrix $\bar{H}_m$ with +plane rotations and, then, solving the triangular system by backward substitutions to compute $y$. Consequently, solving the least-squares +problem on the GPU is not interesting. Indeed, the triangular solves are not easy to parallelize and inefficient on GPUs. However, +the least-squares problem to solve in the GMRES method with restarts has, generally, a very small size $m$. Therefore, we develop +an inexpensive kernel which must be executed in sequential by a single CUDA thread. + +The most important operation in CG and GMRES methods is the sparse matrix-vector multiplication (SpMV), because it is often an +expensive operation in terms of execution time and memory space. Moreover, it requires to take care of the storage format of the +sparse matrix in the memory. Indeed, the naive storage, row-by-row or column-by-column, of a sparse matrix can cause a significant +waste of memory space and execution time. In addition, the sparsity nature of the matrix often leads to irregular memory accesses +to read the matrix nonzero values. So, the computation of the SpMV multiplication on GPUs can involve non coalesced accesses to +the global memory, which slows down even more its performances. One of the most efficient compressed storage formats of sparse +matrices on GPUs is HYB format~\cite{ref7}. It is a combination of ELLpack (ELL) and Coordinate (COO) formats. Indeed, it stores +a typical number of nonzero values per row in ELL format and remaining entries of exceptional rows in COO format. It combines +the efficiency of ELL due to the regularity of its memory accesses and the flexibility of COO which is insensitive to the matrix +structure. Consequently, we use the HYB kernel~\cite{ref8} developed by Nvidia to implement the SpMV multiplication of CG and +GMRES methods on GPUs. Moreover, to avoid the non coalesced accesses to the high-latency global memory, we fill the elements of +the iterate vector $x$ in the cached texture memory. + +%%****************%% +%%****************%% +\subsection{Data communications} +\label{sec:03.03} +All the computing nodes of the GPU cluster execute in parallel the same iterative solver (Algorithm~\ref{alg:01} or Algorithm~\ref{alg:02}) +adapted to GPUs, but on their own portions of the sparse linear system: $M^{-1}_iA_ix_i=M^{-1}_ib_i$, $0\leq i<p$. However in order to solve +the complete sparse linear system~(\ref{eq:11}), synchronizations must be performed between the local computations of the computing nodes +over the cluster. In what follows, two computing nodes sharing data are called neighboring nodes. + +As already mentioned, the most important operation of CG and GMRES methods is the SpMV multiplication. In the parallel implementation of +the iterative methods, each computing node $i$ performs the SpMV multiplication on its own sparse rectangular sub-matrix $A_i$. Locally, it +has only sub-vectors of size $\frac{n}{p}$ corresponding to rows of its sub-matrix $A_i$. However, it also requires the vector elements +of its neighbors, corresponding to the column indices on which its sub-matrix has nonzero values (see Figure~\ref{fig:01}). So, in addition +to the local vectors, each node must also manage vector elements shared with neighbors and required to compute the SpMV multiplication. +Therefore, the iterate vector $x$ managed by each computing node is composed of a local sub-vector $x^{local}$ of size $\frac{n}{p}$ and a +sub-vector of shared elements $x^{shared}$. In the same way, the vector used to construct the orthonormal basis of the Krylov sub-space +(vectors $p$ and $v$ in CG and GMRES methods, respectively) is composed of a local sub-vector and a shared sub-vector. + +Therefore, before computing the SpMV multiplication, the neighboring nodes over the GPU cluster must exchange between them the shared +vector elements necessary to compute this multiplication. First, each computing node determines, in its local sub-vector, the vector +elements needed by other nodes. Then, the neighboring nodes exchange between them these shared vector elements. The data exchanges +are implemented by using the MPI point-to-point communication routines: blocking sends with \verb+MPI_Send()+ and nonblocking receives +with \verb+MPI_Irecv()+. Figure~\ref{fig:02} shows an example of data exchanges between \textit{Node 1} and its neighbors \textit{Node 0}, +\textit{Node 2} and \textit{Node 3}. In this example, the iterate matrix $A$ split between these four computing nodes is that presented +in Figure~\ref{fig:01}. + +\begin{figure} +\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/compress}} +\caption{Data exchanges between \textit{Node 1} and its neighbors \textit{Node 0}, \textit{Node 2} and \textit{Node 3}.} +\label{fig:02} +\end{figure} + +After the synchronization operation, the computing nodes receive, from their respective neighbors, the shared elements in a sub-vector +stored in a compressed format. However, in order to compute the SpMV multiplication, the computing nodes operate on sparse global vectors +(see Figure~\ref{fig:02}). In this case, the received vector elements must be copied to the corresponding indices in the global vector. +So as not to need to perform this at each iteration, we propose to reorder the columns of each sub-matrix $\{A_i\}_{0\leq i<p}$, so that +the shared sub-vectors could be used in their compressed storage formats. Figure~\ref{fig:03} shows a reordering of a sparse sub-matrix +(sub-matrix of \textit{Node 1}). Furthermore, we use the texture memory to cache the global vector. This allows to avoid the non coalesced +accesses to the GPU global memory. + +\begin{figure} +\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/reorder}} +\caption{Columns reordering of a sparse sub-matrix.} +\label{fig:03} +\end{figure} + +A GPU cluster is a parallel platform with a distributed memory. So, the synchronizations and communication data between GPU nodes are +carried out by passing messages. However, GPUs can not communicate between them in direct way. Then, CPUs via MPI processes are in charge +of the synchronizations within the GPU cluster. Consequently, the vector elements to be exchanged must be copied from the GPU memory +to the CPU memory and vice-versa before and after the synchronization operation between CPUs. We have used the CBLAS communication subroutines +to perform the data transfers between a CPU core and its GPU: \verb+cublasGetVector()+ and \verb+cublasSetVector()+. Finally, in addition +to the data exchanges, GPU nodes perform reduction operations to compute in parallel the dot products and Euclidean norms. This is implemented +by using the MPI global communication \verb+MPI_Allreduce()+. + + +%%--------------------------%% +%% SECTION 4 %% +%%--------------------------%% +\section{Experimental results} +\label{sec:04} +In this section, we present the performances of the parallel CG and GMRES linear solvers obtained on a cluster of $12$ GPUs. Indeed, this GPU +cluster of tests is composed of six machines connected by $20$Gbps InfiniBand network. Each machine is a Quad-Core Xeon E5530 CPU running at +$2.4$GHz and providing $12$GB of RAM with a memory bandwidth of $25.6$GB/s. In addition, two Tesla C1060 GPUs are connected to each machine via +a PCI-Express 16x Gen 2.0 interface with a throughput of $8$GB/s. A Tesla C1060 GPU contains $240$ cores running at $1.3$GHz and providing a +global memory of $4$GB with a memory bandwidth of $102$GB/s. Figure~\ref{fig:04} shows the general scheme of the GPU cluster that we used in +the experimental tests. + +\begin{figure} +\centerline{\includegraphics[scale=0.25]{Chapters/chapter12/figures/cluster}} +\caption{General scheme of the GPU cluster of tests composed of six machines, each with two GPUs.} +\label{fig:04} +\end{figure} + +Linux cluster version 2.6.39 OS is installed on CPUs. C programming language is used for coding the parallel algorithms of both methods on the +GPU cluster. CUDA version 4.0~\cite{ref9} is used for programming GPUs, using CUBLAS library~\cite{ref6} to deal with vector operations in GPUs +and, finally, MPI routines of OpenMPI 1.3.3 are used to carry out the communications between CPU cores. Indeed, the experiments are done on a +cluster of $12$ computing nodes, where each node is managed by a MPI process and it is composed of one CPU core and one GPU card. + +All tests are made on double-precision floating point operations. The parameters of both linear solvers are initialized as follows: the residual +tolerance threshold $\varepsilon=10^{-12}$, the maximum number of iterations $maxiter=500$, the right-hand side $b$ is filled with $1.0$ and the +initial guess $x_0$ is filled with $0.0$. In addition, we limited the Arnoldi process used in the GMRES method to $16$ iterations ($m=16$). For +the sake of simplicity, we have chosen the preconditioner $M$ as the main diagonal of the sparse matrix $A$. Indeed, it allows to easily compute +the required inverse matrix $M^{-1}$ and it provides a relatively good preconditioning for not too ill-conditioned matrices. In the GPU computing, +the size of thread blocks is fixed to $512$ threads. Finally, the performance results, presented hereafter, are obtained from the mean value over +$10$ executions of the same parallel linear solver and for the same input data. + +To get more realistic results, we tested the CG and GMRES algorithms on sparse matrices of the Davis's collection~\cite{ref10}, that arise in a wide +spectrum of real-world applications. We chose six symmetric sparse matrices and six nonsymmetric ones from this collection. In Figure~\ref{fig:05}, +we show structures of these matrices and in Table~\ref{tab:01} we present their main characteristics which are the number of rows, the total number +of nonzero values (nnz) and the maximal bandwidth. In the present chapter, the bandwidth of a sparse matrix is defined as the number of matrix columns +separating the first and the last nonzero value on a matrix row. + +\begin{figure} +\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/matrices}} +\caption{Sketches of sparse matrices chosen from the Davis's collection.} +\label{fig:05} +\end{figure} + +\begin{table}[!h] +\centering +\begin{tabular}{|c|c|c|c|c|} +\hline +{\bf Matrix type} & {\bf Matrix name} & {\bf \# rows} & {\bf \# nnz} & {\bf Bandwidth} \\ \hline \hline + +\multirow{6}{*}{Symmetric} & 2cubes\_sphere & $101,492$ & $1,647,264$ & $100,464$ \\ + + & ecology2 & $999,999$ & $4,995,991$ & $2,001$ \\ + + & finan512 & $74,752$ & $596,992$ & $74,725$ \\ + + & G3\_circuit & $1,585,478$ & $7,660,826$ & $1,219,059$ \\ + + & shallow\_water2 & $81,920$ & $327,680$ & $58,710$ \\ + + & thermal2 & $1,228,045$ & $8,580,313$ & $1,226,629$ \\ \hline \hline + +\multirow{6}{*}{Nonsymmetric} & cage13 & $445,315$ & $7,479,343$ & $318,788$\\ + + & crashbasis & $160,000$ & $1,750,416$ & $120,202$ \\ + + & FEM\_3D\_thermal2 & $147,900$ & $3,489.300$ & $117,827$ \\ + + & language & $399,130$ & $1,216,334$ & $398,622$\\ + + & poli\_large & $15,575$ & $33,074$ & $15,575$ \\ + + & torso3 & $259,156$ & $4,429,042$ & $216,854$ \\ \hline +\end{tabular} +\vspace{0.5cm} +\caption{Main characteristics of sparse matrices chosen from the Davis's collection.} +\label{tab:01} +\end{table} + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|c|} +\hline +{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$ \\ \hline \hline + +2cubes\_sphere & $0.132s$ & $0.069s$ & $1.93$ & $12$ & $1.14e$-$09$ & $3.47e$-$18$ \\ + +ecology2 & $0.026s$ & $0.017s$ & $1.52$ & $13$ & $5.06e$-$09$ & $8.33e$-$17$ \\ + +finan512 & $0.053s$ & $0.036s$ & $1.49$ & $12$ & $3.52e$-$09$ & $1.66e$-$16$ \\ + +G3\_circuit & $0.704s$ & $0.466s$ & $1.51$ & $16$ & $4.16e$-$10$ & $4.44e$-$16$ \\ + +shallow\_water2 & $0.017s$ & $0.010s$ & $1.68$ & $5$ & $2.24e$-$14$ & $3.88e$-$26$ \\ + +thermal2 & $1.172s$ & $0.622s$ & $1.88$ & $15$ & $5.11e$-$09$ & $3.33e$-$16$ \\ \hline +\end{tabular} +\caption{Performances of the parallel CG method on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.} +\label{tab:02} +\end{center} +\end{table} + +\begin{table}[!h] +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|c|} +\hline +{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$ \\ \hline \hline + +2cubes\_sphere & $0.234s$ & $0.124s$ & $1.88$ & $21$ & $2.10e$-$14$ & $3.47e$-$18$ \\ + +ecology2 & $0.076s$ & $0.035s$ & $2.15$ & $21$ & $4.30e$-$13$ & $4.38e$-$15$ \\ + +finan512 & $0.073s$ & $0.052s$ & $1.40$ & $17$ & $3.21e$-$12$ & $5.00e$-$16$ \\ + +G3\_circuit & $1.016s$ & $0.649s$ & $1.56$ & $22$ & $1.04e$-$12$ & $2.00e$-$15$ \\ + +shallow\_water2 & $0.061s$ & $0.044s$ & $1.38$ & $17$ & $5.42e$-$22$ & $2.71e$-$25$ \\ + +thermal2 & $1.666s$ & $0.880s$ & $1.89$ & $21$ & $6.58e$-$12$ & $2.77e$-$16$ \\ \hline \hline + +cage13 & $0.721s$ & $0.338s$ & $2.13$ & $26$ & $3.37e$-$11$ & $2.66e$-$15$ \\ + +crashbasis & $1.349s$ & $0.830s$ & $1.62$ & $121$ & $9.10e$-$12$ & $6.90e$-$12$ \\ + +FEM\_3D\_thermal2 & $0.797s$ & $0.419s$ & $1.90$ & $64$ & $3.87e$-$09$ & $9.09e$-$13$ \\ + +language & $2.252s$ & $1.204s$ & $1.87$ & $90$ & $1.18e$-$10$ & $8.00e$-$11$ \\ + +poli\_large & $0.097s$ & $0.095s$ & $1.02$ & $69$ & $4.98e$-$11$ & $1.14e$-$12$ \\ + +torso3 & $4.242s$ & $2.030s$ & $2.09$ & $175$ & $2.69e$-$10$ & $1.78e$-$14$ \\ \hline +\end{tabular} +\caption{Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.} +\label{tab:03} +\end{center} +\end{table} + +Tables~\ref{tab:02} and~\ref{tab:03} shows the performances of the parallel CG and GMRES solvers, respectively, for solving linear systems associated to +the sparse matrices presented in Tables~\ref{tab:01}. They allow to compare the performances obtained on a cluster of $24$ CPU cores and on a cluster +of $12$ GPUs. However, Table~\ref{tab:02} shows only the solving performances of symmetric sparse linear systems, due to the inability of the CG method +to solve the nonsymmetric systems. In both tables, the second and third columns give, respectively, the execution times in seconds obtained on $24$ CPU +cores($Time_{gpu}$) and that obtained on $12$ GPUs ($Time_{gpu}$). Moreover, we take into account the relative gains $\tau$ of a solver implemented on the +GPU cluster compared to the same solver implemented on the CPU cluster. The relative gains, presented in the fourth column, are computed as a ratio of +the CPU execution time over the GPU execution time: +\begin{equation} +\tau = \frac{Time_{cpu}}{Time_{gpu}}. +\label{eq:20} +\end{equation} +In addition, Tables~\ref{tab:02} and~\ref{tab:03} give the number of iterations ($iter$), the precision $prec$ of the solution computed on the GPU cluster +and the difference $\Delta$ between the solution computed on the CPU cluster and that computed on the GPU cluster. Both parameters $prec$ and $\Delta$ +allow to validate and verify the accuracy of the solution computed on the GPU cluster. We have computed them as follows: +\begin{eqnarray} +\Delta = max|x^{cpu}-x^{gpu}|,\\ +prec = max|M^{-1}r^{gpu}|, +\end{eqnarray} +where $\Delta$ is the maximum vector element, in absolute value, of the difference between the two solutions $x^{cpu}$ and $x^{gpu}$ computed, respectively, +on CPU and GPU cluster and $prec$ is the maximum element, in absolute value, of the residual vector $r^{gpu}\in\mathbb{R}^{n}$ of the solution $x^{gpu}$. +Thus, we can see that the solutions obtained on the GPU cluster were computed with a sufficient accuracy (about $10^{-10}$) and they are, more or less, +equivalent to those computed on the CPU cluster with a small difference ranging from $10^{-10}$ and $10^{-26}$. However, we can notice from the relative +gains $\tau$ that is not interesting to use multiple GPUs for solving small sparse linear systems. in fact, a small sparse matrix does not allow to maximize +utilization of GPU cores. In addition, the communications required to synchronize the computations over the cluster increase the idle times of GPUs and +slow down further the parallel computations. + +Consequently, in order to test the performances of the parallel solvers, we developed in C programming language a generator of large sparse matrices. +This generator takes a matrix from the Davis's collection~\cite{ref10} as an initial matrix to construct large sparse matrices exceeding ten million +of rows. It must be executed in parallel by the MPI processes of the computing nodes, so that each process could construct its sparse sub-matrix. In +first experimental tests, we are focused on sparse matrices having a banded structure, because they are those arise in the most of numerical problems. +So to generate the global sparse matrix, each MPI process constructs its sub-matrix by performing several copies of an initial sparse matrix chosen +from the Davis's collection. Then, it puts all these copies on the main diagonal of the global matrix (see Figure~\ref{fig:06}). Moreover, the empty +spaces between two successive copies in the main diagonal are filled with sub-copies (left-copy and right-copy in Figure~\ref{fig:06}) of the same +initial matrix. + +\begin{figure} +\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/generation}} +\caption{Parallel generation of a large sparse matrix by four computing nodes.} +\label{fig:06} +\end{figure} + +We have used the parallel CG and GMRES algorithms for solving sparse linear systems of $25$ million of unknown values. The sparse matrices associated +to these linear systems are generated from those presented in Table~\ref{tab:01}. Their main characteristics are given in Table~\ref{tab:04}. Tables~\ref{tab:05} +and~\ref{tab:06} shows the performances of the parallel CG and GMRES solvers, respectively, obtained on a cluster of $24$ CPU cores and on a cluster +of $12$ GPUs. Obviously, we can notice from these tables that solving large sparse linear systems on a GPU cluster is more efficient than on a CPU +cluster (see relative gains $\tau$). We can also notice that the execution times of the CG method, whether in a CPU cluster or on a GPU cluster, are +better than those the GMRES method for solving large symmetric linear systems. In fact, the CG method is characterized by a better convergence rate +and a shorter execution time of an iteration than those of the GMRES method. Moreover, an iteration of the parallel GMRES method requires more data +exchanges between computing nodes compared to the parallel CG method. + +\begin{table} +\centering +\begin{tabular}{|c|c|c|c|} +\hline +{\bf Matrix type} & {\bf Matrix name} & {\bf \# nnz} & {\bf Bandwidth} \\ \hline \hline + +\multirow{6}{*}{Symmetric} & 2cubes\_sphere & $413,703,602$ & $198,836$ \\ + + & ecology2 & $124,948,019$ & $2,002$ \\ + + & finan512 & $278,175,945$ & $123,900$ \\ + + & G3\_circuit & $125,262,292$ & $1,891,887$ \\ + + & shallow\_water2 & $100,235,292$ & $62,806$ \\ + + & thermal2 & $175,300,284$ & $2,421,285$ \\ \hline \hline + +\multirow{6}{*}{Nonsymmetric} & cage13 & $435,770,480$ & $352,566$ \\ + + & crashbasis & $409,291,236$ & $200,203$ \\ + + & FEM\_3D\_thermal2 & $595,266,787$ & $206,029$ \\ + + & language & $76,912,824$ & $398,626$ \\ + + & poli\_large & $53,322,580$ & $15,576$ \\ + + & torso3 & $433,795,264$ & $328,757$ \\ \hline +\end{tabular} +\vspace{0.5cm} +\caption{Main characteristics of sparse banded matrices generated from those of the Davis's collection.} +\label{tab:04} +\end{table} + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|c|} +\hline +{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$ \\ \hline \hline + +2cubes\_sphere & $1.625s$ & $0.401s$ & $4.05$ & $14$ & $5.73e$-$11$ & $5.20e$-$18$ \\ + +ecology2 & $0.856s$ & $0.103s$ & $8.27$ & $15$ & $3.75e$-$10$ & $1.11e$-$16$ \\ + +finan512 & $1.210s$ & $0.354s$ & $3.42$ & $14$ & $1.04e$-$10$ & $2.77e$-$16$ \\ + +G3\_circuit & $1.346s$ & $0.263s$ & $5.12$ & $17$ & $1.10e$-$10$ & $5.55e$-$16$ \\ + +shallow\_water2 & $0.397s$ & $0.055s$ & $7.23$ & $7$ & $3.43e$-$15$ & $5.17e$-$26$ \\ + +thermal2 & $1.411s$ & $0.244s$ & $5.78$ & $16$ & $1.67e$-$09$ & $3.88e$-$16$ \\ \hline +\end{tabular} +\caption{Performances of the parallel CG method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. +on a cluster of 12 GPUs.