X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/18a590b4593843d38f5012c20a06f000388301cf..cad2447f35593e377ab1f1b13d88247c31fc43d3:/paper.tex?ds=inline diff --git a/paper.tex b/paper.tex index b9d11d4..886390b 100644 --- a/paper.tex +++ b/paper.tex @@ -21,7 +21,6 @@ \usepackage{algpseudocode} %\usepackage{amsthm} \usepackage{graphicx} -\usepackage[american]{babel} % Extension pour les liens intra-documents (tagged PDF) % et l'affichage correct des URL (commande \url{http://example.com}) %\usepackage{hyperref} @@ -71,27 +70,30 @@ -\begin{document} \RCE{Titre a confirmer.} \title{Comparative performance -analysis of simulated grid-enabled numerical iterative algorithms} +\begin{document} +\title{Grid-enabled simulation of large-scale linear iterative solvers} %\itshape{\journalnamelc}\footnotemark[2]} -\author{ Charles Emile Ramamonjisoa and - David Laiymani and - Arnaud Giersch and - Lilia Ziane Khodja and - Raphaël Couturier +\author{Charles Emile Ramamonjisoa\affil{1}, + David Laiymani\affil{1}, + Arnaud Giersch\affil{1}, + Lilia Ziane Khodja\affil{2} and + Raphaël Couturier\affil{1} } \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} + \affilnum{1}% + Femto-ST Institute, DISC Department, + University of Franche-Comté, + Belfort, France. + Email:~\email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}\break + \affilnum{2} + Department of Aerospace \& Mechanical Engineering, + Non Linear Computational Mechanics, + University of Liege, Liege, Belgium. + Email:~\email{l.zianekhodja@ulg.ac.be} } -%% 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} The behavior of multi-core applications is always a challenge to predict, especially with a new architecture for which no experiment has been performed. With some applications, it is difficult, if not impossible, to build @@ -134,7 +136,7 @@ are often very important. So, in this context it is difficult to optimize a given application for a given architecture. In this way and in order to reduce the access cost to these computing resources it seems very interesting to use a simulation environment. The advantages are numerous: development life cycle, -code debugging, ability to obtain results quickly~\ldots. In counterpart, the simulation results need to be consistent with the real ones. +code debugging, ability to obtain results quickly\dots{} In counterpart, the simulation results need to be consistent with the real ones. In this paper we focus on a class of highly efficient parallel algorithms called \emph{iterative algorithms}. The parallel scheme of iterative methods is quite @@ -163,22 +165,30 @@ application on a given multi-core architecture. Finding good resource allocations policies under varying CPU power, network speeds and loads is very challenging and labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This problematic is even more difficult for the asynchronous scheme where a small -parameter variation of the execution platform can lead to very different numbers -of iterations to reach the converge and so to very different execution times. In -this challenging context we think that the use of a simulation tool can greatly -leverage the possibility of testing various platform scenarios. - -The main contribution of this paper is to show that the use of a simulation tool -(i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real parallel -applications (i.e. large linear system solvers) can help developers to better -tune their application for a given multi-core architecture. To show the validity -of this approach we first compare the simulated execution of the multisplitting -algorithm with the GMRES (Generalized Minimal Residual) -solver~\cite{saad86} in synchronous mode. The obtained results on different -simulated multi-core architectures confirm the real results previously obtained -on non simulated architectures. We also confirm the efficiency of the -asynchronous multisplitting algorithm compared to the synchronous GMRES. In -this way and with a simple computing architecture (a laptop) SimGrid allows us +parameter variation of the execution platform and of the application data can +lead to very different numbers of iterations to reach the converge and so to +very different execution times. In this challenging context we think that the +use of a simulation tool can greatly leverage the possibility of testing various +platform scenarios. + +The {\bf main contribution of this paper} is to show that the use of a +simulation tool (i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real +parallel applications (i.e. large linear system solvers) can help developers to +better tune their application for a given multi-core architecture. To show the +validity of this approach we first compare the simulated execution of the Krylov +multisplitting algorithm with the GMRES (Generalized Minimal Residual) +solver~\cite{saad86} in synchronous mode. The simulation results allow us to +determine which method to choose given a specified multi-core architecture. +Moreover the obtained results on different simulated multi-core architectures +confirm the real results previously obtained on non simulated architectures. +More precisely the simulated results are in accordance (i.e. with the same order +of magnitude) with the works presented in~\cite{couturier15}, which show that +the synchronous multisplitting method is more efficient than GMRES for large +scale clusters. Simulated results also confirm the efficiency of the +asynchronous