X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/f463119047e7f3b45c777e90a8702e74aba0f520..663c29e4f7bb8e7d7d9d75ef19a6be1fed40f48a:/paper.tex diff --git a/paper.tex b/paper.tex index f3ef835..3120437 100644 --- a/paper.tex +++ b/paper.tex @@ -1,4 +1,13 @@ -\documentclass[conference]{IEEEtran} +\documentclass[times]{cpeauth} + +\usepackage{moreverb} + +%\usepackage[dvips,colorlinks,bookmarksopen,bookmarksnumbered,citecolor=red,urlcolor=red]{hyperref} + +%\newcommand\BibTeX{{\rmfamily B\kern-.05em \textsc{i\kern-.025em b}\kern-.08em +%T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} + +\def\volumeyear{2015} \usepackage{graphicx} \usepackage{wrapfig} @@ -27,6 +36,7 @@ \usepackage{xspace} \usepackage[textsize=footnotesize]{todonotes} + \newcommand{\AG}[2][inline]{% \todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace} \newcommand{\RC}[2][inline]{% @@ -53,36 +63,36 @@ \definecolor{Gray}{gray}{0.9} + \begin{document} \RCE{Titre a confirmer.} - \title{Comparative performance analysis of simulated grid-enabled numerical iterative algorithms} +%\itshape{\journalnamelc}\footnotemark[2]} -\author{% - \IEEEauthorblockN{% - Charles Emile Ramamonjisoa and +\author{ Charles Emile Ramamonjisoa and David Laiymani and Arnaud Giersch and Lilia Ziane Khodja and Raphaël Couturier - } - \IEEEauthorblockA{% +} + +\address{ + \centering Femto-ST Institute - DISC Department\\ Université de Franche-Comté\\ Belfort\\ Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr} - } } -\maketitle +%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be \begin{abstract} ABSTRACT +\end{abstract} +\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance} -Keywords : Algorithm distributed iterative asynchronous simulation simgrid performance - -\end{abstract} +\maketitle \section{Introduction} @@ -90,7 +100,46 @@ Keywords : Algorithm distributed iterative asynchronous simulation simgrid perfo \section{SimGrid} -\section{Simulation of the multisplitting method} +%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Two-stage splitting methods} +\label{sec:04} +\subsection{Multisplitting methods for sparse linear systems} +\label{sec:04.01} +Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$ +\begin{equation} +Ax=b, +\label{eq:01} +\end{equation} +where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows +\begin{equation} +x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots +\label{eq:02} +\end{equation} +where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix. The iterations of the multisplitting methods can naturally be computed in parallel such that each processor or a group of processors is responsible for solving one splitting as a linear sub-system +\begin{equation} +M_\ell y_\ell^{k+1} = R_\ell^k,\mbox{~such that~} R_\ell^k = N_\ell x^k_\ell + b, +\label{eq:03} +\end{equation} +then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01}) +\begin{equation} +x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y^{k+1}_\ell. +\label{eq:04} +\end{equation} +The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors. It is dependent on the condition +\begin{equation} +\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1, +\label{eq:05} +\end{equation} +where $\rho$ is the spectral radius of the square matrix. The different linear splittings~(\ref{eq:03}) arising from the multisplitting of matrix $A$can be solved exactly with a direct method or approximated with an iterative method. When the inner method used to solve the linear sub-systems is iterative, the multisplitting method is called {\it inner-outer iterative method} or {\it two-stage multisplitting method}. + +In this paper we are focused on two-stage multisplitting methods where the well-known iterative method GMRES is used as an inner iteration. + +\subsection{Simulation of two-stage methods using SimGrid framework} + +%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%% \section{Experimental, Results and Comments} @@ -194,7 +243,7 @@ and our comments. \textit{3.a Executing the algorithms on various computational grid architecture scaling up the input matrix size} - +\\ % environment \begin{footnotesize} @@ -209,18 +258,24 @@ architecture scaling up the input matrix size} Table 1 : Clusters x Nodes with NX=150 or NX=170 -\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger} -\begin{wrapfigure}{l}{50mm} -\centering -\includegraphics[width=50mm]{Cluster x Nodes NX=150 and NX=170.jpg} -\caption{Cluster x Nodes NX=150 and NX=170 \label{overflow}} -\end{wrapfigure} +\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger} The results in figure 1 show the non-variation of the number of iterations of classical GMRES for a given input matrix size; it is not -the case for the multisplitting method. Unless the 8x8 cluster, the time +the case for the multisplitting method. + +%\begin{wrapfigure}{l}{60mm} +\begin{figure} [ht!] +\centering +\includegraphics[width=60mm]{cluster_x_nodes_nx_150_and_nx_170.pdf} +\caption{Cluster x Nodes NX=150 and NX=170} +%\label{overflow}} +\end{figure} +%\end{wrapfigure} + +Unless the 8x8 cluster, the time execution difference between the two algorithms is important when comparing between different grid architectures, even with the same number of processors (like 2x16 and 4x8 = 32 processors for example). The @@ -237,7 +292,7 @@ matrix size. Grid & 2x16, 4x8\\ %\hline Network & N1 : bw=10Gbs-lat=8E-06 \\ %\hline - & N2 : bw=1Gbs-lat=5E-05 \\ - Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline + Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline \\ \end{tabular} \end{footnotesize} @@ -245,11 +300,14 @@ matrix size. %\RCE{idem pour tous les tableaux de donnees} -\begin{wrapfigure}{l}{45mm} +%\begin{wrapfigure}{l}{60mm} +\begin{figure} [ht!] \centering -\includegraphics[width=50mm]{Cluster x Nodes N1 x N2.jpg} -\caption{Cluster x Nodes N1 x N2\label{overflow}} -\end{wrapfigure} +\includegraphics[width=60mm]{cluster_x_nodes_n1_x_n2.pdf} +\caption{Cluster x Nodes N1 x N2} +%\label{overflow}} +\end{figure} +%\end{wrapfigure} The experiments compare the behavior of the algorithms running first on speed inter- cluster network (N1) and a less performant network (N2). @@ -259,7 +317,7 @@ performance was increased in a factor of 2. The results depict also that when the network speed drops down, the difference between the execution times can reach more than 25\%. -\textit{3.c Network latency impacts on performance} +\textit{\\\\\\\\\\\\\\\\\\3.c Network latency impacts on performance} % environment \begin{footnotesize} @@ -267,18 +325,19 @@ times can reach more than 25\%. \hline Grid & 2x16\\ %\hline Network & N1 : bw=1Gbs \\ %\hline - Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline + Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline\\ \end{tabular} \end{footnotesize} Table 3 : Network latency impact -\begin{wrapfigure}{l}{60mm} +\begin{figure} [ht!] \centering -\includegraphics[width=60mm]{Network latency impact on execution time.jpg} -\caption{Network latency impact on execution time\label{overflow}} -\end{wrapfigure} +\includegraphics[width=60mm]{network_latency_impact_on_execution_time.pdf} +\caption{Network latency impact on execution time} +%\label{overflow}} +\end{figure} According the results in table and figure 3, degradation of the network @@ -305,11 +364,12 @@ of magnitude with a latency of 8.10$^{-6}$. Table 4 : Network bandwidth impact -\begin{wrapfigure}{l}{60mm} +\begin{figure} [ht!] \centering -\includegraphics[width=60mm]{Network bandwith impact on execution time.jpg} -\caption{Network bandwith impact on execution time\label{overflow}} -\end{wrapfigure} +\includegraphics[width=60mm]{network_bandwith_impact_on_execution_time.pdf} +\caption{Network bandwith impact on execution time} +%\label{overflow} +\end{figure} @@ -333,11 +393,12 @@ a gain of 40\% which is only around 24\% for classical GMRES. Table 5 : Input matrix size impact -\begin{wrapfigure}{l}{50mm} +\begin{figure} [ht!] \centering -\includegraphics[width=60mm]{Pb size impact on execution time.jpg} -\caption{Pb size impact on execution time\label{overflow}} -\end{wrapfigure} +\includegraphics[width=60mm]{pb_size_impact_on_execution_time.pdf} +\caption{Pb size impact on execution time} +%\label{overflow}} +\end{figure} In this experimentation, the input matrix size has been set from Nx=Ny=Nz=40 to 200 side elements that is from 40$^{3}$ = 64.000 to @@ -367,11 +428,12 @@ same test has been done with the grid 2x16 getting the same conclusion. Table 6 : CPU Power impact -\begin{wrapfigure}{l}{60mm} +\begin{figure} [ht!] \centering -\includegraphics[width=60mm]{CPU Power impact on execution time.jpg} -\caption{CPU Power impact on execution time\label{overflow}} -\end{wrapfigure} +\includegraphics[width=60mm]{cpu_power_impact_on_execution_time.pdf} +\caption{CPU Power impact on execution time} +%\label{overflow}} +\end{figure} Using the SIMGRID simulator flexibility, we have tried to determine the impact on the algorithms performance in varying the CPU power of the