} +\label{tab:05} +\end{center} +\end{table} + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|c|} +\hline +{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$ \\ \hline \hline + +2cubes\_sphere & $3.597s$ & $0.514s$ & $6.99$ & $21$ & $2.11e$-$14$ & $8.67e$-$18$ \\ + +ecology2 & $2.549s$ & $0.288s$ & $8.83$ & $21$ & $4.88e$-$13$ & $2.08e$-$14$ \\ + +finan512 & $2.660s$ & $0.377s$ & $7.05$ & $17$ & $3.22e$-$12$ & $8.82e$-$14$ \\ + +G3\_circuit & $3.139s$ & $0.480s$ & $6.53$ & $22$ & $1.04e$-$12$ & $5.00e$-$15$ \\ + +shallow\_water2 & $2.195s$ & $0.253s$ & $8.68$ & $17$ & $5.54e$-$21$ & $7.92e$-$24$ \\ + +thermal2 & $3.206s$ & $0.463s$ & $6.93$ & $21$ & $8.89e$-$12$ & $3.33e$-$16$ \\ \hline \hline + +cage13 & $5.560s$ & $0.663s$ & $8.39$ & $26$ & $3.29e$-$11$ & $1.59e$-$14$ \\ + +crashbasis & $25.802s$ & $3.511s$ & $7.35$ & $135$ & $6.81e$-$11$ & $4.61e$-$15$ \\ + +FEM\_3D\_thermal2 & $13.281s$ & $1.572s$ & $8.45$ & $64$ & $3.88e$-$09$ & $1.82e$-$12$ \\ + +language & $12.553s$ & $1.760s$ & $7.13$ & $89$ & $2.11e$-$10$ & $1.60e$-$10$ \\ + +poli\_large & $8.515s$ & $1.053s$ & $8.09$ & $69$ & $5.05e$-$11$ & $6.59e$-$12$ \\ + +torso3 & $31.463s$ & $3.681s$ & $8.55$ & $175$ & $2.69e$-$10$ & $2.66e$-$14$ \\ \hline +\end{tabular} +\caption{Performances of the parallel GMRES method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. +on a cluster of 12 GPUs.} +\label{tab:06} +\end{center} +\end{table} + + +%%--------------------------%% +%% SECTION 5 %% +%%--------------------------%% +\section{Hypergraph partitioning} +\label{sec:05} +In this section, we present the performances of both parallel CG and GMRES solvers for solving linear systems associated to sparse matrices having +large bandwidths. Indeed, we are interested on sparse matrices having the nonzero values distributed along their bandwidths. + +We have developed in C programming language a generator of large sparse matrices having five bands distributed along their bandwidths (see Figure~\ref{fig:07}). +The principle of this generator is equivalent to that in Section~\ref{sec:04}. However, the copies performed on the initial matrix (chosen from the +Davis's collection) are placed on the main diagonal and on four off-diagonals, two on the right and two on the left of the main diagonal. Figure~\ref{fig:07} +shows an example of a generation of a sparse five-bands matrix by four computing nodes. Table~\ref{tab:07} shows the main characteristics of sparse +five-bands matrices generated from those presented in Table~\ref{tab:01} and associated to linear systems of $25$ million of unknown values. + +\begin{figure}[!h] +\centerline{\includegraphics[scale=0.23]{Chapters/chapter12/figures/generation_1}} +\caption{Parallel generation of a large sparse five-bands matrix by four computing nodes.} +\label{fig:07} +\end{figure} + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|c|c|} +\hline +{\bf Matrix type} & {\bf Matrix name} & {\bf \# nnz} & {\bf Bandwidth} \\ \hline \hline + +\multirow{6}{*}{Symmetric} & 2cubes\_sphere & $829,082,728$ & $24,999,999$ \\ + + & ecology2 & $254,892,056$ & $25,000,000$ \\ + + & finan512 & $556,982,339$ & $24,999,973$ \\ + + & G3\_circuit & $257,982,646$ & $25,000,000$ \\ + + & shallow\_water2 & $200,798,268$ & $25,000,000$ \\ + + & thermal2 & $359,340,179$ & $24,999,998$ \\ \hline \hline + +\multirow{6}{*}{Nonsymmetric} & cage13 & $879,063,379$ & $24,999,998$ \\ + + & crashbasis & $820,373,286$ & $24,999,803$ \\ + + & FEM\_3D\_thermal2 & $1,194,012,703$ & $24,999,998$ \\ + + & language & $155,261,826$ & $24,999,492$ \\ + + & poli\_large & $106,680,819$ & $25,000,000$ \\ + + & torso3 & $872,029,998$ & $25,000,000$\\ \hline +\end{tabular} +\caption{Main characteristics of sparse five-bands matrices generated from those of the Davis's collection.} +\label{tab:07} +\end{center} +\end{table} + + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|c|} +\hline +{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$ \\ \hline \hline + +2cubes\_sphere & $6.041s$ & $3.338s$ & $1.81$ & $30$ & $6.77e$-$11$ & $3.25e$-$19$ \\ + +ecology2 & $1.404s$ & $1.301s$ & $1.08$ & $13$ & $5.22e$-$11$ & $2.17e$-$18$ \\ + +finan512 & $1.822s$ & $1.299s$ & $1.40$ & $12$ & $3.52e$-$11$ & $3.47e$-$18$ \\ + +G3\_circuit & $2.331s$ & $2.129s$ & $1.09$ & $15$ & $1.36e$-$11$ & $5.20e$-$18$ \\ + +shallow\_water2 & $0.541s$ & $0.504s$ & $1.07$ & $6$ & $2.12e$-$16$ & $5.05e$-$28$ \\ + +thermal2 & $2.549s$ & $1.705s$ & $1.49$ & $14$ & $2.36e$-$10$ & $5.20e$-$18$ \\ \hline +\end{tabular} +\caption{Performances of parallel CG solver for solving linear systems associated to sparse five-bands matrices +on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs} +\label{tab:08} +\end{center} +\end{table} + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|c|} +\hline +{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$ \\ \hline \hline + +2cubes\_sphere & $15.963s$ & $7.250s$ & $2.20$ & $58$ & $6.23e$-$16$ & $3.25e$-$19$ \\ + +ecology2 & $3.549s$ & $2.176s$ & $1.63$ & $21$ & $4.78e$-$15$ & $1.06e$-$15$ \\ + +finan512 & $3.862s$ & $1.934s$ & $1.99$ & $17$ & $3.21e$-$14$ & $8.43e$-$17$ \\ + +G3\_circuit & $4.636s$ & $2.811s$ & $1.65$ & $22$ & $1.08e$-$14$ & $1.77e$-$16$ \\ + +shallow\_water2 & $2.738s$ & $1.539s$ & $1.78$ & $17$ & $5.54e$-$23$ & $3.82e$-$26$ \\ + +thermal2 & $5.017s$ & $2.587s$ & $1.94$ & $21$ & $8.25e$-$14$ & $4.34e$-$18$ \\ \hline \hline + +cage13 & $9.315s$ & $3.227s$ & $2.89$ & $26$ & $3.38e$-$13$ & $2.08e$-$16$ \\ + +crashbasis & $35.980s$ & $14.770s$ & $2.43$ & $127$ & $1.17e$-$12$ & $1.56e$-$17$ \\ + +FEM\_3D\_thermal2 & $24.611s$ & $7.749s$ & $3.17$ & $64$ & $3.87e$-$11$ & $2.84e$-$14$ \\ + +language & $16.859s$ & $9.697s$ & $1.74$ & $89$ & $2.17e$-$12$ & $1.70e$-$12$ \\ + +poli\_large & $10.200s$ & $6.534s$ & $1.56$ & $69$ & $5.14e$-$13$ & $1.63e$-$13$ \\ + +torso3 & $49.074s$ & $19.397s$ & $2.53$ & $175$ & $2.69e$-$12$ & $2.77e$-$16$ \\ \hline +\end{tabular} +\caption{Performances of parallel GMRES solver for solving linear systems associated to sparse five-bands matrices +on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs} +\label{tab:09} +\end{center} +\end{table} + +Tables~\ref{tab:08} and~\ref{tab:09} shows the performaces of the parallel CG and GMRES solvers, respectively, obtained on +a cluster of $24$ CPU cores and on a cluster of $12$ GPUs. The linear systems solved in these tables are associated to the +sparse five-bands matrices presented on Table~\ref{tab:07}. We can notice from both Tables~\ref{tab:08} and~\ref{tab:09} that +using a GPU cluster is not efficient for solving these kind of sparse linear systems. We can see that the execution times obtained +on the GPU cluster are almost equivalent to those obtained on the CPU cluster (see the relative gains presented in column~$4$ +of each table). This is due to the large number of communications necessary to synchronize the computations over the cluster. +Indeed, the naive partitioning, row-by-row or column-by-column, of sparse matrices having large bandwidths can link a computing +node to many neighbors and then generate a large number of data dependencies between these computing nodes in the cluster. + +Therefore, we have chosen to use a hypergraph partitioning method, which is well-suited to numerous kinds of sparse matrices~\cite{ref11}. +Indeed, it can well model the communications between the computing nodes, particularly in the case of nonsymmetric and irregular +matrices, and it gives good reduction of the total communication volume. In contrast, it is an expensive operation in terms of +execution time and memory space. + +The sparse matrix $A$ of the linear system to be solved is modeled as a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$ as +follows: +\begin{itemize*} +\item each matrix row $\{i\}_{0\leq i<n}$ corresponds to a vertex $v_i\in\mathcal{V}$ and, +\item each matrix column $\{j\}_{0\leq j<n}$ corresponds to a hyperedge $e_j\in\mathcal{E}$, where: +\begin{equation} +\forall a_{ij} \neq 0 \mbox{~is a nonzero value of matrix~} A \mbox{~:~} v_i \in pins[e_j], +\end{equation} +\item $w_i$ is the weight of vertex $v_i$ and, +\item $c_j$ is the cost of hyperedge $e_j$. +\end{itemize*} +A $K$-way partitioning of a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$ is defined as $\mathcal{P}=\{\mathcal{V}_1,\ldots,\mathcal{V}_K\}$ +a set of pairwise disjoint non-empty subsets (or parts) of the vertex set $\mathcal{V}$, so that each subset is attributed to a computing node. +Figure~\ref{fig:08} shows an example of the hypergraph model of a $(9\times 9)$ sparse matrix in three parts. The circles and squares correspond, +respectively, to the vertices and hyperedges of the hypergraph. The solid squares define the cut hyperedges connecting at least two different parts. +The connectivity $\lambda_j$ of a cut hyperedge $e_j$ denotes the number of different parts spanned by $e_j$. + +\begin{figure} +\centerline{\includegraphics[scale=0.5]{Chapters/chapter12/figures/hypergraph}} +\caption{An example of the hypergraph partitioning of a sparse matrix decomposed between three computing nodes.} +\label{fig:08} +\end{figure} + +The cut hyperedges model the total communication volume between the different computing nodes in the cluster, necessary to perform the parallel SpMV +multiplication. Indeed, each hyperedge $e_j$ defines a set of atomic computations $b_i\leftarrow b_i+a_{ij}x_j$, $0\leq i,j<n$, of the SpMV multiplication +$Ax=b$ that need the $j^{th}$ unknown value of solution vector $x$. Therefore, pins of hyperedge $e_j$, $pins[e_j]$, are the set of matrix rows sharing +and requiring the same unknown value $x_j$. For example in Figure~\ref{fig:08}, hyperedge $e_9$ whose pins are: $pins[e_9]=\{v_2,v_5,v_9\}$ represents the +dependency of matrix rows $2$, $5$ and $9$ to unknown $x_9$ needed to perform in parallel the atomic operations: $b_2\leftarrow b_2+a_{29}x_9$, +$b_5\leftarrow b_5+a_{59}x_9$ and $b_9\leftarrow b_9+a_{99}x_9$. However, unknown $x_9$ is the third entry of the sub-solution vector $x$ of part (or node) $3$. +So the computing node $3$ must exchange this value with nodes $1$ and $2$, which leads to perform two communications. + +The hypergraph partitioning allows to reduce the total communication volume required to perform the parallel SpMV multiplication, while maintaining the +load balancing between the computing nodes. In fact, it allows to minimize at best the following amount: +\begin{equation} +\mathcal{X}(\mathcal{P})=\sum_{e_{j}\in\mathcal{E}_{C}}c_{j}(\lambda_{j}-1), +\end{equation} +where $\mathcal{E}_{C}$ denotes the set of the cut hyperedges coming from the hypergraph partitioning $\mathcal{P}$ and $c_j$ and $\lambda_j$ are, respectively, +the cost and the connectivity of cut hyperedge $e_j$. Moreover, it also ensures the load balancing between the $K$ parts as follows: +\begin{equation} + W_{k}\leq (1+\epsilon)W_{avg}, \hspace{0.2cm} (1\leq k\leq K) \hspace{0.2cm} \text{and} \hspace{0.2cm} (0<\epsilon<1), +\end{equation} +where $W_{k}$ is the sum of all vertex weights ($w_{i}$) in part $\mathcal{V}_{k}$, $W_{avg}$ is the average weight of all $K$ parts and $\epsilon$ is the +maximum allowed imbalanced ratio. + +The hypergraph partitioning is a NP-complete problem but software tools using heuristics are developed, for example: hMETIS~\cite{ref12}, PaToH~\cite{ref13} +and Zoltan~\cite{ref14}. Since our objective is solving large sparse linear systems, we use the parallel hypergraph partitioning which must be performed by +at least two MPI processes. It allows us to accelerate the data partitioning of large sparse matrices. For this, the hypergraph $\mathcal{H}$ must be partitioned +in $p$ (number of MPI processes) sub-hypergraphs $\mathcal{H}_k=(\mathcal{V}_k,\mathcal{E}_k)$, $0\leq k<p$, and then we performed the parallel hypergraph +partitioning method using some functions of the MPI library between the $p$ processes. + +Tables~\ref{tab:10} and~\ref{tab:11} shows the performances of the parallel CG and GMRES solvers, respectively, using the hypergraph partitioning for solving +large linear systems associated to the sparse five-bands matrices presented in Table~\ref{tab:07}. For these experimental tests, we have applied the parallel +hypergraph partitioning~\cite{ref15} developed in Zoltan tool~\ref{ref14}. We have initialized the parameters of the partitioning as follows: +\begin{itemize*} +\item the weight $w_{i}$ of each vertex $v_{j}\in\mathcal{V}$ is set to the number of nonzero values on matrix row $i$, +\item for the sake of simplicity, the cost $c_{j}$ of each hyperedge $e_{j}\in\mathcal{E}$ is fixed to $1$, +\item the maximum imbalanced load ratio $\epsilon$ is limited to $10\%$.\\ +\end{itemize*} + +\begin{table}[!h] +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline +{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{Gains \%}$ \\ \hline \hline + +2cubes\_sphere & $5.935s$ & $1.213s$ & $4.89$ & $63.66\%$ \\ + +ecology2 & $1.093s$ & $0.136s$ & $8.00$ & $89.55\%$ \\ + +finan512 & $1.762s$ & $0.475s$ & $3.71$ & $63.43\%$ \\ + +G3\_circuit & $2.095s$ & $0.558s$ & $3.76$ & $73.79\%$ \\ + +shallow\_water2 & $0.498s$ & $0.068s$ & $7.31$ & $86.51\%$ \\ + +thermal2 & $1.889s$ & $0.348s$ & $5.43$ & $79.59\%$ \\ \hline +\end{tabular} +\caption{Performances of the parallel CG solver using hypergraph partitioning for solving linear systems associated to +sparse five-bands matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPU.} +\label{tab:10} +\end{center} +\end{table} + +\begin{table}[!h] +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline +{\bf Matrix} & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{Gains \%}$ \\ \hline \hline + +2cubes\_sphere & $16.430s$ & $2.840s$ & $5.78$ & $60.83\%$ \\ + +ecology2 & $3.152s$ & $0.367s$ & $8.59$ & $83.13\%$ \\ + +finan512 & $3.672s$ & $0.723s$ & $5.08$ & $62.62\%$ \\ + +G3\_circuit & $4.468s$ & $0.971s$ & $4.60$ & $65.46\%$ \\ + +shallow\_water2 & $2.647s$ & $0.312s$ & $8.48$ & $79.73\%$ \\ + +thermal2 & $4.190s$ & $0.666s$ & $6.29$ & $74.25\%$ \\ \hline \hline + +cage13 & $8.077s$ & $1.584s$ & $5.10$ & $50.91\%$ \\ + +crashbasis & $35.173s$ & $5.546s$ & $6.34$ & $62.43\%$ \\ + +FEM\_3D\_thermal2 & $24.825s$ & $3.113s$ & $7.97$ & $59.83\%$ \\ + +language & $16.706s$ & $2.522s$ & $6.62$ & $73.99\%$ \\ + +poli\_large & $12.715s$ & $3.989s$ & $3.19$ & $38.95\%$ \\ + +torso3 & $48.459s$ & $6.234s$ & $7.77$ & $67.86\%$ \\ \hline +\end{tabular} +\caption{Performances of the parallel GMRES solver using hypergraph partitioning for solving linear systems associated to +sparse five-bands matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPU.} +\label{tab:11} +\end{center} +\end{table} + +We can notice from both Tables~\ref{tab:10} and~\ref{tab:11} that the hypergraph partitioning has improved the performances of both +parallel CG and GMRES algorithms for solving large linear systems associated to matrices having large bandwidths. The execution times +on the GPU cluster of both parallel solvers are significantly improved compared to those obtained by using the partitioning row-by-row. +For these examples of sparse matrices, the execution times of CG and GMRES solvers are reduced on average about $76\%$ and $65\%$ respectively +(see column~$5$ of each table) compared to those obtained in Tables~\ref{tab:08} and~\ref{tab:09}. + +In fact, the hypergraph partitioning applied to sparse matrices having large bandwidths allows to reduce the total communication volume +necessary to synchronize the computations between the computing nodes in the GPU cluster. Table~\ref{tab:12} presents, for each sparse +matrix, the total communication volume between $12$ GPU computing nodes obtained by using the partitioning row-by-row (column~$2$), the +total communication volume obtained by using the hypergraph partitioning (column~$3$) and the execution times in minutes of the hypergraph +partitioning operation performed by $12$ MPI processes (column~$4$). The total communication volume defines the total number of the vector +elements exchanged by the computing nodes. Then, Table~\ref{tab:12} shows that the hypergraph partitioning method can split the sparse +matrix so as to minimize the data dependencies between the computing nodes and thus to reduce the total communication volume. + + +\begin{table}[!h] +\begin{center} +\begin{tabular}{|c|c|c|c|} +\hline +\multirow{4}{*}{\bf Matrix} & {\bf Total comms.} & {\bf Total comms.} & {\bf Execution} \\ + & {\bf volume without} & {\bf volume with} & {\bf trime} \\ + & {\bf hypergraph} & {\bf hypergraph } & {\bf of the parti.} \\ + & {\bf parti.} & {\bf parti.} & {\bf in minutes}\\ \hline \hline + +2cubes\_sphere & $25,360,543$ & $240,679$ & $68.98$ \\ + +ecology2 & $26,044,002$ & $73,021$ & $4.92$ \\ + +finan512 & $26,087,431$ & $900,729$ & $33.72$ \\ + +G3\_circuit & $31,912,003$ & $5,366,774$ & $11.63$ \\ + +shallow\_water2 & $25,105,108$ & $60,899$ & $5.06$ \\ + +thermal2 & $30,012,846$ & $1,077,921$ & $17.88$ \\ \hline \hline + +cage13 & $28,254,282$ & $3,845,440$ & $196.45$ \\ + +crashbasis & $29,020,060$ & $2,401,876$ & $33.39$ \\ + +FEM\_3D\_thermal2 & $25,263,767$ & $250,105$ & $49.89$ \\ + +language & $27,291,486$ & $1,537,835$ & $9.07$ \\ + +poli\_large & $25,053,554$ & $7,388,883$ & $5.92$ \\ + +torso3 & $25,682,514$ & $613,250$ & $61.51$ \\ \hline +\end{tabular} +\caption{The total communication volume between 12 GPU computing nodes without and with the hypergraph partitioning method.} +\label{tab:12} +\end{center} +\end{table} + +Nevertheless, as we can see from the fourth column of Table~\ref{tab:12}, the hypergraph partitioning takes longer compared +to the execution times of the resolutions. As previously mentioned, the hypergraph partitioning method is less efficient in +terms of memory consumption and partitioning time than its graph counterpart, but the hypergraph well models the nonsymmetric +and irregular problems. So for the applications which often use the same sparse matrices, we can perform the hypergraph partitioning +on these matrices only once for each and then, we save the traces of these partitionings in files to be reused several times. +Therefore, this allows us to avoid the partitioning of the sparse matrices at each resolution of the linear systems. + +However, the most important performance parameter is the scalability of the parallel CG and GMRES solvers on a GPU cluster. +Particularly, we have taken into account the weak-scaling of both parallel algorithms on a cluster of one to 12 GPU computing +nodes. We have performed a set of experimental tests on both matrix structures: band matrices and five-bands matrices. The +sparse matrices of tests are generated from the symmetric sparse matrix {\it thermal2} chosen from the Davis's collection. +Figures~\ref{fig:09}-$(a)$ and~\ref{fig:09}-$(b)$ show the execution times of both parallel methods for solving large linear +systems associated to band matrices and those associated to five-bands matrices, respectively. The size of a sparse sub-matrix +per computing node, for each matrix structure, is fixed as follows: +\begin{itemize*} +\item band matrix: $15$ million of rows and $105,166,557$ of nonzero values, +\item five-bands matrix: $5$ million of rows and $78,714,492$ of nonzero values. +\end{itemize*} +We can see from these figures that both parallel solvers are quite scalable on a GPU cluster. Indeed, the execution times remains +almost constant while the size of the size of the sparse linear systems to be solved increases proportionally with the number of +the GPU computing nodes. This means that the communication cost is relatively constant regardless of the number the computing nodes +in the GPU cluster. + +\begin{figure} +\centering +\begin{tabular}{c} +\includegraphics[scale=0.7]{Chapters/chapter12/figures/scale_band} \\ +\small{(a) Sparse band matrices} \\ +\\ +\includegraphics[scale=0.7]{Chapters/chapter12/figures/scale_5band} \\ +\small{(b) Sparse five-bands matrices} +\end{tabular} +\caption{Weak-scaling of the parallel CG and GMRES solvers on a GPU cluster for solving large sparse linear systems.