multisplitting algorithm compared to the synchronous GMRES +especially in case of geographically distant clusters. + +In this way and with a simple computing architecture (a laptop) SimGrid allows us to run a test campaign of a real parallel iterative applications on different simulated multi-core architectures. To our knowledge, there is no related work on the large-scale multi-core simulation of a real synchronous and @@ -192,7 +202,7 @@ experimental results are presented in section~\ref{sec:expe} followed by some concluding remarks and perspectives. -\section{The asynchronous iteration model} +\section{The asynchronous iteration model and the motivations of our work} \label{sec:asynchro} Asynchronous iterative methods have been studied for many years theoritecally and @@ -216,10 +226,77 @@ point. In the asynchronous model, the convergence detection is more tricky as it must not synchronize all the processors. Interested readers can consult~\cite{myBCCV05c,bahi07,ccl09:ij}. +The number of iterations required to reach the convergence is generally greater +for the asynchronous scheme (this number depends depends on the delay of the +messages). Note that, it is not the case in the synchronous mode where the +number of iterations is the same than in the sequential mode. In this way, the +set of the parameters of the platform (number of nodes, power of nodes, +inter and intra clusters bandwidth and latency, \ldots) and of the +application can drastically change the number of iterations required to get the +convergence. It follows that asynchronous iterative algorithms are difficult to +optimize since the financial and deployment costs on large scale multi-core +architecture are often very important. So, prior to delpoyment and tests it +seems very promising to be able to simulate the behavior of asynchronous +iterative algorithms. The problematic is then to show that the results produce +by simulation are in accordance with reality i.e. of the same order of +magnitude. To our knowledge, there is no study on this problematic. + \section{SimGrid} - \label{sec:simgrid} +\label{sec:simgrid} +SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile} is a discrete event simulation framework to study the behavior of large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds and High Performance Computation systems. It is widely used to simulate and evaluate heuristics, prototype applications or even assess legacy MPI applications. It is still actively developed by the scientific community and distributed as an open source software. %%%%%%%%%%%%%%%%%%%%%%%%% +% SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile} +% is a simulation framework to study the behavior of large-scale distributed +% systems. As its name suggests, it emanates from the grid computing community, +% but is nowadays used to study grids, clouds, HPC or peer-to-peer systems. The +% early versions of SimGrid date back from 1999, but it is still actively +% developed and distributed as an open source software. Today, it is one of the +% major generic tools in the field of simulation for large-scale distributed +% systems. + +SimGrid provides several programming interfaces: MSG to simulate Concurrent +Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to +run real applications written in MPI~\cite{MPI}. Apart from the native C +interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming +languages. SMPI is the interface that has been used for the work described in +this paper. The SMPI interface implements about \np[\%]{80} of the MPI 2.0 +standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports +applications written in C or Fortran, with little or no modifications (cf Section IV - paragraph B). + +Within SimGrid, the execution of a distributed application is simulated by a +single process. The application code is really executed, but some operations, +like communications, are intercepted, and their running time is computed +according to the characteristics of the simulated execution platform. The +description of this target platform is given as an input for the execution, by +means of an XML file. It describes the properties of the platform, such as +the computing nodes with their computing power, the interconnection links with +their bandwidth and latency, and the routing strategy. The scheduling of the +simulated processes, as well as the simulated running time of the application +are computed according to these properties. + +To compute the durations of the operations in the simulated world, and to take +into account resource sharing (e.g. bandwidth sharing between competing +communications), SimGrid uses a fluid model. This allows users to run relatively fast +simulations, while still keeping accurate +results~\cite{bedaride+degomme+genaud+al.2013.toward, + velho+schnorr+casanova+al.2013.validity}. Moreover, depending on the +simulated application, SimGrid/SMPI allows to skip long lasting computations and +to only take their duration into account. When the real computations cannot be +skipped, but the results are unimportant for the simulation results, it is +also possible to share dynamically allocated data structures between +several simulated processes, and thus to reduce the whole memory consumption. +These two techniques can help to run simulations on a very large scale. + +The validity of simulations with SimGrid has been asserted by several studies. +See, for example, \cite{velho+schnorr+casanova+al.2013.validity} and articles +referenced therein for the validity of the network models. Comparisons between +real execution of MPI applications on the one hand, and their simulation with +SMPI on the other hand, are presented in~\cite{guermouche+renard.2010.first, + clauss+stillwell+genaud+al.2011.single, + bedaride+degomme+genaud+al.2013.toward}. All these works conclude that +SimGrid is able to simulate pretty accurately the real behavior of the +applications. %%%%%%%%%%%%%%%%%%%%%%%%% \section{Two-stage multisplitting methods} @@ -349,13 +426,13 @@ In addition, the following arguments are given to the programs at runtime: \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$, \item inner precision $\TOLG$ and outer precision $\TOLM$, \item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively, - \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments, \RC{CE tu vérifies, je dis ca de tête} + \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments, \item matrix off-diagonal value is fixed to $-1.0$, \item number of vectors in matrix $S$ (i.e. value of $s$), \item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method, \item maximum number of iterations and precision for the classical GMRES method, \item maximum number of restarts for the Arnorldi process in GMRES method, - \item execution mode: synchronous or asynchronous, + \item execution mode: synchronous or asynchronous. \end{itemize} It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine. @@ -432,13 +509,13 @@ input data. \\ a grid environment} When running a distributed application in a computational grid, many factors may -have a strong impact on the performances. First of all, the architecture of the +have a strong impact on the performance. First of all, the architecture of the grid itself can obviously influence the performance results of the program. The performance gain might be important theoretically when the number of clusters and/or the number of nodes (processors/cores) in each individual cluster increase. -Another important factor impacting the overall performances of the application +Another important factor impacting the overall performance of the application is the network configuration. Two main network parameters can modify drastically the program output results: \begin{enumerate} @@ -464,8 +541,8 @@ and between distant clusters. This parameter is application dependent. \subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode} In the scope of this paper, our first objective is to analyze when the Krylov -Multisplitting method has better performances than the classical GMRES -method. With a synchronous iterative method, better performances mean a +Multisplitting method has better performance than the classical GMRES +method. With a synchronous iterative method, better performance means a smaller number of iterations and execution time before reaching the convergence. For a systematic study, the experiments should figure out that, for various grid parameters values, the simulator will confirm the targeted outcomes, @@ -481,26 +558,26 @@ architectures and scaling up the input matrix size} \ \\ % environment -\begin{figure} [ht!] +\begin{table} [ht!] \begin{center} \begin{tabular}{r c } \hline - Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline + Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline - & N$_{x}$ x N$_{y}$ x N$_{z}$ =170 x 170 x 170 \\ \hline \end{tabular} -\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}} +\caption{Test conditions: various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?} +\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}. Idem dans le texte, les figures, etc.}} +\label{tab:01} \end{center} -\end{figure} +\end{table} -%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger} - -In this section, we analyze the performences of algorithms running on various +In this section, we analyze the performance of algorithms running on various grid configurations (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01} show for all grid configurations the non-variation of the number of iterations of classical GMRES for a given input matrix size; it is not the case for the @@ -514,7 +591,8 @@ multisplitting method. \begin{center} \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf} \end{center} - \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170} + \caption{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170\RC{idem} +\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}} \label{fig:01} \end{figure} @@ -524,23 +602,29 @@ grid architectures, even with the same number of processors (for example, 2x16 and 4x8). We can observ the low sensitivity of the Krylov multisplitting method (compared with the classical GMRES) when scaling up the number of the processors in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs -$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. +$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?} -\subsubsection{Running on two different inter-clusters network speed} -\ \\ +\subsubsection{Running on two different inter-clusters network speeds \\} -\begin{figure} [ht!] +\begin{table} [ht!] \begin{center} \begin{tabular}{r c } \hline - Grid & 2x16, 4x8\\ %\hline + Grid Architecture & 2x16, 4x8\\ %\hline Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \end{tabular} -\caption{Clusters x Nodes - Networks N1 x N2} +\caption{Test conditions: grid 2x16 and 4x8 with networks N1 vs N2} +\label{tab:02} \end{center} -\end{figure} +\end{table} + +These experiments compare the behavior of the algorithms running first on a +speed inter-cluster network (N1) and also on a less performant network (N2). \RC{Il faut définir cela avant...