} +\label{fig:09} +\end{figure} + +%%--------------------------%% +%% SECTION 6 %% +%%--------------------------%% +\section{Conclusion} +\label{sec:06} +In this chapter, we have aimed at harnessing the computing power of a cluster of GPUs for solving large sparse linear systems. +For this, we have used two Krylov sub-space iterative methods: the CG and GMRES methods. The first method is well-known to its +efficiency for solving symmetric linear systems and the second one is used, particularly, for solving nonsymmetric linear systems. + +We have presentend the parallel implementation of both iterative methods on a GPU cluster. Particularly, the operations dealing with +the vectors and/or matrices, of these methods, are parallelized between the different GPU computing nodes of the cluster. Indeed, +the data-parallel vector operations are accelerated by GPUs and the communications required to synchronize the parallel computations +are carried out by CPU cores. For this, we have used a heterogeneous CUDA/MPI programming to implement the parallel iterative +algorithms. + +In the experimental tests, we have shown that using a GPU cluster is efficient for solving linear systems associated to very large +sparse matrices. The experimental results, obtained in the present chapter, showed that a cluster of $12$ GPUs is about $7$ +times faster than a cluster of $24$ CPU cores for solving large sparse linear systems of $25$ million unknown values. This is due +to the GPU ability to compute the data-parallel operations faster than the CPUs. However, we have shown that solving linear systems +associated to matrices having large bandwidths uses many communications to synchronize the computations of GPUs, which slow down +even more the resolution. Moreover, there two kinds of communications: between a CPU and its GPU and between CPUs of the computing +nodes, such that the first ones are the slowest communications on a GPU cluster. So, we have proposed to use the hypergraph partitioning +instead of the row-by-row partitioning. This allows to minimize the data dependencies between the GPU computing nodes and thus to +reduce the total communication volume. The experimental results showed that using the hypergraph partitioning technique improve the +execution times on average of $76\%$ to the CG method and of $65\%$ to the GMRES method on a cluster of $12$ GPUs. + +In the recent GPU hardware and software architectures, the GPU-Direct system with CUDA version 5.0 is used so that two GPUs located on +the same node or on distant nodes can communicate between them directly without CPUs. 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file mode 100755 index 0000000..635181d --- /dev/null +++ b/BookGPU/Chapters/chapter13/ch13.tex @@ -0,0 +1,710 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% CHAPTER 13 %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\chapterauthor{}{} +\newcommand{\scalprod}[2]% +{\ensuremath{\langle #1 \, , #2 \rangle}} +\chapter{Solving sparse nonlinear systems of obstacle problems on GPU clusters} + +%%--------------------------%% +%% SECTION 1 %% +%%--------------------------%% +\section{Introduction} +\label{sec:01} +The obstacle problem is one kind of free boundary problems. It allows to model, for example, an elastic membrane covering a solid obstacle. +In this case, the objective is to find an equilibrium position of this membrane constrained to be above the obstacle and which tends to minimize +its surface and/or its energy. The study of such problems occurs in many applications, for example: fluid mechanics, bio-mathematics (tumour growth +process) or financial mathematics (American or European option pricing). + +In this chapter, we focus on solutions of large obstacle problems defined in a three-dimensional domain. Particularly, the present study consists +in solving large nonlinear systems derived from the spatial discretization of these problems. Owing to the great size of such systems, in order to +reduce computation times, we proceed by solving them by parallel synchronous or asynchronous iterative algorithms. Moreover, we aim at harnessing +the computing power of GPUs to accelerate computations of these parallel algorithms. For this, we use an iterative method involving a projection +on a convex set, which is: the projected Richardson method. We choose this method among other iterative methods because it is easy to implement on +parallel computers and easy to adapt to GPU architectures. + +In Section~\ref{sec:02}, we present the mathematical model of obstacle problems then, in Section~\ref{sec:03}, we describe the general principle of +the parallel projected Richardson method. Next, in Section~\ref{sec:04}, we give the main key points of the parallel implementation of both synchronous +and asynchronous algorithms of the projected Richardson method on a GPU cluster. In Section~\ref{sec:05}, we present the performances of both parallel +algorithms obtained from simulations carried out on a CPU and GPU clusters. Finally, in Section~\ref{sec:06}, we use the read-black ordering technique +to improve the convergence and, thus, the execution times of the parallel projected Richardson algorithms on the GPU cluster. + + +%%--------------------------%% +%% SECTION 2 %% +%%--------------------------%% +\section{Obstacle problems} +\label{sec:02} +In this section, we present the mathematical model of obstacle problems defined in a three-dimensional domain. +This model is based on that presented in~\cite{ref1}. + +%%******************* +%%******************* +\subsection{Mathematical model} +\label{sec:02.01} +An obstacle problem, arising for example in mechanics or financial derivatives, consists in solving a time dependent +nonlinear equation: +\begin{equation} +\left\{ +\begin{array}{l} +\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0,\\ +(\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega,\\ +u(0,x,y,z)=u_0(x,y,z),\\ +\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega, +\end{array} +\right. +\label{eq:01} +\end{equation} +where $u_0$ is the initial condition, $c\geq 0$, $b$ and $\eta$ are physical parameters, $T$ is the final time, $u=u(t,x,y,z)$ +is an element of the solution vector $U$ to compute, $f$ is the right-hand right that could represent, for example, the external +forces, B.C. describes the boundary conditions on the boundary $\partial\Omega$ of the domain $\Omega$, $\phi$ models a constraint +imposed to $u$, $\Delta$ is the Laplacian operator, $\nabla$ is the gradient operator, a.e.w. means almost every where and ``.'' +defines the products between two scalars, a scalar and a vector or a matrix and a vector. In practice the boundary condition, +generally considered, is the Dirichlet condition (where $u$ is fixed on $\partial\Omega$) or the Neumann condition (where the +normal derivative of $u$ is fixed on $\partial\Omega$). + +The time dependent equation~(\ref{eq:01}) is numerically solved by considering an implicit or a semi-implicit time marching, +where at each time step $k$ a stationary nonlinear problem is solved: +\begin{equation} +\left\{ +\begin{array}{l} +b^t.\nabla u-\eta.\Delta u+(c+\delta).u-g\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0, \\ +(b^t.\nabla u-\eta.\Delta u+(c+\delta).u- g)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega, \\ +\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega, +\end{array} +\right. +\label{eq:02} +\end{equation} +where $\delta=\frac{1}{k}$ is the inverse of the time step $k$, $g=f+\delta u^{prev}$ and $u^{prev}$ is the solution computed at the +previous time step. + + +%%******************* +%%******************* +\subsection{Discretization} +\label{sec:02.02} +First, we note that the spatial discretization of the previous stationary problem~(\ref{eq:02}) does not provide a symmetric matrix, +because the convection-diffusion operator is not self-adjoint. Moreover, the fact that the operator is self-adjoint or not plays an +important role in the choice of the appropriate algorithm for solving nonlinear systems derived from the discretization of the obstacle +problem. Nevertheless, since the convection coefficients arising in the operator~(\ref{eq:02}) are constant, we can formulate the same +problem by an self-adjoint operator by performing a classical change of variables. Then, we can replace the stationary convection-diffusion +problem: +\begin{equation} +b^{t}.\nabla v-\eta.\Delta v+(c+\delta).v=g\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}c\geq 0\mbox{,~}\delta\geq~0, +\label{eq:03} +\end{equation} +by the following stationary diffusion operator: +\begin{equation} +-\eta.\Delta u+(\frac{\|b\|^{2}_{2}}{4\eta}+c+\delta).u=e^{-a}g=f, +\label{eq:04} +\end{equation} +where $b=\{b_{1},b_{2},b_{3}\}$, $\|b\|_{2}$ denotes the Euclidean norm of $b$ and $v=e^{-a}.u$ represents the general change of variables +such that $a=\frac{b^{t}(x,y,z)}{2\eta}$. Consequently, the numerical resolution of the diffusion problem (the self-adjoint operator~(\ref{eq:04})) +is done by optimization algorithms, in contrast, that of the convection-diffusion problem (non self-adjoint operator~(\ref{eq:03})) is +done by relaxation algorithms. In the case of our studied algorithm, the convergence is ensured by M-matrix property then, the performance +is linked to the magnitude of the spectral radius of the iteration matrix, which is independent of the condition number. + +Next, the three-dimensional domain $\Omega\subset\mathbb{R}^{3}$ is set to $\Omega=\lbrack 0,1\rbrack^{3}$ and discretized with an uniform +Cartesian mesh constituted by $M=m^3$ discretization points, where $m$ related to the spatial discretization step by $h=\frac{1}{m+1}$. This +is carried out by using a classical order 2 finite difference approximation of the Laplacian. So, the complete discretization of both stationary +boundary value problems~(\ref{eq:03}) and~(\ref{eq:04}) leads to the solution of a large discrete complementary problem of the following +form, when both Dirichlet or Neumann boundary conditions are used: +\begin{equation} +\left\{ +\begin{array}{l} +\mbox{Find~}U^{*}\in\mathbb{R}^{M}\mbox{~such~that} \\ +(A+\delta I)U^{*}-G\geq 0\mbox{,~}U^{*}\geq\bar{\Phi},\\ +((A+\delta I)U^{*}-G)^{T}(U^{*}-\bar{\Phi})=0,\\ +\end{array} +\right. +\label{eq:05} +\end{equation} +where $A$ is a matrix obtained after the spatial discretization by a finite difference method, $G$ is derived from the Euler first order implicit time +marching scheme and from the discretized right-hand side of the obstacle problem, $\delta$ is the inverse of the time step $k$ and $I$ is the identity +matrix. The matrix $A$ is symmetric when the self-adjoint operator is considered and nonsymmetric otherwise. + +According to the chosen discretization scheme of the Laplacian, $A$ is an M-matrix (irreducibly diagonal dominant, see~\cite{ref2}) and, consequently, +the matrix $(A+\delta I)$ is also an M-matrix. This property is important to the convergence of iterative methods. + + +%%--------------------------%% +%% SECTION 3 %% +%%--------------------------%% +\section{Parallel iterative method} +\label{sec:03} +Owing to the large size of the previous discrete complementary problem~(\ref{eq:05}), we will solve it by parallel synchronous or asynchronous iterative +algorithms (see~\cite{ref3,ref4,ref5}). In this chapter, we aim at harnessing the computing power of GPU clusters for solving these large nonlinear systems. +Then, we choose to use the projected Richardson iterative method for solving the diffusion problem~(\ref{eq:04}). Indeed, this method is based on the iterations +of the Jacobi method which are easy to parallelize on parallel computers and easy to adapt to GPU architectures. Then, according to the boundary value problem +formulation with a self-adjoint operator~(\ref{eq:04}), we can consider here the equivalent optimization problem and the fixed point mapping associated to +its solution. + +Assume that $E=\mathbb{R}^{M}$ is a Hilbert space, in which $\scalprod{.}{.}$ is the scalar product and $\|.\|$ its associated norm. So, the general fixed +point problem to be solved is defined as follows: +\begin{equation} +\left\{ +\begin{array}{l} +\mbox{Find~} U^{*} \in E \mbox{~such that} \\ +U^{*} = F(U^{*}), \\ +\end{array} +\right. +\label{eq:06} +\end{equation} +where $U\mapsto F(U)$ is an application from $E$ to $E$. + +Let $K$ be a closed convex set defined by: +\begin{equation} +K = \{U | U \geq \Phi \mbox{~everywhere in~} E\}, +\label{eq:07} +\end{equation} +where $\Phi$ is the discrete obstacle function. In fact, the obstacle problem~(\ref{eq:05}) is formulated as the following constrained optimization problem: +\begin{equation} +\left\{ +\begin{array}{l} +\mbox{Find~} U^{*} \in K \mbox{~such that} \\ +\forall V \in K, J(U^{*}) \leq J(V), \\ +\end{array} +\right. +\label{eq:08} +\end{equation} +where the cost function is given by: +\begin{equation} +J(U) = \frac{1}{2}\scalprod{\mathcal{A}.U}{U} - \scalprod{G}{U}, +\label{eq:09} +\end{equation} +in which $\scalprod{.}{.}$ denotes the scalar product in $E$, $\mathcal{A}=A+\delta I$ is a symmetric positive definite, $A$ is the discretization matrix +associated with the self-adjoint operator~(\ref{eq:04}) after change of variables. + +For any $U\in E$, let $P_K(U)$ be the projection of $U$ on $K$. For any $\gamma\in\mathbb{R}$, $\gamma>0$, the fixed point mapping $F_{\gamma}$ of the projected +Richardson method is defined as follows: +\begin{equation} +U^{*} = F_{\gamma}(U^{*}) = P_K(U^{*} - \gamma(\mathcal{A}.U^{*} - G)). +\label{eq:10} +\end{equation} +In order to reduce the computation time, the large optimization problem is solved in a numerical way by using a parallel asynchronous algorithm of the projected +Richardson method on the convex set $K$. Particularly, we will consider an asynchronous parallel adaptation of the projected Richardson method~\cite{ref6}. + +Let $\alpha\in\mathbb{N}$ be a positive integer. We consider that the space $E=\displaystyle\prod_{i=1}^{\alpha} E_i$ is a product of $\alpha$ subspaces $E_i$ +where $i\in\{1,\ldots,\alpha\}$. Note that $E_i=\mathbb{R}^{m_i}$, where $\displaystyle\sum_{i=1}^{\alpha} m_{i}=M$, is also a Hilbert space in which $\scalprod{.}{.}_i$ +denotes the scalar product and $|.|_i$ the associated norm, for all $i\in\{1,\ldots,\alpha\}$. Then, for all $u,v\in E$, $\scalprod{u}{v}=\displaystyle\sum_{i=1}^{\alpha}\scalprod{u_i}{v_i}_i$ +is the scalar product on $E$. + +Let $U\in E$, we consider the following decomposition of $U$ and the corresponding decomposition of $F_\gamma$ into $\alpha$ blocks: +\begin{equation} +\begin{array}{rcl} +U & = & (U_1,\ldots,U_{\alpha}), \\ +F_{\gamma}(U) & = & (F_{1,\gamma}(U),\ldots,F_{\alpha,\gamma}(U)). \\ +\end{array} +\label{eq:11} +\end{equation} +Assume that the convex set $K=\displaystyle\prod_{i=1}^{\alpha}K_{i}$, such that $\forall i\in\{1,\ldots,\alpha\},K_i\subset E_i$ and $K_i$ is a closed convex set. +Let also $G=(G_1,\ldots,G_{\alpha})\in E$ and, for any $U\in E$, $P_K(U)=(P_{K_1}(U_1),\ldots,P_{K_{\alpha}}(U_{\alpha}))$ is the projection of $U$ on $K$ where $\forall i\in\{1,\ldots,\alpha\},P_{K_i}$ +is the projector from $E_i$ onto $K_i$. So, the fixed point mapping of the projected Richardson method~(\ref{eq:10}) can be written in the following way: +\begin{equation} +\forall U\in E\mbox{,~}\forall i\in\{1,\ldots,\alpha\}\mbox{,~}F_{i,\gamma}(U) = P_{K_i}(U_i - \gamma(\mathcal{A}_i.U - G_i)). +\label{eq:12} +\end{equation} +Note that $\displaystyle\mathcal{A}_i.U= \sum_{j=1}^{\alpha}\mathcal{A}_{i,j}.U_j$, where $\mathcal{A}_{i,j}$ denote block matrices of $\mathcal{A}$. + +The parallel asynchronous iterations of the projected Richardson method for solving the obstacle problem~(\ref{eq:08}) are defined as follows: let $U^0\in E,U^0=(U^0_1,\ldots,U^0_\alpha)$ be +the initial solution, then for all $p\in\mathbb{N}$, the iterate $U^{p+1}=(U^{p+1}_1,\ldots,U^{p+1}_{\alpha})$ is recursively defined by: +\begin{equation} +U_i^{p+1} = +\left\{ +\begin{array}{l} +F_{i,\gamma}(U_1^{\rho_1(p)}, \ldots, U_{\alpha}^{\rho_{\alpha}(p)}) \mbox{~if~} i\in s(p), \\ +U_i^p \mbox{~otherwise}, \\ +\end{array} +\right. +\label{eq:13} +\end{equation} +where +\begin{equation} +\left\{ +\begin{array}{l} +\forall p\in\mathbb{N}, s(p)\subset\{1,\ldots,\alpha\}\mbox{~and~} s(p)\ne\emptyset, \\ +\forall i\in\{1,\ldots,\alpha\},\{p \ | \ i \in s(p)\}\mbox{~is denombrable}, +\end{array} +\right. +\label{eq:14} +\end{equation} +and $\forall j\in\{1,\ldots,\alpha\}$, +\begin{equation} +\left\{ +\begin{array}{l} +\forall p\in\mathbb{N}, \rho_j(p)\in\mathbb{N}, 0\leq\rho_j(p)\leq p\mbox{~and~}\rho_j(p)=p\mbox{~if~} j\in s(p),\\ +\displaystyle\lim_{p\to\infty}\rho_j(p) = +\infty.\\ +\end{array} +\right. +\label{eq:15} +\end{equation} + +The previous asynchronous scheme of the projected Richardson method models computations that are carried out in parallel +without order nor synchronization (according to the behavior of the parallel iterative method) and describes a subdomain +method without overlapping. It is a general model that takes into account all possible situations of parallel computations +and non-blocking message passing. So, the synchronous iterative scheme is defined by: +\begin{equation} +\forall j\in\{1,\ldots,\alpha\} \mbox{,~} \forall p\in\mathbb{N} \mbox{,~} \rho_j(p)=p. +\label{eq:16} +\end{equation} +The values of $s(p)$ and $\rho_j(p)$ are defined dynamically and not explicitly by the parallel asynchronous or synchronous +execution of the algorithm. Particularly, it enables one to consider distributed computations whereby processors compute at +their own pace according to their intrinsic characteristics and computational load. The parallelism between the processors is +well described by the set $s(p)$ which contains at each step $p$ the index of the components relaxed by each processor on a +parallel way while the use of delayed components in~(\ref{eq:13}) permits one to model nondeterministic behavior and does not +imply inefficiency of the considered distributed scheme of computation. Note that, according to~\cite{ref7}, theoretically, +each component of the vector must be relaxed an infinity of time. The choice of the relaxed components to be used in the +computational process may be guided by any criterion and, in particular, a natural criterion is to pick-up the most recently +available values of the components computed by the processors. Furthermore, the asynchronous iterations are implemented by +means of non-blocking MPI communication subroutines (asynchronous communications). + +The important property ensuring the convergence of the parallel projected Richardson method, both synchronous and asynchronous +algorithms, is the fact that $\mathcal{A}$ is an M-matrix. Moreover, the convergence proceeds from a result of~\cite{ref6}. +Indeed, there exists a value $\gamma_0>0$, such that $\forall\gamma\in ]0,\gamma_0[$, the parallel iterations~(\ref{eq:13}), +(\ref{eq:14}) and~(\ref{eq:15}), associated to the fixed point mapping $F_\gamma$~(\ref{eq:12}), converge to the unique solution +$U^{*}$ of the discretized problem. + + +%%--------------------------%% +%% SECTION 4 %% +%%--------------------------%% +\section{Parallel implementation on a GPU cluster} +\label{sec:04} +In this section, we give the main key points of the parallel implementation of the projected Richardson method, both synchronous +and asynchronous versions, on a GPU cluster, for solving the nonlinear systems derived from the discretization of large obstacle +problems. More precisely, each nonlinear system is solved iteratively using the whole cluster. We use a heteregeneous CUDA/MPI +programming. Indeed, the communication of data, at each iteration between the GPU computing nodes, can be either synchronous +or asynchronous using the MPI communication subroutines, whereas inside each GPU node, a CUDA parallelization is performed. + +\begin{figure}[!h] +\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitCPU}} +\caption{Data partitioning of a problem to be solved among $S=3\times 4$ computing nodes.} +\label{fig:01} +\end{figure} + +Let $S$ denote the number of computing nodes on the GPU cluster, where a computing node is composed of CPU core holding one MPI +process and a GPU card. So, before starting computations, the obstacle problem of size $(NX\times NY\times NZ)$ is split into $S$ +parallelepipedic sub-problems, each for a node (MPI process, GPU), as is shown in Figure~\ref{fig:01}. Indeed, the $NY$ and $NZ$ +dimensions (according to the $y$ and $z$ axises) of the three-dimensional problem are, respectively, split into $Sy$ and $Sz$ parts, +such that $S=Sy\times Sz$. In this case, each computing node has at most four neighboring nodes. This kind of the data partitioning +reduces the data exchanges at subdomain boundaries compared to a naive $z$-axis-wise partitioning. + +\begin{algorithm}[!t] +%\SetLine +%\linesnumbered +Initialization of the parameters of the sub-problem\; +Allocate and fill the data in the global memory GPU\; +\For{$i=1$ {\bf to} $NbSteps$}{ + $G = \frac{1}{k}.U^0 + F$\; + Solve($A$, $U^0$, $G$, $U$, $\varepsilon$, $MaxRelax$)\; + $U^0 = U$\; +} +Copy the solution $U$ back from GPU memory\; +\caption{Parallel solving of the obstacle problem on a GPU cluster} +\label{alg:01} +\end{algorithm} + +All the computing nodes of the GPU cluster execute in parallel the same Algorithm~\ref{alg:01} but on different three-dimensional +sub-problems of size $(NX\times ny\times nz)$. This algorithm gives the main key points for solving an obstacle problem defined in +a three-dimensional domain, where $A$ is the discretization matrix, $G$ is the right-hand side and $U$ is the solution vector. After +the initialization step, all the data generated from the partitioning operation are copied from the CPU memories to the GPU global +memories, to be processed on the GPUs. Next, the algorithm uses $NbSteps$ time steps to solve the global obstacle problem. In fact, +it uses a parallel algorithm adapted to GPUs of the projected Richardson iterative method for solving the nonlinear systems of the +obstacle problem. This function is defined by {\it Solve()} in Algorithm~\ref{alg:01}. At every time step, the initial guess $U^0$ +for the iterative algorithm is set to the solution found at the previous time step. Moreover, the right-hand side $G$ is computed +as follows: \[G = \frac{1}{k}.U^{prev} + F\] where $k$ is the time step, $U^{prev}$ is the solution computed in the previous time +step and each element $f(x, y, z)$ of the vector $F$ is computed as follows: +\begin{equation} +f(x,y,z)=\cos(2\pi x)\cdot\cos(4\pi y)\cdot\cos(6\pi z). +\label{eq:18} +\end{equation} +Finally, the solution $U$ of the obstacle problem is copied back from the GPU global memories to the CPU memories. We use the +communication subroutines of the CUBLAS library~\cite{ref8} (CUDA Basic Linear Algebra Subroutines) for the memory allocations in +the GPU (\verb+cublasAlloc()+) and the data transfers between the CPU and its GPU: \verb+cublasSetVector()+ and \verb+cublasGetVector()+. + +\begin{algorithm}[!t] +% \SetLine +% \linesnumbered + $p = 0$\; + $conv = false$\; + $U = U^{0}$\; + \Repeat{$(conv=true)$}{ + Determine\_Bordering\_Vector\_Elements($U$)\; + Compute\_New\_Vector\_Elements($A$, $G$, $U$)\; + $tmp = U^{0} - U$\; + $error = \|tmp\|_{2}$\; + $U^{0} = U$\; + $p = p + 1$\; + $conv$ = Convergence($error$, $p$, $\varepsilon$, $MaxRelax$)\; + } +\caption{Parallel iterative solving of the nonlinear systems on a GPU cluster ($Solve()$ function)} +\label{alg:02} +\end{algorithm} + +As many other iterative methods, the algorithm of the projected Richardson method is based on algebraic functions operating on vectors +and/or matrices, which are more efficient on parallel computers when they work on large vectors. Its parallel implementation on the GPU +cluster is carried out so that the GPUs execute the vector operations as kernels and the CPUs execute the serial codes, supervise the +kernel executions and the data exchanges with the neighboring nodes and supply the GPUs with data. Algorithm~\ref{alg:02} shows the +main key points of the parallel iterative algorithm (function $Solve()$ in Algorithm~\ref{alg:01}). All the vector operations inside +the main loop ({\bf repeat} ... {\bf until}) are executed by the GPU. We use the following functions of the CUBLAS library: +\begin{itemize} +\item \verb+cublasDaxpy()+ to compute the difference between the solution vectors $U^{p}$ and $U^{p+1}$ computed in two successive relaxations +$p$ and $p+1$ (line~$7$ in Algorithm~\ref{alg:02}), +\item \verb+cublasDnrm2()+ to perform the Euclidean norm (line~$8$) and, +\item \verb+cublasDcpy()+ for the data copy of a vector to another one in the GPU memory (lines~$3$ and~$9$). +\end{itemize} + +The dimensions of the grid and blocks of threads that execute a given kernel depend on the resources of the GPU multiprocessor and the +resource requirements of the kernel. So, if $block$ defines the size of a thread block, which must not exceed the maximum size of a thread +block, then the number of thread blocks in the grid, denoted by $grid$, can be computed according to the size of the local sub-problem +as follows: \[grid = \frac{(NX\times ny\times nz)+block-1}{block}.\] However, when solving very large problems, the size of the thread +grid can exceed the maximum number of thread blocks that can be executed on the GPUs (up-to $65.535$ thread blocks) and, thus, the kernel +will fail to launch. Therefore, for each kernel, we decompose the three-dimensional sub-problem into $nz$ two-dimensional slices of size +($NX\times ny$), as is shown in Figure~\ref{fig:02}. All slices of the same kernel are executed using {\bf for} loop by $NX\times ny$ parallel +threads organized in a two-dimensional grid of two-dimensional thread blocks, as is shown in Listing~\ref{list:01}. Each thread is in charge +of $nz$ discretization points (one from each slice), accessed in the GPU memory with a constant stride $(NX\times ny)$. + +\begin{figure} +\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitGPU}} +\caption{Decomposition of a sub-problem in a GPU into $nz$ slices.} +\label{fig:02} +\end{figure} + +\begin{center} +\lstinputlisting[label=list:01,caption=Skeleton codes of a GPU kernel and a CPU function]{Chapters/chapter13/ex1.cu} +\end{center} +The function $Determine\_Bordering\_Vector\_Elements()$ (line~$5$ in Algorithm~\ref{alg:02}) determines the values of the vector +elements shared at the boundaries with neighboring computing nodes. Its main operations are defined as follows: +\begin{enumerate} +\item define the values associated to the bordering points needed by the neighbors, +\item copy the values associated to the bordering points from the GPU to the CPU, +\item exchange the values associated to the bordering points between the neighboring CPUs, +\item copy the received values associated to the bordering points from the CPU to the GPU, +\end{enumerate} +The first operation of this function is implemented as kernels to be performed by the GPU: +\begin{itemize} +\item a kernel executed by $NX\times nz$ threads to define the values associated to the bordering vector elements along $y$-axis and, +\item a kernel executed by $NX\times ny$ threads to define the values associated to the bordering vector elements along $z$-axis. +\end{itemize} +As mentioned before, we develop the \emph{synchronous} and \emph{asynchronous} algorithms of the projected Richardson method. Obviously, +in this scope, the synchronous or asynchronous communications refer to the communications between the CPU cores (MPI processes) on the +GPU cluster, in order to exchange the vector elements associated to subdomain boundaries. For the memory copies between a CPU core and +its GPU, we use the synchronous communication routines of the CUBLAS library: \verb+cublasSetVector()+ and \verb+cublasGetVector()+ +in the synchronous algorithm and the asynchronous ones: \verb+cublasSetVectorAsync()+ and \verb+cublasGetVectorAsync()+ in the +asynchronous algorithm. Moreover, we use the communication routines of the MPI library to carry out the data exchanges between the neighboring +nodes. We use the following communication routines: \verb+MPI_Isend()+ and \verb+MPI_Irecv()+ to perform non-blocking sends and receptions, +respectively. For the synchronous algorithm, we use the MPI routine \verb+MPI_Waitall()+ which puts the MPI process of a computing node +in blocking status until all data exchanges with neighboring nodes (sends and receptions) are completed. In contrast, for the asynchronous +algorithms, we use the MPI routine \verb+MPI_Test()+ which tests the completion of a data exchange (send or reception) without putting the +MPI process in blocking status. + +The function $Compute\_New\_Vector\_Elements()$ (line~$6$ in Algorithm~\ref{alg:02}) computes, at each iteration, the new elements +of the iterate vector $U$. Its general code is presented in Listing~\ref{list:01} (CPU function). The iterations of the projected +Richardson method, based on those of the Jacobi method, are defined as follows: +\begin{equation} +\begin{array}{ll} +u^{p+1}(x,y,z) =& \frac{1}{Center}(g(x,y,z) - (Center\cdot u^{p}(x,y,z) + \\ +& West\cdot u^{p}(x-h,y,z) + East\cdot u^{p}(x+h,y,z) + \\ +& South\cdot u^{p}(x,y-h,z) + North\cdot u^{p}(x,y+h,z) + \\ +& Rear\cdot u^{p}(x,y,z-h) + Front\cdot u^{p}(x,y,z+h))), +\end{array} +\label{eq:17} +\end{equation} +where $u^{p}(x,y,z)$ is an element of the iterate vector $U$ computed at the iteration $p$ and $g(x,y,z)$ is a vector element of the +right-hand side $G$. The scalars $Center$, $West$, $East$, $South$, $North$, $Rear$ and $Front$ define constant coefficients of the +block matrix $A$. Figure~\ref{fig:03} shows the positions of these coefficients in a three-dimensional domain. + +\begin{figure} +\centerline{\includegraphics[scale=0.35]{Chapters/chapter13/figures/matrix}} +\caption{Matrix constant coefficients in a three-dimensional domain.} +\label{fig:03} +\end{figure} + +The kernel implementations of the projected Richardson method on GPUs uses a perfect fine-grain multithreading parallelism. Since the +projected Richardson algorithm is implemented as a fixed point method, each kernel is executed by a large number of GPU threads such +that each thread is in charge of the computation of one element of the iterate vector $U$. Moreover, this method uses the vector elements +updates of the Jacobi method, which means that each thread $i$ computes the new value of its element $u_{i}^{p+1}$ independently of the +new values $u_{j}^{p+1}$, where $j\neq i$, of those computed in parallel by other threads at the same iteration $p+1$. Listing~\ref{list:02} +shows the GPU implementations of the main kernels of the projected Richardson method, which are: the matrix-vector multiplication +(\verb+MV_Multiplication()+) and the vector elements updates (\verb+Vector_Updates()+). The codes of these kernels are based on +that presented in Listing~\ref{list:01}. + +\lstinputlisting[label=list:02,caption=GPU kernels of the projected Richardson method]{Chapters/chapter13/ex2.cu} + +\begin{figure} +\centerline{\includegraphics[scale=0.3]{Chapters/chapter13/figures/points3D}} +\caption{Computation of a vector element with the projected Richardson method.} +\label{fig:04} +\end{figure} + +Each kernel is executed by $NX\times ny$ GPU threads so that $nz$ slices of $(NX\times ny)$ vector elements are computed in +a {\bf for} loop. In this case, each thread is in charge of one vector element from each slice (in total $nz$ vector elements +along $z$-axis). We can notice from the formula~(\ref{eq:17}) that the computation of a vector element $u^{p+1}(x,y,z)$, by +a thread at iteration $p+1$, requires seven vector elements computed at the previous iteration $p$: two vector elements in +each dimension plus the vector element at the intersection of the three axises $x$, $y$ and $z$ (see Figure~\ref{fig:04}). +So, to reduce the memory accesses to the high-latency global memory, the vector elements of the current slice can be stored +in the low-latency shared memories of thread blocks, as is described in~\cite{ref9}. Nevertheless, the fact that the computation +of a vector element requires only two elements in each dimension does not allow to maximize the data reuse from the shared memories. +The computation of a slice involves in total $(bx+2)\times(by+2)$ accesses to the global memory per thread block, to fill the +required vector elements in the shared memory where $bx$ and $by$ are the dimensions of a thread block. Then, in order to optimize +the memory accesses on GPUs, the elements of the iterate vector $U$ are filled in the cache texture memory (see~\cite{ref10}). +In new GPU generations as Fermi or Kepler, the global memory accesses are always cached in L1 and L2 caches. For example, for +a given kernel, we can favour the use of the L1 cache to that of the shared memory by using the function \verb+cudaFuncSetCacheConfig(Kernel,cudaFuncCachePreferL1)+. +So, the initial access to the global memory loads the vector elements required by the threads of a block into the cache memory +(texture or L1/L2 caches). Then, all the following memory accesses read from this cache memory. In Listing~\ref{list:02}, the +function \verb+fetch_double(v,i)+ is used to read from the texture memory the $i^{th}$ element of the double-precision vector +\verb+v+ (see Listing~\ref{list:03}). Moreover, the seven constant coefficients of matrix $A$ can be stored in the constant memory +but, since they are reused $nz$ times by each thread, it is more interesting to fill them on the low-latency registers of each thread. + +\lstinputlisting[label=list:03,caption=Memory access to the cache texture memory]{Chapters/chapter13/ex3.cu} + +The function $Convergence()$ (line~$11$ in Algorithm~\ref{alg:02}) allows to detect the convergence of the parallel iterative algorithm +and is based on the tolerance threshold $\varepsilon$ and the maximum number of relaxations $MaxRelax$. We take into account the number +of relaxations since that of iterations cannot be computed in the asynchronous case. Indeed, a relaxation is the update~(\ref{eq:13}) of +a local iterate vector $U_i$ according to $F_i$. Then, counting the number of relaxations is possible in both synchronous and asynchronous +cases. On the other hand, an iteration is the update of at least all vector components with $F_i$. + +In the synchronous algorithm, the global convergence is detected when the maximal value of the absolute error, $error$, is sufficiently small +and/or the maximum number of relaxations, $MaxRelax$, is reached, as follows: +$$ +\begin{array}{l} +error=\|U^{p}-U^{p+1}\|_{2}; \\ +AllReduce(error,\hspace{0.1cm}maxerror,\hspace{0.1cm}MAX); \\ +\text{if}((maxerror<\varepsilon)\hspace{0.2cm}\text{or}\hspace{0.2cm}(p\geq MaxRelax)) \\ +conv \leftarrow true; +\end{array} +$$ +where the function $AllReduce()$ uses the MPI reduction subroutine \verb+MPI_Allreduce()+ to compute the maximal value, $maxerror$, among the +local absolute errors, $error$, of all computing nodes and $p$ (in Algorithm~\ref{alg:02}) is used as a counter of the local relaxations carried +out by a computing node. In the asynchronous algorithms, the global convergence is detected when all computing nodes locally converge. For this, +we use a token ring architecture around which a boolean token travels, in one direction, from a computing node to another. Starting from node $0$, +the boolean token is set to $true$ by node $i$ if the local convergence is reached or to $false$ otherwise and, then, it is sent to node $i+1$. +Finally, the global convergence is detected when node $0$ receives from its neighbor node $S-1$, in the ring architecture, a token set to $true$. +In this case, node $0$ sends a stop message (end of parallel solving) to all computing nodes in the cluster. + + +%%--------------------------%% +%% SECTION 5 %% +%%--------------------------%% +\section{Experimental tests on a GPU cluster} +\label{sec:05} +The GPU cluster of tests, that we used in this chapter, is an $20Gbps$ Infiniband network of six machines. Each machine is a Quad-Core Xeon +E5530 CPU running at $2.4$GHz. It provides a RAM memory of $12$GB with a memory bandwidth of $25.6$GB/s and it is equipped with two Nvidia +Tesla C1060 GPUs. A Tesla GPU contains in total $240$ cores running at $1.3$GHz. It provides $4$GB of global memory with a memory bandwidth +of $102$GB/s, accessible by all its cores and also by the CPU through the PCI-Express 16x Gen 2.0 interface with a throughput of $8$GB/s. +Hence, the memory copy operations between the GPU and the CPU are about $12$ times slower than those of the Tesla GPU memory. We have performed +our simulations on a cluster of $24$ CPU cores and on a cluster of $12$ GPUs. Figure~\ref{fig:05} describes the components of the GPU cluster +of tests. + +\begin{figure} +\centerline{\includegraphics[scale=0.25]{Chapters/chapter13/figures/cluster}} +\caption{GPU cluster of tests composed of 12 computing nodes (six machines, each with two GPUs.} +\label{fig:05} +\end{figure} + +Linux cluster version 2.6.39 OS is installed on CPUs. C programming language is used for coding the parallel algorithms of the methods on both +GPU cluster and CPU cluster. CUDA version 4.0~\cite{ref12} is used for programming GPUs, using CUBLAS library~\cite{ref8} to deal with vector +operations in GPUs and, finally, MPI functions of OpenMPI 1.3.3 are used to carry out the synchronous and asynchronous communications between +CPU cores. Indeed, in our experiments, a computing node is managed by a MPI process and it is composed of one CPU core and one GPU card. + +All experimental results of the parallel projected Richardson algorithms are obtained from simulations made in double precision data. The obstacle +problems to be solved are defined in constant three-dimensional domain $\Omega\subset\mathbb{R}^{3}$. The numerical values of the parameters of the +obstacle problems are: $\eta=0.2$, $c=1.1$, $f$ is computed by formula~(\ref{eq:18}) and final time $T=0.02$. Moreover, three time steps ($NbSteps=3$) +are computed with $k=0.0066$. As the discretization matrix is constant along the time steps, the convergence properties of the iterative algorithms +do not change. Thus, the performance characteristics obtained with three time steps will still be valid for more time steps. The initial function +$u(0,x,y,z)$ of the obstacle problem~(\ref{eq:01}) is set to $0$, with a constraint $u\geq\phi=0$. The relaxation parameter $\gamma$ used by the +projected Richardson method is computed automatically thanks to the diagonal entries of the discretization matrix. The formula and its proof can be +found in~\cite{ref11}, Section~2.3. The convergence tolerance threshold $\varepsilon$ is set to $1e$-$04$ and the maximum number of relaxations is +limited to $10^{6}$ relaxations. Finally, the number of threads per block is set to $256$ threads, which gives, in general, good performances for +most GPU applications. We have performed some tests for the execution configurations and we have noticed that the best configuration of the $256$ +threads per block is an organization into two dimensions of sizes $(64,4)$. + +\begin{table}[!h] +\centering +\begin{tabular}{|c|c|c|c|c|c|} +\hline +\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-5} + + & $\mathbf{T_{cpu}}$ & {\bf \#relax.} & $\mathbf{T_{cpu}}$ & {\bf \#relax.} & \\ \hline \hline + +$256^{3}$ & $575.22$ & $198,288$ & $539.25$ & $198,613$ & $6.25$ \\ \hline \hline + +$512^{3}$ & $19,250.25$ & $750,912$ & $18,237.14$ & $769,611$ & $5.26$ \\ \hline \hline + +$768^{3}$ & $206,159.44$ & $1,635,264$ & $183,582.60$ & $1,577,004$ & $10.95$ \\ \hline \hline + +$800^{3}$ & $222,108.09$ & $1,769,232$ & $188,790.04$ & $1,701,735$ & $15.00$ \\ \hline +\end{tabular} +\vspace{0.5cm} +\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 24 CPU cores.} +\label{tab:01} +\end{table} + +\begin{table}[!h] +\centering +\begin{tabular}{|c|c|c|c|c|c|c|c|} +\hline +\multirow{2}{*}{\bf Pb. size} & \multicolumn{3}{c|}{\bf Synchronous} & \multicolumn{3}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-7} + + & $\mathbf{T_{gpu}}$ & {\bf \#relax.} & $\mathbf{\tau}$ & $\mathbf{T_{gpu}}$ & {\bf \#relax.} & $\mathbf{\tau}$ & \\ \hline \hline + +$256^{3}$ & $29.67$ & $100,692$ & $19.39$ & $18.00$ & $94,215$ & $29.96$ & $39.33$ \\ \hline \hline + +$512^{3}$ & $521.83$ & $381,300$ & $36.89$ & $425.15$ & $347,279$ & $42.89$ & $18.53$ \\ \hline \hline + +$768^{3}$ & $4,112.68$ & $831,144$ & $50.13$ & $3,313.87$ & $750,232$ & $55.40$ & $19.42$ \\ \hline \hline + +$800^{3}$ & $3,950.87$ & $899,088$ & $56.22$ & $3,636.57$ & $834,900$ & $51.91$ & $7.95$ \\ \hline +\end{tabular} +\vspace{0.5cm} +\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 12 GPUs.} +\label{tab:02} +\end{table} + +The performance measures that we took into account are the execution times and the number of relaxations performed by the parallel iterative algorithms, +both synchronous and asynchronous versions, on the GPU and CPU clusters. These algorithms are used for solving nonlinear systems derived from the discretization +of obstacle problems of sizes $256^{3}$, $512^{3}$, $768^{3}$ and $800^{3}$. In Table~\ref{tab:01} and Table~\ref{tab:02}, we show the performances +of the parallel synchronous and asynchronous algorithms of the projected Richardson method implemented, respectively, on a cluster of $24$ CPU cores +and on a cluster of $12$ GPUs. In these tables, the execution time defines the time spent by the slowest computing node and the number of relaxations +is computed as the summation of those carried out by all computing nodes. + +In the sixth column of Table~\ref{tab:01} and in the eighth column of Table~\ref{tab:02}, we give the gains in $\%$ obtained by using an +asynchronous algorithm compared to a synchronous one. We can notice that the asynchronous version on CPU and GPU clusters is slightly faster +than the synchronous one for both methods. Indeed, the cluster of tests is composed of local and homogeneous nodes communicating via low-latency +connections. So, in the case of distant and/or heterogeneous nodes (or even with geographically distant clusters) the asynchronous version +would be faster than the synchronous one. However, the gains obtained on the GPU cluster are better than those obtained on the CPU cluster. +In fact, the computation times are reduced by accelerating the computations on GPUs while the communication times still unchanged. + +The fourth and seventh columns of Table~\ref{tab:02} show the relative gains obtained by executing the parallel algorithms on the cluster +of $12$ GPUs instead on the cluster of $24$ CPU cores. We compute the relative gain $\tau$ as a ratio between the execution time $T_{cpu}$ +spent on the CPU cluster over that $T_{gpu}$ spent on the GPU cluster: \[\tau=\frac{T_{cpu}}{T_{gpu}}.\] We can see from these ratios that +solving large obstacle problems is faster on the GPU cluster than on the CPU cluster. Indeed, the GPUs are more efficient than their +counterpart CPUs to execute large data-parallel operations. In addition, the projected Richardson method is implemented as a fixed point-based +iteration and uses the Jacobi vector updates that allow a well thread-parallelization on GPUs, such that each GPU thread is in charge +of one vector component at a time without being dependent on other vector components computed by other threads. Then, this allow to exploit +at best the high performance computing of the GPUs by using all the GPU resources and avoiding the idle cores. + +Finally, the number of relaxations performed by the parallel synchronous algorithm is different in the CPU and GPU versions, because the number +of computing nodes involved in the GPU cluster and in the CPU cluster is different. In the CPU case, $24$ computing nodes ($24$ CPU cores) are +considered, whereas in the GPU case, $12$ computing nodes ($12$ GPUs) are considered. As the number of relaxations depends on the domain decomposition, +consequently it also depends on the number of computing nodes. + + +%%--------------------------%% +%% SECTION 6 %% +%%--------------------------%% +\section{Red-Black ordering technique} +\label{sec:06} +As is well-known, the Jacobi method is characterized by a slow convergence rate compared to some iterative methods (for example Gauss-Seidel method). +So, in this section, we present some solutions to reduce the execution time and the number of relaxations and, more specifically, to speed up the +convergence of the parallel projected Richardson method on the GPU cluster. We propose to use the point red-black ordering technique to accelerate +the convergence. This technique is often used to increase the parallelism of iterative methods for solving linear systems~\cite{ref13,ref14,ref15}. +We apply it to the projected Richardson method as a compromise between the Jacobi and Gauss-Seidel iterative methods. + +The general principle of the red-black technique is as follows. Let $t$ be the summation of the integer $x$-, $y$- and $z$-coordinates of a vector +element $u(x,y,z)$ on a three-dimensional domain: $t=x+y+z$. As is shown in Figure~\ref{fig:06.01}, the red-black ordering technique consists in the +parallel computing of the red vector elements having even value $t$ by using the values of the black ones then, the parallel computing of the black +vector elements having odd values $t$ by using the new values of the red ones. + +\begin{figure} +\centering + \mbox{\subfigure[Red-black ordering on x, y and z axises]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir}\label{fig:06.01}}\quad + \subfigure[Red-black ordering on y axis]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir-y}\label{fig:06.02}}} +\caption{Red-black ordering for computing the iterate vector elements in a three-dimensional space.} +\end{figure} + +This technique can be implemented on the GPU in two different manners: +\begin{itemize} +\item among all launched threads ($NX\times ny$ threads), only one thread out of two computes its red or black vector element at a time or, +\item all launched threads (on average half of $NX\times ny$ threads) compute the red vector elements first and, then, the black ones. +\end{itemize} +However, in both solutions, for each memory transaction, only half of the memory segment addressed by a half-warp is used. So, the computation of the +red and black vector elements leads to use twice the initial number of memory transactions. Then, we apply the point red-black ordering accordingly to +the $y$-coordinate, as is shown in Figure~\ref{fig:06.02}. In this case, the vector elements having even $y$-coordinate are computed in parallel using +the values of those having odd $y$-coordinate and then vice-versa. Moreover, in the GPU implementation of the parallel projected Richardson method (Section~\ref{sec:04}), +we have shown that a sub-problem of size $(NX\times ny\times nz)$ is decomposed into $nz$ grids of size $(NX\times ny)$. Then, each kernel is executed +in parallel by $NX\times ny$ GPU threads, so that each thread is in charge of $nz$ vector elements along $z$-axis (one vector element in each grid of +the sub-problem). So, we propose to use the new values of the vector elements computed in grid $i$ to compute those of the vector elements in grid $i+1$. +Listing~\ref{list:04} describes the kernel of the matrix-vector multiplication and the kernel of the vector elements updates of the parallel projected +Richardson method using the red-black ordering technique. + +\lstinputlisting[label=list:04,caption=GPU kernels of the projected Richardson method using the red-black technique]{Chapters/chapter13/ex4.cu} + +Finally, we exploit the concurrent executions between the host functions and the GPU kernels provided by the GPU hardware and software. In fact, the kernel +launches are asynchronous (when this environment variable is not disabled on the GPUs), such that the control is returned to the host (MPI process) before +the GPU has completed the requested task (kernel)~\cite{ref12}. Therefore, all the kernels necessary to update the local vector elements, $u(x,y,z)$ where +$0<y<(ny-1)$ and $0<z<(nz-1)$, are executed first. Then, the values associated to the bordering vector elements are exchanged between the neighbors. Finally, +the values of the vector elements associated to the bordering vector elements are updated. In this case, the computation of the local vector elements is +performed concurrently with the data exchanges between neighboring CPUs and this in both synchronous and asynchronous cases. + +\begin{table}[!h] +\centering +\begin{tabular}{|c|c|c|c|c|c|} +\hline +\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-5} + + & $\mathbf{T_{gpu}}$ & {\bf \#relax.} & $\mathbf{T_{gpu}}$ & {\bf \#relax.} & \\ \hline \hline + +$256^{3}$ & $18.37$ & $71,988$ & $12.58$ & $67,638$ & $31.52$ \\ \hline \hline + +$512^{3}$ & $349.23$ & $271,188$ & $289.41$ & $246,036$ & $17.13$ \\ \hline \hline + +$768^{3}$ & $2,773.65$ & $590,652$ & $2,222.22$ & $532,806$ & $19.88$ \\ \hline \hline + +$800^{3}$ & $2,748.23$ & $638,916$ & $2,502.61$ & $592,525$ & $8.92$ \\ \hline +\end{tabular} +\vspace{0.5cm} +\caption{Execution times in seconds of the parallel projected Richardson method using read-black ordering technique implemented on a cluster of 12 GPUs.} +\label{tab:03} +\end{table} + +In Table~\ref{tab:03}, we report the execution times and the number of relaxations performed on a cluster of $12$ GPUs by the parallel projected Richardson +algorithms; it can be noted that the performances of the projected Richardson are improved by using the point read-black ordering. We compare the performances +of the parallel projected Richardson method with and without this later ordering (Tables~\ref{tab:02} and~\ref{tab:03}). We can notice that both parallel synchronous +and asynchronous algorithms are faster when they use the red-black ordering. Indeed, we can see in Table~\ref{tab:03} that the execution times of these algorithms +are reduced, on average, by $32\%$ compared to those shown in Table~\ref{tab:02}. + +\begin{figure} +\centerline{\includegraphics[scale=0.9]{Chapters/chapter13/figures/scale}} +\caption{Weak scaling of both synchronous and asynchronous algorithms of the projected Richardson method using red-black ordering technique.} +\label{fig:07} +\end{figure} + +In Figure~\ref{fig:07}, we study the ratio between the computation time and the communication time of the parallel projected Richardson algorithms on a GPU cluster. +The experimental tests are carried out on a cluster composed of one to ten Tesla GPUs. We have focused on the weak scaling of both parallel, synchronous and asynchronous, +algorithms using the red-black ordering technique. For this, we have fixed the size of a sub-problem to $256^{3}$ per computing node (a CPU core and a GPU). Then, +Figure~\ref{fig:07} shows the number of relaxations performed, on average, per second by a computing node. We can see from this figure that the efficiency of the +asynchronous algorithm is almost stable, while that of the synchronous algorithm decreases (down to $81\%$ in this example) with the increasing of the number of +computing nodes on the cluster. This is due to the fact that the ratio between the time of the computation over that of the communication is reduced when the computations +are performed on GPUs. Indeed, GPUs compute faster than CPUs and communications are more time consuming. In this context, asynchronous algorithms are more scalable +than synchronous ones. So, with large scale GPU clusters, synchronous algorithms might be more penalized by communications, as can be deduced from Figure~\ref{fig:07}. +That is why we think that asynchronous iterative algorithms are all the more interesting in this case. + + +%%--------------------------%% +%% SECTION 7 %% +%%--------------------------%% +\section{Conclusion} +\label{sec:07} +Our main contribution, in this chapter, is the parallel implementation of an asynchronous iterative method on GPU clusters for solving large scale nonlinear +systems derived from the spatial discretization of three-dimensional obstacle problems. For this, we have implemented both synchronous and asynchronous algorithms of the +Richardson iterative method using a projection on a convex set. Indeed, this method uses point-based iterations of the Jacobi method that are very easy to parallelize on +parallel computers. We have shown that its adapted parallel algorithms to GPU architectures allows to exploit at best the computing power of the GPUs and to accelerate the +resolution of large nonlinear systems. Consequently, the experimental results have shown that solving nonlinear systems of large obstacle problems with this method is about +fifty times faster on a cluster of $12$ GPUs than on a cluster of $24$ CPU cores. Moreover, we have applied to this projected Richardson method the red-black ordering technique +which allows it to improve its convergence rate. Thus, the execution times of both parallel algorithms performed on the cluster of $12$ GPUs are reduced on average of $32\%$. + +Afterwards, the experiments have shown that the asynchronous version is slightly more efficient than the synchronous one. In fact, the computations are accelerated by using GPUs +while the communication times still unchanged. In addition, we have studied the weak-scaling in the synchronous and asynchronous cases, which has confirmed that the ratio between +the computations and the communications are reduced when using a cluster of GPUs. We highlight that asynchronous iterative algorithms are more scalable than synchronous ones. +Therefore, we can conclude that asynchronous iterations are well suited to tackle scalability issues on GPU clusters. + +In future works, we plan to perform experiments on large scale GPU clusters and on geographically distant GPU clusters, because we expect that asynchronous versions would +be faster and more scalable on such architectures. Furthermore, we want to study the performance behavior and the scalability of other numerical algorithms which support, +if possible, the model of asynchronous iterations. + +\putbib[Chapters/chapter13/biblio13] + diff --git a/BookGPU/Chapters/chapter13/ch13.tex~ b/BookGPU/Chapters/chapter13/ch13.tex~ new file mode 100755 index 0000000..65a9980 --- /dev/null +++ b/BookGPU/Chapters/chapter13/ch13.tex~ @@ -0,0 +1,710 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% CHAPTER 13 %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\chapterauthor{}{} +\newcommand{\scalprod}[2]% +{\ensuremath{\langle #1 \, , #2 \rangle}} +\chapter{Solving sparse nonlinear systems of obstacle problems on GPU clusters} + +%%--------------------------%% +%% SECTION 1 %% +%%--------------------------%% +\section{Introduction} +\label{sec:01} +The obstacle problem is one kind of free boundary problems. It allows to model, for example, an elastic membrane covering a solid obstacle. +In this case, the objective is to find an equilibrium position of this membrane constrained to be above the obstacle and which tends to minimize +its surface and/or its energy. The study of such problems occurs in many applications, for example: fluid mechanics, bio-mathematics (tumour growth +process) or financial mathematics (American or European option pricing). + +In this chapter, we focus on solutions of large obstacle problems defined in a three-dimensional domain. Particularly, the present study consists +in solving large nonlinear systems derived from the spatial discretization of these problems. Owing to the great size of such systems, in order to +reduce computation times, we proceed by solving them by parallel synchronous or asynchronous iterative algorithms. Moreover, we aim at harnessing +the computing power of GPUs to accelerate computations of these parallel algorithms. For this, we use an iterative method involving a projection +on a convex set, which is: the projected Richardson method. We choose this method among other iterative methods because it is easy to implement on +parallel computers and easy to adapt to GPU architectures. + +In Section~\ref{sec:02}, we present the mathematical model of obstacle problems then, in Section~\ref{sec:03}, we describe the general principle of +the parallel projected Richardson method. Next, in Section~\ref{sec:04}, we give the main key points of the parallel implementation of both synchronous +and asynchronous algorithms of the projected Richardson method on a GPU cluster. In Section~\ref{sec:05}, we present the performances of both parallel +algorithms obtained from simulations carried out on a CPU and GPU clusters. Finally, in Section~\ref{sec:06}, we use the read-black ordering technique +to improve the convergence and, thus, the execution times of the parallel projected Richardson algorithms on the GPU cluster. + + +%%--------------------------%% +%% SECTION 2 %% +%%--------------------------%% +\section{Obstacle problems} +\label{sec:02} +In this section, we present the mathematical model of obstacle problems defined in a three-dimensional domain. +This model is based on that presented in~\cite{ref1}. + +%%******************* +%%******************* +\subsection{Mathematical model} +\label{sec:02.01} +An obstacle problem, arising for example in mechanics or financial derivatives, consists in solving a time dependent +nonlinear equation: +\begin{equation} +\left\{ +\begin{array}{l} +\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0,\\ +(\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega,\\ +u(0,x,y,z)=u_0(x,y,z),\\ +\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega, +\end{array} +\right. +\label{eq:01} +\end{equation} +where $u_0$ is the initial condition, $c\geq 0$, $b$ and $\eta$ are physical parameters, $T$ is the final time, $u=u(t,x,y,z)$ +is an element of the solution vector $U$ to compute, $f$ is the right-hand right that could represent, for example, the external +forces, B.C. describes the boundary conditions on the boundary $\partial\Omega$ of the domain $\Omega$, $\phi$ models a constraint +imposed to $u$, $\Delta$ is the Laplacian operator, $\nabla$ is the gradient operator, a.e.w. means almost every where and ``.'' +defines the products between two scalars, a scalar and a vector or a matrix and a vector. In practice the boundary condition, +generally considered, is the Dirichlet condition (where $u$ is fixed on $\partial\Omega$) or the Neumann condition (where the +normal derivative of $u$ is fixed on $\partial\Omega$). + +The time dependent equation~(\ref{eq:01}) is numerically solved by considering an implicit or a semi-implicit time marching, +where at each time step $k$ a stationary nonlinear problem is solved: +\begin{equation} +\left\{ +\begin{array}{l} +b^t.\nabla u-\eta.\Delta u+(c+\delta).u-g\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0, \\ +(b^t.\nabla u-\eta.\Delta u+(c+\delta).u- g)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega, \\ +\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega, +\end{array} +\right. +\label{eq:02} +\end{equation} +where $\delta=\frac{1}{k}$ is the inverse of the time step $k$, $g=f+\delta u^{prev}$ and $u^{prev}$ is the solution computed at the +previous time step. + + +%%******************* +%%******************* +\subsection{Discretization} +\label{sec:02.02} +First, we note that the spatial discretization of the previous stationary problem~(\ref{eq:02}) does not provide a symmetric matrix, +because the convection-diffusion operator is not self-adjoint. Moreover, the fact that the operator is self-adjoint or not plays an +important role in the choice of the appropriate algorithm for solving nonlinear systems derived from the discretization of the obstacle +problem. Nevertheless, since the convection coefficients arising in the operator~(\ref{eq:02}) are constant, we can formulate the same +problem by an self-adjoint operator by performing a classical change of variables. Then, we can replace the stationary convection-diffusion +problem: +\begin{equation} +b^{t}.\nabla v-\eta.\Delta v+(c+\delta).v=g\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}c\geq 0\mbox{,~}\delta\geq~0, +\label{eq:03} +\end{equation} +by the following stationary diffusion operator: +\begin{equation} +-\eta.\Delta u+(\frac{\|b\|^{2}_{2}}{4\eta}+c+\delta).u=e^{-a}g=f, +\label{eq:04} +\end{equation} +where $b=\{b_{1},b_{2},b_{3}\}$, $\|b\|_{2}$ denotes the Euclidean norm of $b$ and $v=e^{-a}.u$ represents the general change of variables +such that $a=\frac{b^{t}(x,y,z)}{2\eta}$. Consequently, the numerical resolution of the diffusion problem (the self-adjoint operator~(\ref{eq:04})) +is done by optimization algorithms, in contrast, that of the convection-diffusion problem (non self-adjoint operator~(\ref{eq:03})) is +done by relaxation algorithms. In the case of our studied algorithm, the convergence is ensured by M-matrix property then, the performance +is linked to the magnitude of the spectral radius of the iteration matrix, which is independent of the condition number. + +Next, the three-dimensional domain $\Omega\subset\mathbb{R}^{3}$ is set to $\Omega=\lbrack 0,1\rbrack^{3}$ and discretized with an uniform +Cartesian mesh constituted by $M=m^3$ discretization points, where $m$ related to the spatial discretization step by $h=\frac{1}{m+1}$. This +is carried out by using a classical order 2 finite difference approximation of the Laplacian. So, the complete discretization of both stationary +boundary value problems~(\ref{eq:03}) and~(\ref{eq:04}) leads to the solution of a large discrete complementary problem of the following +form, when both Dirichlet or Neumann boundary conditions are used: +\begin{equation} +\left\{ +\begin{array}{l} +\mbox{Find~}U^{*}\in\mathbb{R}^{M}\mbox{~such~that} \\ +(A+\delta I)U^{*}-G\geq 0\mbox{,~}U^{*}\geq\bar{\Phi},\\ +((A+\delta I)U^{*}-G)^{T}(U^{*}-\bar{\Phi})=0,\\ +\end{array} +\right. +\label{eq:05} +\end{equation} +where $A$ is a matrix obtained after the spatial discretization by a finite difference method, $G$ is derived from the Euler first order implicit time +marching scheme and from the discretized right-hand side of the obstacle problem, $\delta$ is the inverse of the time step $k$ and $I$ is the identity +matrix. The matrix $A$ is symmetric when the self-adjoint operator is considered and nonsymmetric otherwise. + +According to the chosen discretization scheme of the Laplacian, $A$ is an M-matrix (irreducibly diagonal dominant, see~\cite{ref2}) and, consequently, +the matrix $(A+\delta I)$ is also an M-matrix. This property is important to the convergence of iterative methods. + + +%%--------------------------%% +%% SECTION 3 %% +%%--------------------------%% +\section{Parallel iterative method} +\label{sec:03} +Owing to the large size of the previous discrete complementary problem~(\ref{eq:05}), we will solve it by parallel synchronous or asynchronous iterative +algorithms (see~\cite{ref3,ref4,ref5}). In this chapter, we aim at harnessing the computing power of GPU clusters for solving these large nonlinear systems. +Then, we choose to use the projected Richardson iterative method for solving the diffusion problem~(\ref{eq:04}). Indeed, this method is based on the iterations +of the Jacobi method which are easy to parallelize on parallel computers and easy to adapt to GPU architectures. Then, according to the boundary value problem +formulation with a self-adjoint operator~(\ref{eq:04}), we can consider here the equivalent optimization problem and the fixed point mapping associated to +its solution. + +Assume that $E=\mathbb{R}^{M}$ is a Hilbert space, in which $\scalprod{.}{.}$ is the scalar product and $\|.\|$ its associated norm. So, the general fixed +point problem to be solved is defined as follows: +\begin{equation} +\left\{ +\begin{array}{l} +\mbox{Find~} U^{*} \in E \mbox{~such that} \\ +U^{*} = F(U^{*}), \\ +\end{array} +\right. +\label{eq:06} +\end{equation} +where $U\mapsto F(U)$ is an application from $E$ to $E$. + +Let $K$ be a closed convex set defined by: +\begin{equation} +K = \{U | U \geq \Phi \mbox{~everywhere in~} E\}, +\label{eq:07} +\end{equation} +where $\Phi$ is the discrete obstacle function. In fact, the obstacle problem~(\ref{eq:05}) is formulated as the following constrained optimization problem: +\begin{equation} +\left\{ +\begin{array}{l} +\mbox{Find~} U^{*} \in K \mbox{~such that} \\ +\forall V \in K, J(U^{*}) \leq J(V), \\ +\end{array} +\right. +\label{eq:08} +\end{equation} +where the cost function is given by: +\begin{equation} +J(U) = \frac{1}{2}\scalprod{\mathcal{A}.U}{U} - \scalprod{G}{U}, +\label{eq:09} +\end{equation} +in which $\scalprod{.}{.}$ denotes the scalar product in $E$, $\mathcal{A}=A+\delta I$ is a symmetric positive definite, $A$ is the discretization matrix +associated with the self-adjoint operator~(\ref{eq:04}) after change of variables. + +For any $U\in E$, let $P_K(U)$ be the projection of $U$ on $K$. For any $\gamma\in\mathbb{R}$, $\gamma>0$, the fixed point mapping $F_{\gamma}$ of the projected +Richardson method is defined as follows: +\begin{equation} +U^{*} = F_{\gamma}(U^{*}) = P_K(U^{*} - \gamma(\mathcal{A}.U^{*} - G)). +\label{eq:10} +\end{equation} +In order to reduce the computation time, the large optimization problem is solved in a numerical way by using a parallel asynchronous algorithm of the projected +Richardson method on the convex set $K$. Particularly, we will consider an asynchronous parallel adaptation of the projected Richardson method~\cite{ref6}. + +Let $\alpha\in\mathbb{N}$ be a positive integer. We consider that the space $E=\displaystyle\prod_{i=1}^{\alpha} E_i$ is a product of $\alpha$ subspaces $E_i$ +where $i\in\{1,\ldots,\alpha\}$. Note that $E_i=\mathbb{R}^{m_i}$, where $\displaystyle\sum_{i=1}^{\alpha} m_{i}=M$, is also a Hilbert space in which $\scalprod{.}{.}_i$ +denotes the scalar product and $|.|_i$ the associated norm, for all $i\in\{1,\ldots,\alpha\}$. Then, for all $u,v\in E$, $\scalprod{u}{v}=\displaystyle\sum_{i=1}^{\alpha}\scalprod{u_i}{v_i}_i$ +is the scalar product on $E$. + +Let $U\in E$, we consider the following decomposition of $U$ and the corresponding decomposition of $F_\gamma$ into $\alpha$ blocks: +\begin{equation} +\begin{array}{rcl} +U & = & (U_1,\ldots,U_{\alpha}), \\ +F_{\gamma}(U) & = & (F_{1,\gamma}(U),\ldots,F_{\alpha,\gamma}(U)). \\ +\end{array} +\label{eq:11} +\end{equation} +Assume that the convex set $K=\displaystyle\prod_{i=1}^{\alpha}K_{i}$, such that $\forall i\in\{1,\ldots,\alpha\},K_i\subset E_i$ and $K_i$ is a closed convex set. +Let also $G=(G_1,\ldots,G_{\alpha})\in E$ and, for any $U\in E$, $P_K(U)=(P_{K_1}(U_1),\ldots,P_{K_{\alpha}}(U_{\alpha}))$ is the projection of $U$ on $K$ where $\forall i\in\{1,\ldots,\alpha\},P_{K_i}$ +is the projector from $E_i$ onto $K_i$. So, the fixed point mapping of the projected Richardson method~(\ref{eq:10}) can be written in the following way: +\begin{equation} +\forall U\in E\mbox{,~}\forall i\in\{1,\ldots,\alpha\}\mbox{,~}F_{i,\gamma}(U) = P_{K_i}(U_i - \gamma(\mathcal{A}_i.U - G_i)). +\label{eq:12} +\end{equation} +Note that $\displaystyle\mathcal{A}_i.U= \sum_{j=1}^{\alpha}\mathcal{A}_{i,j}.U_j$, where $\mathcal{A}_{i,j}$ denote block matrices of $\mathcal{A}$. + +The parallel asynchronous iterations of the projected Richardson method for solving the obstacle problem~(\ref{eq:08}) are defined as follows: let $U^0\in E,U^0=(U^0_1,\ldots,U^0_\alpha)$ be +the initial solution, then for all $p\in\mathbb{N}$, the iterate $U^{p+1}=(U^{p+1}_1,\ldots,U^{p+1}_{\alpha})$ is recursively defined by: +\begin{equation} +U_i^{p+1} = +\left\{ +\begin{array}{l} +F_{i,\gamma}(U_1^{\rho_1(p)}, \ldots, U_{\alpha}^{\rho_{\alpha}(p)}) \mbox{~if~} i\in s(p), \\ +U_i^p \mbox{~otherwise}, \\ +\end{array} +\right. +\label{eq:13} +\end{equation} +where +\begin{equation} +\left\{ +\begin{array}{l} +\forall p\in\mathbb{N}, s(p)\subset\{1,\ldots,\alpha\}\mbox{~and~} s(p)\ne\emptyset, \\ +\forall i\in\{1,\ldots,\alpha\},\{p \ | \ i \in s(p)\}\mbox{~is denombrable}, +\end{array} +\right. +\label{eq:14} +\end{equation} +and $\forall j\in\{1,\ldots,\alpha\}$, +\begin{equation} +\left\{ +\begin{array}{l} +\forall p\in\mathbb{N}, \rho_j(p)\in\mathbb{N}, 0\leq\rho_j(p)\leq p\mbox{~and~}\rho_j(p)=p\mbox{~if~} j\in s(p),\\ +\displaystyle\lim_{p\to\infty}\rho_j(p) = +\infty.\\ +\end{array} +\right. +\label{eq:15} +\end{equation} + +The previous asynchronous scheme of the projected Richardson method models computations that are carried out in parallel +without order nor synchronization (according to the behavior of the parallel iterative method) and describes a subdomain +method without overlapping. It is a general model that takes into account all possible situations of parallel computations +and non-blocking message passing. So, the synchronous iterative scheme is defined by: +\begin{equation} +\forall j\in\{1,\ldots,\alpha\} \mbox{,~} \forall p\in\mathbb{N} \mbox{,~} \rho_j(p)=p. +\label{eq:16} +\end{equation} +The values of $s(p)$ and $\rho_j(p)$ are defined dynamically and not explicitly by the parallel asynchronous or synchronous +execution of the algorithm. Particularly, it enables one to consider distributed computations whereby processors compute at +their own pace according to their intrinsic characteristics and computational load. The parallelism between the processors is +well described by the set $s(p)$ which contains at each step $p$ the index of the components relaxed by each processor on a +parallel way while the use of delayed components in~(\ref{eq:13}) permits one to model nondeterministic behavior and does not +imply inefficiency of the considered distributed scheme of computation. Note that, according to~\cite{ref7}, theoretically, +each component of the vector must be relaxed an infinity of time. The choice of the relaxed components to be used in the +computational process may be guided by any criterion and, in particular, a natural criterion is to pick-up the most recently +available values of the components computed by the processors. Furthermore, the asynchronous iterations are implemented by +means of non-blocking MPI communication subroutines (asynchronous communications). + +The important property ensuring the convergence of the parallel projected Richardson method, both synchronous and asynchronous +algorithms, is the fact that $\mathcal{A}$ is an M-matrix. Moreover, the convergence proceeds from a result of~\cite{ref6}. +Indeed, there exists a value $\gamma_0>0$, such that $\forall\gamma\in ]0,\gamma_0[$, the parallel iterations~(\ref{eq:13}), +(\ref{eq:14}) and~(\ref{eq:15}), associated to the fixed point mapping $F_\gamma$~(\ref{eq:12}), converge to the unique solution +$U^{*}$ of the discretized problem. + + +%%--------------------------%% +%% SECTION 4 %% +%%--------------------------%% +\section{Parallel implementation on a GPU cluster} +\label{sec:04} +In this section, we give the main key points of the parallel implementation of the projected Richardson method, both synchronous +and asynchronous versions, on a GPU cluster, for solving the nonlinear systems derived from the discretization of large obstacle +problems. More precisely, each nonlinear system is solved iteratively using the whole cluster. We use a heteregeneous CUDA/MPI +programming. Indeed, the communication of data, at each iteration between the GPU computing nodes, can be either synchronous +or asynchronous using the MPI communication subroutines, whereas inside each GPU node, a CUDA parallelization is performed. + +\begin{figure}[!h] +\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitCPU}} +\caption{Data partitioning of a problem to be solved among $S=3\times 4$ computing nodes.} +\label{fig:01} +\end{figure} + +Let $S$ denote the number of computing nodes on the GPU cluster, where a computing node is composed of CPU core holding one MPI +process and a GPU card. So, before starting computations, the obstacle problem of size $(NX\times NY\times NZ)$ is split into $S$ +parallelepipedic sub-problems, each for a node (MPI process, GPU), as is shown in Figure~\ref{fig:01}. Indeed, the $NY$ and $NZ$ +dimensions (according to the $y$ and $z$ axises) of the three-dimensional problem are, respectively, split into $Sy$ and $Sz$ parts, +such that $S=Sy\times Sz$. In this case, each computing node has at most four neighboring nodes. This kind of the data partitioning +reduces the data exchanges at subdomain boundaries compared to a naive $z$-axis-wise partitioning. + +\begin{algorithm}[!t] +\SetLine +\linesnumbered +Initialization of the parameters of the sub-problem\; +Allocate and fill the data in the global memory GPU\; +\For{$i=1$ {\bf to} $NbSteps$}{ + $G = \frac{1}{k}.U^0 + F$\; + Solve($A$, $U^0$, $G$, $U$, $\varepsilon$, $MaxRelax$)\; + $U^0 = U$\; +} +Copy the solution $U$ back from GPU memory\; +\caption{Parallel solving of the obstacle problem on a GPU cluster} +\label{alg:01} +\end{algorithm} + +All the computing nodes of the GPU cluster execute in parallel the same Algorithm~\ref{alg:01} but on different three-dimensional +sub-problems of size $(NX\times ny\times nz)$. This algorithm gives the main key points for solving an obstacle problem defined in +a three-dimensional domain, where $A$ is the discretization matrix, $G$ is the right-hand side and $U$ is the solution vector. After +the initialization step, all the data generated from the partitioning operation are copied from the CPU memories to the GPU global +memories, to be processed on the GPUs. Next, the algorithm uses $NbSteps$ time steps to solve the global obstacle problem. In fact, +it uses a parallel algorithm adapted to GPUs of the projected Richardson iterative method for solving the nonlinear systems of the +obstacle problem. This function is defined by {\it Solve()} in Algorithm~\ref{alg:01}. At every time step, the initial guess $U^0$ +for the iterative algorithm is set to the solution found at the previous time step. Moreover, the right-hand side $G$ is computed +as follows: \[G = \frac{1}{k}.U^{prev} + F\] where $k$ is the time step, $U^{prev}$ is the solution computed in the previous time +step and each element $f(x, y, z)$ of the vector $F$ is computed as follows: +\begin{equation} +f(x,y,z)=\cos(2\pi x)\cdot\cos(4\pi y)\cdot\cos(6\pi z). +\label{eq:18} +\end{equation} +Finally, the solution $U$ of the obstacle problem is copied back from the GPU global memories to the CPU memories. We use the +communication subroutines of the CUBLAS library~\cite{ref8} (CUDA Basic Linear Algebra Subroutines) for the memory allocations in +the GPU (\verb+cublasAlloc()+) and the data transfers between the CPU and its GPU: \verb+cublasSetVector()+ and \verb+cublasGetVector()+. + +\begin{algorithm}[!t] + \SetLine + \linesnumbered + $p = 0$\; + $conv = false$\; + $U = U^{0}$\; + \Repeat{$(conv=true)$}{ + Determine\_Bordering\_Vector\_Elements($U$)\; + Compute\_New\_Vector\_Elements($A$, $G$, $U$)\; + $tmp = U^{0} - U$\; + $error = \|tmp\|_{2}$\; + $U^{0} = U$\; + $p = p + 1$\; + $conv$ = Convergence($error$, $p$, $\varepsilon$, $MaxRelax$)\; + } +\caption{Parallel iterative solving of the nonlinear systems on a GPU cluster ($Solve()$ function)} +\label{alg:02} +\end{algorithm} + +As many other iterative methods, the algorithm of the projected Richardson method is based on algebraic functions operating on vectors +and/or matrices, which are more efficient on parallel computers when they work on large vectors. Its parallel implementation on the GPU +cluster is carried out so that the GPUs execute the vector operations as kernels and the CPUs execute the serial codes, supervise the +kernel executions and the data exchanges with the neighboring nodes and supply the GPUs with data. Algorithm~\ref{alg:02} shows the +main key points of the parallel iterative algorithm (function $Solve()$ in Algorithm~\ref{alg:01}). All the vector operations inside +the main loop ({\bf repeat} ... {\bf until}) are executed by the GPU. We use the following functions of the CUBLAS library: +\begin{itemize*} +\item \verb+cublasDaxpy()+ to compute the difference between the solution vectors $U^{p}$ and $U^{p+1}$ computed in two successive relaxations +$p$ and $p+1$ (line~$7$ in Algorithm~\ref{alg:02}), +\item \verb+cublasDnrm2()+ to perform the Euclidean norm (line~$8$) and, +\item \verb+cublasDcpy()+ for the data copy of a vector to another one in the GPU memory (lines~$3$ and~$9$). +\end{itemize*} + +The dimensions of the grid and blocks of threads that execute a given kernel depend on the resources of the GPU multiprocessor and the +resource requirements of the kernel. So, if $block$ defines the size of a thread block, which must not exceed the maximum size of a thread +block, then the number of thread blocks in the grid, denoted by $grid$, can be computed according to the size of the local sub-problem +as follows: \[grid = \frac{(NX\times ny\times nz)+block-1}{block}.\] However, when solving very large problems, the size of the thread +grid can exceed the maximum number of thread blocks that can be executed on the GPUs (up-to $65.535$ thread blocks) and, thus, the kernel +will fail to launch. Therefore, for each kernel, we decompose the three-dimensional sub-problem into $nz$ two-dimensional slices of size +($NX\times ny$), as is shown in Figure~\ref{fig:02}. All slices of the same kernel are executed using {\bf for} loop by $NX\times ny$ parallel +threads organized in a two-dimensional grid of two-dimensional thread blocks, as is shown in Listing~\ref{list:01}. Each thread is in charge +of $nz$ discretization points (one from each slice), accessed in the GPU memory with a constant stride $(NX\times ny)$. + +\begin{figure} +\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitGPU}} +\caption{Decomposition of a sub-problem in a GPU into $nz$ slices.} +\label{fig:02} +\end{figure} + +\begin{center} +\lstinputlisting[label=list:01,caption=Skeleton codes of a GPU kernel and a CPU function]{Chapters/chapter13/ex1.cu} +\end{center} +The function $Determine\_Bordering\_Vector\_Elements()$ (line~$5$ in Algorithm~\ref{alg:02}) determines the values of the vector +elements shared at the boundaries with neighboring computing nodes. Its main operations are defined as follows: +\begin{enumerate*} +\item define the values associated to the bordering points needed by the neighbors, +\item copy the values associated to the bordering points from the GPU to the CPU, +\item exchange the values associated to the bordering points between the neighboring CPUs, +\item copy the received values associated to the bordering points from the CPU to the GPU, +\end{enumerate*} +The first operation of this function is implemented as kernels to be performed by the GPU: +\begin{itemize*} +\item a kernel executed by $NX\times nz$ threads to define the values associated to the bordering vector elements along $y$-axis and, +\item a kernel executed by $NX\times ny$ threads to define the values associated to the bordering vector elements along $z$-axis. +\end{itemize*} +As mentioned before, we develop the \emph{synchronous} and \emph{asynchronous} algorithms of the projected Richardson method. Obviously, +in this scope, the synchronous or asynchronous communications refer to the communications between the CPU cores (MPI processes) on the +GPU cluster, in order to exchange the vector elements associated to subdomain boundaries. For the memory copies between a CPU core and +its GPU, we use the synchronous communication routines of the CUBLAS library: \verb+cublasSetVector()+ and \verb+cublasGetVector()+ +in the synchronous algorithm and the asynchronous ones: \verb+cublasSetVectorAsync()+ and \verb+cublasGetVectorAsync()+ in the +asynchronous algorithm. Moreover, we use the communication routines of the MPI library to carry out the data exchanges between the neighboring +nodes. We use the following communication routines: \verb+MPI_Isend()+ and \verb+MPI_Irecv()+ to perform non-blocking sends and receptions, +respectively. For the synchronous algorithm, we use the MPI routine \verb+MPI_Waitall()+ which puts the MPI process of a computing node +in blocking status until all data exchanges with neighboring nodes (sends and receptions) are completed. In contrast, for the asynchronous +algorithms, we use the MPI routine \verb+MPI_Test()+ which tests the completion of a data exchange (send or reception) without putting the +MPI process in blocking status. + +The function $Compute\_New\_Vector\_Elements()$ (line~$6$ in Algorithm~\ref{alg:02}) computes, at each iteration, the new elements +of the iterate vector $U$. Its general code is presented in Listing~\ref{list:01} (CPU function). The iterations of the projected +Richardson method, based on those of the Jacobi method, are defined as follows: +\begin{equation} +\begin{array}{ll} +u^{p+1}(x,y,z) =& \frac{1}{Center}(g(x,y,z) - (Center\cdot u^{p}(x,y,z) + \\ +& West\cdot u^{p}(x-h,y,z) + East\cdot u^{p}(x+h,y,z) + \\ +& South\cdot u^{p}(x,y-h,z) + North\cdot u^{p}(x,y+h,z) + \\ +& Rear\cdot u^{p}(x,y,z-h) + Front\cdot u^{p}(x,y,z+h))), +\end{array} +\label{eq:17} +\end{equation} +where $u^{p}(x,y,z)$ is an element of the iterate vector $U$ computed at the iteration $p$ and $g(x,y,z)$ is a vector element of the +right-hand side $G$. The scalars $Center$, $West$, $East$, $South$, $North$, $Rear$ and $Front$ define constant coefficients of the +block matrix $A$. Figure~\ref{fig:03} shows the positions of these coefficients in a three-dimensional domain. + +\begin{figure} +\centerline{\includegraphics[scale=0.35]{Chapters/chapter13/figures/matrix}} +\caption{Matrix constant coefficients in a three-dimensional domain.} +\label{fig:03} +\end{figure} + +The kernel implementations of the projected Richardson method on GPUs uses a perfect fine-grain multithreading parallelism. Since the +projected Richardson algorithm is implemented as a fixed point method, each kernel is executed by a large number of GPU threads such +that each thread is in charge of the computation of one element of the iterate vector $U$. Moreover, this method uses the vector elements +updates of the Jacobi method, which means that each thread $i$ computes the new value of its element $u_{i}^{p+1}$ independently of the +new values $u_{j}^{p+1}$, where $j\neq i$, of those computed in parallel by other threads at the same iteration $p+1$. Listing~\ref{list:02} +shows the GPU implementations of the main kernels of the projected Richardson method, which are: the matrix-vector multiplication +(\verb+MV_Multiplication()+) and the vector elements updates (\verb+Vector_Updates()+). The codes of these kernels are based on +that presented in Listing~\ref{list:01}. + +\lstinputlisting[label=list:02,caption=GPU kernels of the projected Richardson method]{Chapters/chapter13/ex2.cu} + +\begin{figure} +\centerline{\includegraphics[scale=0.3]{Chapters/chapter13/figures/points3D}} +\caption{Computation of a vector element with the projected Richardson method.} +\label{fig:04} +\end{figure} + +Each kernel is executed by $NX\times ny$ GPU threads so that $nz$ slices of $(NX\times ny)$ vector elements are computed in +a {\bf for} loop. In this case, each thread is in charge of one vector element from each slice (in total $nz$ vector elements +along $z$-axis). We can notice from the formula~(\ref{eq:17}) that the computation of a vector element $u^{p+1}(x,y,z)$, by +a thread at iteration $p+1$, requires seven vector elements computed at the previous iteration $p$: two vector elements in +each dimension plus the vector element at the intersection of the three axises $x$, $y$ and $z$ (see Figure~\ref{fig:04}). +So, to reduce the memory accesses to the high-latency global memory, the vector elements of the current slice can be stored +in the low-latency shared memories of thread blocks, as is described in~\cite{ref9}. Nevertheless, the fact that the computation +of a vector element requires only two elements in each dimension does not allow to maximize the data reuse from the shared memories. +The computation of a slice involves in total $(bx+2)\times(by+2)$ accesses to the global memory per thread block, to fill the +required vector elements in the shared memory where $bx$ and $by$ are the dimensions of a thread block. Then, in order to optimize +the memory accesses on GPUs, the elements of the iterate vector $U$ are filled in the cache texture memory (see~\cite{ref10}). +In new GPU generations as Fermi or Kepler, the global memory accesses are always cached in L1 and L2 caches. For example, for +a given kernel, we can favour the use of the L1 cache to that of the shared memory by using the function \verb+cudaFuncSetCacheConfig(Kernel,cudaFuncCachePreferL1)+. +So, the initial access to the global memory loads the vector elements required by the threads of a block into the cache memory +(texture or L1/L2 caches). Then, all the following memory accesses read from this cache memory. In Listing~\ref{list:02}, the +function \verb+fetch_double(v,i)+ is used to read from the texture memory the $i^{th}$ element of the double-precision vector +\verb+v+ (see Listing~\ref{list:03}). Moreover, the seven constant coefficients of matrix $A$ can be stored in the constant memory +but, since they are reused $nz$ times by each thread, it is more interesting to fill them on the low-latency registers of each thread. + +\lstinputlisting[label=list:03,caption=Memory access to the cache texture memory]{Chapters/chapter13/ex3.cu} + +The function $Convergence()$ (line~$11$ in Algorithm~\ref{alg:02}) allows to detect the convergence of the parallel iterative algorithm +and is based on the tolerance threshold $\varepsilon$ and the maximum number of relaxations $MaxRelax$. We take into account the number +of relaxations since that of iterations cannot be computed in the asynchronous case. Indeed, a relaxation is the update~(\ref{eq:13}) of +a local iterate vector $U_i$ according to $F_i$. Then, counting the number of relaxations is possible in both synchronous and asynchronous +cases. On the other hand, an iteration is the update of at least all vector components with $F_i$. + +In the synchronous algorithm, the global convergence is detected when the maximal value of the absolute error, $error$, is sufficiently small +and/or the maximum number of relaxations, $MaxRelax$, is reached, as follows: +$$ +\begin{array}{l} +error=\|U^{p}-U^{p+1}\|_{2}; \\ +AllReduce(error,\hspace{0.1cm}maxerror,\hspace{0.1cm}MAX); \\ +\text{if}((maxerror<\varepsilon)\hspace{0.2cm}\text{or}\hspace{0.2cm}(p\geq MaxRelax)) \\ +conv \leftarrow true; +\end{array} +$$ +where the function $AllReduce()$ uses the MPI reduction subroutine \verb+MPI_Allreduce()+ to compute the maximal value, $maxerror$, among the +local absolute errors, $error$, of all computing nodes and $p$ (in Algorithm~\ref{alg:02}) is used as a counter of the local relaxations carried +out by a computing node. In the asynchronous algorithms, the global convergence is detected when all computing nodes locally converge. For this, +we use a token ring architecture around which a boolean token travels, in one direction, from a computing node to another. Starting from node $0$, +the boolean token is set to $true$ by node $i$ if the local convergence is reached or to $false$ otherwise and, then, it is sent to node $i+1$. +Finally, the global convergence is detected when node $0$ receives from its neighbor node $S-1$, in the ring architecture, a token set to $true$. +In this case, node $0$ sends a stop message (end of parallel solving) to all computing nodes in the cluster. + + +%%--------------------------%% +%% SECTION 5 %% +%%--------------------------%% +\section{Experimental tests on a GPU cluster} +\label{sec:05} +The GPU cluster of tests, that we used in this chapter, is an $20Gbps$ Infiniband network of six machines. Each machine is a Quad-Core Xeon +E5530 CPU running at $2.4$GHz. It provides a RAM memory of $12$GB with a memory bandwidth of $25.6$GB/s and it is equipped with two Nvidia +Tesla C1060 GPUs. A Tesla GPU contains in total $240$ cores running at $1.3$GHz. It provides $4$GB of global memory with a memory bandwidth +of $102$GB/s, accessible by all its cores and also by the CPU through the PCI-Express 16x Gen 2.0 interface with a throughput of $8$GB/s. +Hence, the memory copy operations between the GPU and the CPU are about $12$ times slower than those of the Tesla GPU memory. We have performed +our simulations on a cluster of $24$ CPU cores and on a cluster of $12$ GPUs. Figure~\ref{fig:05} describes the components of the GPU cluster +of tests. + +\begin{figure} +\centerline{\includegraphics[scale=0.25]{Chapters/chapter13/figures/cluster}} +\caption{GPU cluster of tests composed of 12 computing nodes (six machines, each with two GPUs.} +\label{fig:05} +\end{figure} + +Linux cluster version 2.6.39 OS is installed on CPUs. C programming language is used for coding the parallel algorithms of the methods on both +GPU cluster and CPU cluster. CUDA version 4.0~\cite{ref12} is used for programming GPUs, using CUBLAS library~\cite{ref8} to deal with vector +operations in GPUs and, finally, MPI functions of OpenMPI 1.3.3 are used to carry out the synchronous and asynchronous communications between +CPU cores. Indeed, in our experiments, a computing node is managed by a MPI process and it is composed of one CPU core and one GPU card. + +All experimental results of the parallel projected Richardson algorithms are obtained from simulations made in double precision data. The obstacle +problems to be solved are defined in constant three-dimensional domain $\Omega\subset\mathbb{R}^{3}$. The numerical values of the parameters of the +obstacle problems are: $\eta=0.2$, $c=1.1$, $f$ is computed by formula~(\ref{eq:18}) and final time $T=0.02$. Moreover, three time steps ($NbSteps=3$) +are computed with $k=0.0066$. As the discretization matrix is constant along the time steps, the convergence properties of the iterative algorithms +do not change. Thus, the performance characteristics obtained with three time steps will still be valid for more time steps. The initial function +$u(0,x,y,z)$ of the obstacle problem~(\ref{eq:01}) is set to $0$, with a constraint $u\geq\phi=0$. The relaxation parameter $\gamma$ used by the +projected Richardson method is computed automatically thanks to the diagonal entries of the discretization matrix. The formula and its proof can be +found in~\cite{ref11}, Section~2.3. The convergence tolerance threshold $\varepsilon$ is set to $1e$-$04$ and the maximum number of relaxations is +limited to $10^{6}$ relaxations. Finally, the number of threads per block is set to $256$ threads, which gives, in general, good performances for +most GPU applications. We have performed some tests for the execution configurations and we have noticed that the best configuration of the $256$ +threads per block is an organization into two dimensions of sizes $(64,4)$. + +\begin{table}[!h] +\centering +\begin{tabular}{|c|c|c|c|c|c|} +\hline +\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-5} + + & $\mathbf{T_{cpu}}$ & {\bf \#relax.} & $\mathbf{T_{cpu}}$ & {\bf \#relax.} & \\ \hline \hline + +$256^{3}$ & $575.22$ & $198,288$ & $539.25$ & $198,613$ & $6.25$ \\ \hline \hline + +$512^{3}$ & $19,250.25$ & $750,912$ & $18,237.14$ & $769,611$ & $5.26$ \\ \hline \hline + +$768^{3}$ & $206,159.44$ & $1,635,264$ & $183,582.60$ & $1,577,004$ & $10.95$ \\ \hline \hline + +$800^{3}$ & $222,108.09$ & $1,769,232$ & $188,790.04$ & $1,701,735$ & $15.00$ \\ \hline +\end{tabular} +\vspace{0.5cm} +\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 24 CPU cores.} +\label{tab:01} +\end{table} + +\begin{table}[!h] +\centering +\begin{tabular}{|c|c|c|c|c|c|c|c|} +\hline +\multirow{2}{*}{\bf Pb. size} & \multicolumn{3}{c|}{\bf Synchronous} & \multicolumn{3}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-7} + + & $\mathbf{T_{gpu}}$ & {\bf \#relax.} & $\mathbf{\tau}$ & $\mathbf{T_{gpu}}$ & {\bf \#relax.} & $\mathbf{\tau}$ & \\ \hline \hline + +$256^{3}$ & $29.67$ & $100,692$ & $19.39$ & $18.00$ & $94,215$ & $29.96$ & $39.33$ \\ \hline \hline + +$512^{3}$ & $521.83$ & $381,300$ & $36.89$ & $425.15$ & $347,279$ & $42.89$ & $18.53$ \\ \hline \hline + +$768^{3}$ & $4,112.68$ & $831,144$ & $50.13$ & $3,313.87$ & $750,232$ & $55.40$ & $19.42$ \\ \hline \hline + +$800^{3}$ & $3,950.87$ & $899,088$ & $56.22$ & $3,636.57$ & $834,900$ & $51.91$ & $7.95$ \\ \hline +\end{tabular} +\vspace{0.5cm} +\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 12 GPUs.} +\label{tab:02} +\end{table} + +The performance measures that we took into account are the execution times and the number of relaxations performed by the parallel iterative algorithms, +both synchronous and asynchronous versions, on the GPU and CPU clusters. These algorithms are used for solving nonlinear systems derived from the discretization +of obstacle problems of sizes $256^{3}$, $512^{3}$, $768^{3}$ and $800^{3}$. In Table~\ref{tab:01} and Table~\ref{tab:02}, we show the performances +of the parallel synchronous and asynchronous algorithms of the projected Richardson method implemented, respectively, on a cluster of $24$ CPU cores +and on a cluster of $12$ GPUs. In these tables, the execution time defines the time spent by the slowest computing node and the number of relaxations +is computed as the summation of those carried out by all computing nodes. + +In the sixth column of Table~\ref{tab:01} and in the eighth column of Table~\ref{tab:02}, we give the gains in $\%$ obtained by using an +asynchronous algorithm compared to a synchronous one. We can notice that the asynchronous version on CPU and GPU clusters is slightly faster +than the synchronous one for both methods. Indeed, the cluster of tests is composed of local and homogeneous nodes communicating via low-latency +connections. So, in the case of distant and/or heterogeneous nodes (or even with geographically distant clusters) the asynchronous version +would be faster than the synchronous one. However, the gains obtained on the GPU cluster are better than those obtained on the CPU cluster. +In fact, the computation times are reduced by accelerating the computations on GPUs while the communication times still unchanged. + +The fourth and seventh columns of Table~\ref{tab:02} show the relative gains obtained by executing the parallel algorithms on the cluster +of $12$ GPUs instead on the cluster of $24$ CPU cores. We compute the relative gain $\tau$ as a ratio between the execution time $T_{cpu}$ +spent on the CPU cluster over that $T_{gpu}$ spent on the GPU cluster: \[\tau=\frac{T_{cpu}}{T_{gpu}}.\] We can see from these ratios that +solving large obstacle problems is faster on the GPU cluster than on the CPU cluster. Indeed, the GPUs are more efficient than their +counterpart CPUs to execute large data-parallel operations. In addition, the projected Richardson method is implemented as a fixed point-based +iteration and uses the Jacobi vector updates that allow a well thread-parallelization on GPUs, such that each GPU thread is in charge +of one vector component at a time without being dependent on other vector components computed by other threads. Then, this allow to exploit +at best the high performance computing of the GPUs by using all the GPU resources and avoiding the idle cores. + +Finally, the number of relaxations performed by the parallel synchronous algorithm is different in the CPU and GPU versions, because the number +of computing nodes involved in the GPU cluster and in the CPU cluster is different. In the CPU case, $24$ computing nodes ($24$ CPU cores) are +considered, whereas in the GPU case, $12$ computing nodes ($12$ GPUs) are considered. As the number of relaxations depends on the domain decomposition, +consequently it also depends on the number of computing nodes. + + +%%--------------------------%% +%% SECTION 6 %% +%%--------------------------%% +\section{Red-Black ordering technique} +\label{sec:06} +As is well-known, the Jacobi method is characterized by a slow convergence rate compared to some iterative methods (for example Gauss-Seidel method). +So, in this section, we present some solutions to reduce the execution time and the number of relaxations and, more specifically, to speed up the +convergence of the parallel projected Richardson method on the GPU cluster. We propose to use the point red-black ordering technique to accelerate +the convergence. This technique is often used to increase the parallelism of iterative methods for solving linear systems~\cite{ref13,ref14,ref15}. +We apply it to the projected Richardson method as a compromise between the Jacobi and Gauss-Seidel iterative methods. + +The general principle of the red-black technique is as follows. Let $t$ be the summation of the integer $x$-, $y$- and $z$-coordinates of a vector +element $u(x,y,z)$ on a three-dimensional domain: $t=x+y+z$. As is shown in Figure~\ref{fig:06.01}, the red-black ordering technique consists in the +parallel computing of the red vector elements having even value $t$ by using the values of the black ones then, the parallel computing of the black +vector elements having odd values $t$ by using the new values of the red ones. + +\begin{figure} +\centering + \mbox{\subfigure[Red-black ordering on x, y and z axises]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir}\label{fig:06.01}}\quad + \subfigure[Red-black ordering on y axis]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir-y}\label{fig:06.02}}} +\caption{Red-black ordering for computing the iterate vector elements in a three-dimensional space.} +\end{figure} + +This technique can be implemented on the GPU in two different manners: +\begin{itemize*} +\item among all launched threads ($NX\times ny$ threads), only one thread out of two computes its red or black vector element at a time or, +\item all launched threads (on average half of $NX\times ny$ threads) compute the red vector elements first and, then, the black ones. +\end{itemize*} +However, in both solutions, for each memory transaction, only half of the memory segment addressed by a half-warp is used. So, the computation of the +red and black vector elements leads to use twice the initial number of memory transactions. Then, we apply the point red-black ordering accordingly to +the $y$-coordinate, as is shown in Figure~\ref{fig:06.02}. In this case, the vector elements having even $y$-coordinate are computed in parallel using +the values of those having odd $y$-coordinate and then vice-versa. Moreover, in the GPU implementation of the parallel projected Richardson method (Section~\ref{sec:04}), +we have shown that a sub-problem of size $(NX\times ny\times nz)$ is decomposed into $nz$ grids of size $(NX\times ny)$. Then, each kernel is executed +in parallel by $NX\times ny$ GPU threads, so that each thread is in charge of $nz$ vector elements along $z$-axis (one vector element in each grid of +the sub-problem). So, we propose to use the new values of the vector elements computed in grid $i$ to compute those of the vector elements in grid $i+1$. +Listing~\ref{list:04} describes the kernel of the matrix-vector multiplication and the kernel of the vector elements updates of the parallel projected +Richardson method using the red-black ordering technique. + +\lstinputlisting[label=list:04,caption=GPU kernels of the projected Richardson method using the red-black technique]{Chapters/chapter13/ex4.cu} + +Finally, we exploit the concurrent executions between the host functions and the GPU kernels provided by the GPU hardware and software. In fact, the kernel +launches are asynchronous (when this environment variable is not disabled on the GPUs), such that the control is returned to the host (MPI process) before +the GPU has completed the requested task (kernel)~\cite{ref12}. Therefore, all the kernels necessary to update the local vector elements, $u(x,y,z)$ where +$0<y<(ny-1)$ and $0<z<(nz-1)$, are executed first. Then, the values associated to the bordering vector elements are exchanged between the neighbors. Finally, +the values of the vector elements associated to the bordering vector elements are updated. In this case, the computation of the local vector elements is +performed concurrently with the data exchanges between neighboring CPUs and this in both synchronous and asynchronous cases. + +\begin{table}[!h] +\centering +\begin{tabular}{|c|c|c|c|c|c|} +\hline +\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-5} + + & $\mathbf{T_{gpu}}$ & {\bf \#relax.} & $\mathbf{T_{gpu}}$ & {\bf \#relax.} & \\ \hline \hline + +$256^{3}$ & $18.37$ & $71,988$ & $12.58$ & $67,638$ & $31.52$ \\ \hline \hline + +$512^{3}$ & $349.23$ & $271,188$ & $289.41$ & $246,036$ & $17.13$ \\ \hline \hline + +$768^{3}$ & $2,773.65$ & $590,652$ & $2,222.22$ & $532,806$ & $19.88$ \\ \hline \hline + +$800^{3}$ & $2,748.23$ & $638,916$ & $2,502.61$ & $592,525$ & $8.92$ \\ \hline +\end{tabular} +\vspace{0.5cm} +\caption{Execution times in seconds of the parallel projected Richardson method using read-black ordering technique implemented on a cluster of 12 GPUs.} +\label{tab:03} +\end{table} + +In Table~\ref{tab:03}, we report the execution times and the number of relaxations performed on a cluster of $12$ GPUs by the parallel projected Richardson +algorithms; it can be noted that the performances of the projected Richardson are improved by using the point read-black ordering. We compare the performances +of the parallel projected Richardson method with and without this later ordering (Tables~\ref{tab:02} and~\ref{tab:03}). We can notice that both parallel synchronous +and asynchronous algorithms are faster when they use the red-black ordering. Indeed, we can see in Table~\ref{tab:03} that the execution times of these algorithms +are reduced, on average, by $32\%$ compared to those shown in Table~\ref{tab:02}. + +\begin{figure} +\centerline{\includegraphics[scale=0.9]{Chapters/chapter13/figures/scale}} +\caption{Weak scaling of both synchronous and asynchronous algorithms of the projected Richardson method using red-black ordering technique.} +\label{fig:07} +\end{figure} + +In Figure~\ref{fig:07}, we study the ratio between the computation time and the communication time of the parallel projected Richardson algorithms on a GPU cluster. +The experimental tests are carried out on a cluster composed of one to ten Tesla GPUs. We have focused on the weak scaling of both parallel, synchronous and asynchronous, +algorithms using the red-black ordering technique. For this, we have fixed the size of a sub-problem to $256^{3}$ per computing node (a CPU core and a GPU). Then, +Figure~\ref{fig:07} shows the number of relaxations performed, on average, per second by a computing node. We can see from this figure that the efficiency of the +asynchronous algorithm is almost stable, while that of the synchronous algorithm decreases (down to $81\%$ in this example) with the increasing of the number of +computing nodes on the cluster. This is due to the fact that the ratio between the time of the computation over that of the communication is reduced when the computations +are performed on GPUs. Indeed, GPUs compute faster than CPUs and communications are more time consuming. In this context, asynchronous algorithms are more scalable +than synchronous ones. So, with large scale GPU clusters, synchronous algorithms might be more penalized by communications, as can be deduced from Figure~\ref{fig:07}. +That is why we think that asynchronous iterative algorithms are all the more interesting in this case. + + +%%--------------------------%% +%% SECTION 7 %% +%%--------------------------%% +\section{Conclusion} +\label{sec:07} +Our main contribution, in this chapter, is the parallel implementation of an asynchronous iterative method on GPU clusters for solving large scale nonlinear +systems derived from the spatial discretization of three-dimensional obstacle problems. For this, we have implemented both synchronous and asynchronous algorithms of the +Richardson iterative method using a projection on a convex set. Indeed, this method uses point-based iterations of the Jacobi method that are very easy to parallelize on +parallel computers. We have shown that its adapted parallel algorithms to GPU architectures allows to exploit at best the computing power of the GPUs and to accelerate the +resolution of large nonlinear systems. Consequently, the experimental results have shown that solving nonlinear systems of large obstacle problems with this method is about +fifty times faster on a cluster of $12$ GPUs than on a cluster of $24$ CPU cores. Moreover, we have applied to this projected Richardson method the red-black ordering technique +which allows it to improve its convergence rate. Thus, the execution times of both parallel algorithms performed on the cluster of $12$ GPUs are reduced on average of $32\%$. + +Afterwards, the experiments have shown that the asynchronous version is slightly more efficient than the synchronous one. In fact, the computations are accelerated by using GPUs +while the communication times still unchanged. In addition, we have studied the weak-scaling in the synchronous and asynchronous cases, which has confirmed that the ratio between +the computations and the communications are reduced when using a cluster of GPUs. We highlight that asynchronous iterative algorithms are more scalable than synchronous ones. +Therefore, we can conclude that asynchronous iterations are well suited to tackle scalability issues on GPU clusters. + +In future works, we plan to perform experiments on large scale GPU clusters and on geographically distant GPU clusters, because we expect that asynchronous versions would +be faster and more scalable on such architectures. Furthermore, we want to study the performance behavior and the scalability of other numerical algorithms which support, +if possible, the model of asynchronous iterations. + +\putbib[Chapters/chapter13/biblio13] + diff --git a/BookGPU/Chapters/chapter13/ex1.cu b/BookGPU/Chapters/chapter13/ex1.cu new file mode 100644 index 0000000..546d8ef --- /dev/null +++ b/BookGPU/Chapters/chapter13/ex1.cu @@ -0,0 +1,30 @@ +/* GPU kernel */ +__global__ void kernel(..., int n, int nx, int ny, int slices, int stride, ...) +{ + int tx = blockIdx.x * blockDim.x + threadIdx.x;//x-coordinate + int ty = blockIdx.y * blockDim.y + threadIdx.y;//y-coordinate + int tid = tx + ty * nx; //thread ID + for(int i=0; i<slices; i++){ + if((tx<nx) && (ty<ny) && (tid<n)){ + ... + } + tid += stride; + } +} + +/* CPU function */ +void Function(...) +{ + int n = NX * ny * nz; //size of the sub-problem + int slices = nz; + int stride = NX * ny; + int bx = 64, by = 4; + int gx = (NX + bx - 1) / bx; + int gy = (ny + by - 1) / by; + dim3 block(bx,by); //dimensions of a thread block + dim3 grid(gx,gy); //dimensions of the grid + ... + kernel<<<grid,block>>>(..., n, NX, ny, slices, stride, ...); + ... +} + diff --git a/BookGPU/Chapters/chapter13/ex2.cu b/BookGPU/Chapters/chapter13/ex2.cu new file mode 100644 index 0000000..f5d30ce --- /dev/null +++ b/BookGPU/Chapters/chapter13/ex2.cu @@ -0,0 +1,48 @@ +/* Kernel of the matrix-vector multiplication */ +__global__ void MV_Multiplication (..., double* U, double* Y) +{ + ... + //Matrix coefficients filled in registers + + for(int tz=0; tz<slices; tz++){ + if((tx<NX) && (ty<ny) && (tid<n)){ + double sum = Center * fetch_double(U, tid); + if(tx != 0) sum += West * fetch_double(U,tid-1); + if(tx != (NX-1)) sum += East * fetch_double(U,tid+1); + if(ty != 0) sum += South * fetch_double(U,tid-NX); + if(ty != (ny-1)) sum += North * fetch_double(U,tid+NX); + if(tz != 0) sum += Rear * fetch_double(U,tid-NX*ny); + if(tz != (nz-1)) sum += Front * fetch_double(U,tid+NX*ny); + Y[tid] = sum; + } + tid += stride; + } +} + +/* Kernel of the vector elements updates */ +__global__ void Vector_Updates(..., double* G,double* Y,double* U) +{ + ... + //Matrix coefficient filled in a register: Center + + for(int tz=0; tz<slices; tz++){ + if((tx<NX) && (ty<ny) && (tid<n)){ + double var = (G[tid]-Y[tid]) / Center + fetch_double(U,tid); + if(var < 0.0) var = 0.0; //projection + U[tid] = var; + } + tid += stride; + } +} + +/* Function to be executed by the CPU */ +void Computation_New_Vector_Elements(double*A,double* G,double* U) +{ + double *Y; + //Allocate a GPU memory space for the vector Y + //Configure the kernel execution: grid and block + //Elements of vector U filled in the texture memory + + MV_Multiplication<<<grid,block>>>(..., U, Y); + Vector_Updates<<<grid,block>>>(..., G, Y, U); +} diff --git a/BookGPU/Chapters/chapter13/ex3.cu b/BookGPU/Chapters/chapter13/ex3.cu new file mode 100644 index 0000000..6090907 --- /dev/null +++ b/BookGPU/Chapters/chapter13/ex3.cu @@ -0,0 +1,8 @@ +texture<int2,1> v; + +static __inline__ +__device__ double fetch_double(texture<int2,1> v, int i) +{ + int2 w = tex1Dfetch(v, i); + return __hiloint2double(w.y, w.x); +} diff --git a/BookGPU/Chapters/chapter13/ex4.cu b/BookGPU/Chapters/chapter13/ex4.cu new file mode 100644 index 0000000..738ca3d --- /dev/null +++ b/BookGPU/Chapters/chapter13/ex4.cu @@ -0,0 +1,54 @@ +/* Kernel of the matrix-vector multiplication */ +__global__ void MV_Multiplication(...,double*G,double*U,double*Y) +{ + ... + //Matrix coefficients filled in registers: + //Center, West, East, Front + + for(int tz=0; tz<slices; tz++){ + if((tx<NX) && (ty<ny) && (tid<n)){ + double sum = G[tid] - Center * fetch_double(U,tid); + if(tx != 0) sum -= West * fetch_double(U,tid-1); + if(tx != (NX-1)) sum -= East * fetch_double(U,tid+1); + if(tz != (nz-1)) sum -= Front * fetch_double(U,tid+NX*ny); + Y[tid] = sum; + } + tid += stride; + } +} + +/* Kernel of the vector elements updates */ +__global__ void Vector_Updates(..., int rb, double* Y, double* U) +{ + ... + double val; //vector component computed in previous grid + //Matrix coefficients filled in registers: + //Center, South, North, Rear + + for(int tz=0; tz<slices; tz++){ + if((tx<NX) && (ty<ny) && (tid<n) && ((ty&1)==rb)){ + double sum = Y[tid] - South * fetch_double(U,tid-NX) - + North * fetch_double(U,tid+NX); + //val: computed in previous grid + if(tz != 0) sum -= Rear * val; + sum = (sum / Center) + fetch_double(U,tid); + if(sum < 0) sum = 0; //projection + U[tid] = val = sum; //update of U + } + tid += stride; + } +} + +/* CPU Function */ +void Computation_New_Vector_Components(double*A,double*G,double*U) +{ + double *Y; + int red=0, black=1; + //Allocate a GPU memory space for the vector Y + //Configure the kernel execution: grid and block + //Elements of vector U filled in the texture memory + + MV_Multiplication<<<grid,block>>>(..., G, U, Y); + Vector_Updates<<<grid,block>>>(..., red, Y, U); + Vector_Updates<<<grid,block>>>(..., black, Y, U); +} diff --git a/BookGPU/Chapters/chapter13/figures/bib.bib b/BookGPU/Chapters/chapter13/figures/bib.bib new file mode 100644 index 0000000..0262701 --- /dev/null +++ b/BookGPU/Chapters/chapter13/figures/bib.bib @@ -0,0 +1,358 @@ +@article{Zha07, + author = {Zhao, Y.}, + title = {Lattice {B}oltzmann {B}ased {PDE} {S}olver on the {GPU}}, + journal = {The Visual Computer}, + volume = {24}, + number = {5}, + pages = {323--333}, + year = {2008}, +} + +@article{Mar09, + author = {Maringanti, A. and Athavale, V. and Patkar, S. 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C. and Tseng, P.}, + title = {Convergence {R}ate {A}nalysis of an {A}synchronous {S}pace {D}ecomposition {M}ethod for {C}onvex {M}inimization}, + journal = {Mathematics of Computation}, + volume = {71}, + number = {239}, + pages = {1105--1135}, + year = {2002}, +} + +@article{Tai03, + author = {Tai, X. C.}, + title = {Rate of {C}onvergence for {S}ome {S}onstraint {D}ecomposition {M}ethods for {N}onlinear {V}ariational {I}nequalities}, + journal = {Numer. Math.}, + volume = {93}, + number = {}, + pages = {755--786}, + year = {2003}, +} + +@article{Badea04, + author = {Badea, L. and Tai, X. C. and Wang, J.}, + title = {Convergence {R}ate {A}nalysis of a {M}ultiplicative {S}chwarz {M}ethod for {V}ariational {I}nequalities}, + journal = {SIAM J. on Numer. 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Cosnard et al. eds) Amsterdam: North-Holland}, + volume = {}, + number = {}, + pages = {85--96}, + year = {1986}, +} + +@article{GS91b, + author = {Giraud, L. and Spiteri, P.}, + title = {R\'esolution {P}arall\`ele de {P}robl\`emes aux {L}imites non {L}in\'eaires}, + journal = {M2 AN}, + volume = {25}, + number = {}, + pages = {579--606}, + year = {1991}, +} + +@article{MSE98, + author = {Miellou, J. C. and El Baz, D. and Spiteri, P.}, + title = {A {N}ew {C}lass of {A}synchronous {I}terative {A}lgorithms with {O}rder {I}nterval}, + journal = {Mathematics of Computation}, + volume = {67}, + number = {221}, + pages = {237--255}, + year = {1998}, +} + +@article{CM69, + author = {Chazan, D. and Miranker, W.}, + title = {Chaotic {R}elaxation}, + journal = {Linear Algebra Appl.}, + volume = {2}, + number = {}, + pages = {199--222}, + year = {1969}, +} + +@article{M75a, + author = {Miellou, J. C.}, + title = {Algorithmes de {R}elaxation {C}haotique \`a {R}etards}, + journal = {RAIRO Analyse num\'erique}, + volume = {R1}, + number = {}, + pages = {55--82}, + year = {1975}, +} + +@article{Baudet78, + author = {Baudet, G.}, + title = {Asynchronous {I}terative {M}ethods for {M}ultiprocessors}, + journal = {Journal Assoc. Comput. Mach.}, + volume = {25}, + number = {}, + pages = {226--244}, + year = {1978}, +} + +@article{BT89, + author = {Bertsekas, D. and Tsitsiklis, J.}, + title = {Parallel and {D}istributed {C}omputation, {N}umerical {M}ethods}, + journal = {Prentice Hall Englewood Cliffs N. J. (1989)}, + volume = {}, + number = {}, + pages = {}, + year = {1989}, +} + +@article{Bahi07, + author = {Bahi, J. M. and Contassot-Vivier, S. and Couturier, R.}, + title = {Parallel {I}terative {A}lgorithms: from {S}equential to {G}rid {C}omputing}, + journal = {Chapman \& Hall/CRC, Numerical Analysis \& Scientific Computating 1 (2007)}, + volume = {}, + number = {}, + pages = {}, + year = {2007}, +} + +@article{Nvi10, + author = {Nvidia}, + title = {{NVIDIA} {CUDA} {C} {P}rogramming {G}uide}, + journal = {Version 4.0 (2011)}, + volume = {}, + number = {}, + pages = {}, + year = {2011}, +} + +@article{Nvid10, + author = {Nvidia}, + title = {{CUDA} {CUBLAS} {L}ibrary (2010)}, + journal = {}, + volume = {}, + number = {}, + pages = {}, + year = {2010}, +} + +@article{ref10, + author = {Micikevicius, P}, + title = {{3D} {F}inite {D}ifference {C}omputation on {GPUs} using {CUDA}}, + journal = {Proceedings of 2nd Workshop on General Purpose Processing on Graphics Processing Units}, + volume = {}, + number = {}, + pages = {79--84}, + year = {2009}, +} + +@article{ref11, + author = {Leist, A. and Playne, D. 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L.}, + title = {{A}pplication des {I}néquations {V}ariationnelles en {C}ontr\^ole {O}ptimal {S}tochastique}, + journal = {Dunod, Paris (1978)}, + volume = {}, + number = {}, + pages = {}, + year = {1978}, +} + diff --git a/BookGPU/Chapters/chapter13/figures/cluster.eps b/BookGPU/Chapters/chapter13/figures/cluster.eps new file mode 100644 index 0000000..d143f51 --- /dev/null +++ b/BookGPU/Chapters/chapter13/figures/cluster.eps @@ -0,0 +1,710 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: cluster.fig +%%Creator: fig2dev Version 3.2 Patchlevel 5c +%%CreationDate: Mon Jan 14 10:39:57 2013 +%%BoundingBox: 0 0 1391 706 +%Magnification: 1.0000 +%%EndComments +%%BeginProlog +/$F2psDict 200 dict def +$F2psDict begin +$F2psDict /mtrx matrix put +/col-1 {0 setgray} bind def +/col0 {0.000 0.000 0.000 srgb} bind def +/col1 {0.000 0.000 1.000 srgb} bind def +/col2 {0.000 1.000 0.000 srgb} bind def +/col3 {0.000 1.000 1.000 srgb} bind def +/col4 {1.000 0.000 0.000 srgb} bind def +/col5 {1.000 0.000 1.000 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+%*********************************************************************** +%* * +%* Object: Image decoding PS-routine Date: 01.02.93 * +%* Author: Evgeni CHERNYAEV (chernaev@vxcern.cern.ch) * +%* * +%* Function: Display a run-length encoded color image. * +%* The image is displayed in color on viewers and printers * +%* that support color Postscript, otherwise it is displayed * +%* as grayscale. * +%* * +%*********************************************************************** +/byte 1 string def +/color 3 string def +systemdict /colorimage known { /cnt 3 def } { /cnt 1 def } ifelse +/String 256 cnt mul string def +%*********************************************************************** +/DecodePacket % Decode color packet * +%*********************************************************************** +{ + currentfile byte readhexstring pop 0 get + /Nbyte exch 1 add cnt mul def + /color ColorMap currentfile byte readhexstring pop 0 get get def + String dup + 0 cnt Nbyte 1 sub { color putinterval dup } for + pop 0 Nbyte getinterval +} bind def +%*********************************************************************** +/DisplayImage % Display run-length encoded color image * +%*********************************************************************** +{ + gsave + currentfile String readline pop + token { /columns exch def } { } ifelse + token { /rows exch def pop } { } ifelse + currentfile String readline pop + token { /Ncol exch def pop } { } ifelse + /ColorMap Ncol array def + systemdict /colorimage known { + 0 1 Ncol 1 sub { + ColorMap exch + currentfile 3 string readhexstring pop put + } for + columns rows 8 + [ columns 0 0 rows neg 0 rows ] + { DecodePacket } false 3 colorimage + }{ + 0 1 Ncol 1 sub { + ColorMap exch + 1 string dup 0 + currentfile color readhexstring pop pop + color 0 get 0.299 mul + color 1 get 0.587 mul add + color 2 get 0.114 mul add + cvi put put + } for + columns rows 8 + [ columns 0 0 rows neg 0 rows ] + { DecodePacket } image + } ifelse + 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