} +Figure~\ref{fig:02} shows that end users will reduce the execution time +for both algorithms when using a grid architecture like 4x16 or 8x8: the reduction is about $2$. The results depict also that when +the network speed drops down (variation of 12.5\%), the difference between the two Multisplitting algorithms execution times can reach more than 25\%. @@ -548,71 +632,69 @@ $40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \begin{figure} [ht!] \centering \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf} -\caption{Cluster x Nodes N1 x N2} +\caption{Grid 2x16 and 4x8 with networks N1 vs N2 +\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}} \label{fig:02} \end{figure} %\end{wrapfigure} -These experiments compare the behavior of the algorithms running first on a -speed inter-cluster network (N1) and also on a less performant network (N2). -Figure~\ref{fig:02} shows that end users will gain to reduce the execution time -for both algorithms in using a grid architecture like 4x16 or 8x8: the -performance was increased by a factor of $2$. The results depict also that when -the network speed drops down (12.5\%), the difference between the execution -times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi et quoi?} -\DL{pas clair} \subsubsection{Network latency impacts on performance} \ \\ -\begin{figure} [ht!] +\begin{table} [ht!] \centering \begin{tabular}{r c } \hline - Grid & 2x16\\ %\hline + Grid Architecture & 2x16\\ %\hline Network & N1 : bw=1Gbs \\ %\hline Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \end{tabular} -\caption{Network latency impacts} -\end{figure} +\caption{Test conditions: network latency impacts} +\label{tab:03} +\end{table} \begin{figure} [ht!] \centering \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf} -\caption{Network latency impacts on execution time} +\caption{Network latency impacts on execution time +\AG{\np{E-6}}} \label{fig:03} \end{figure} -According to the results of Figure~\ref{fig:03}, a degradation of the network -latency from $8.10^{-6}$ to $6.10^{-5}$ implies an absolute time increase of more -than $75\%$ (resp. $82\%$) of the execution for the classical GMRES (resp. Krylov -multisplitting) algorithm. In addition, it appears that the Krylov -multisplitting method tolerates more the network latency variation with a less -rate increase of the execution time. Consequently, in the worst case -($lat=6.10^{-5 }$), the execution time for GMRES is almost the double than the -time of the Krylov multisplitting, even though, the performance was on the same -order of magnitude with a latency of $8.10^{-6}$. +According to the results of Figure~\ref{fig:03}, a degradation of the network +latency from $8.10^{-6}$ to $6.10^{-5}$ implies an absolute time increase of +more than $75\%$ (resp. $82\%$) of the execution for the classical GMRES +(resp. Krylov multisplitting) algorithm. In addition, it appears that the +Krylov multisplitting method tolerates more the network latency variation with a +less rate increase of the execution time.\RC{Les 2 précédentes phrases me + semblent en contradiction....} Consequently, in the worst case ($lat=6.10^{-5 +}$), the execution time for GMRES is almost the double than the time of the +Krylov multisplitting, even though, the performance was on the same order of +magnitude with a latency of $8.10^{-6}$. \subsubsection{Network bandwidth impacts on performance} \ \\ -\begin{figure} [ht!] +\begin{table} [ht!] \centering \begin{tabular}{r c } \hline - Grid & 2x16\\ %\hline + Grid Architecture & 2x16\\ %\hline Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\ \end{tabular} -\caption{Network bandwidth impacts} -\end{figure} +\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}} +\label{tab:04} +\end{table} \begin{figure} [ht!] \centering \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf} -\caption{Network bandwith impacts on execution time} +\caption{Network bandwith impacts on execution time +\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.} \label{fig:04} \end{figure} @@ -624,16 +706,17 @@ of $40\%$ which is only around $24\%$ for the classical GMRES. \subsubsection{Input matrix size impacts on performance} \ \\ -\begin{figure} [ht!] +\begin{table} [ht!] \centering \begin{tabular}{r c } \hline - Grid & 4x8\\ %\hline + Grid Architecture & 4x8\\ %\hline Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \end{tabular} -\caption{Input matrix size impacts} -\end{figure} +\caption{Test conditions: Input matrix size impacts} +\label{tab:05} +\end{table} \begin{figure} [ht!] @@ -649,9 +732,9 @@ In these experiments, the input matrix size has been set from $N_{x} = N_{y} time for both algorithms increases when the input matrix size also increases. But the interesting results are: \begin{enumerate} - \item the drastic increase ($300$ times) \RC{Je ne vois pas cela sur la figure} -of the number of iterations needed to reach the convergence for the classical -GMRES algorithm when the matrix size go beyond $N_{x}=150$; + \item the drastic increase ($10$ times) of the number of iterations needed to + reach the convergence for the classical GMRES algorithm when the matrix size + go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itérations est 10 fois plus long mais la suite de la phrase ne veut rien dire} \item the classical GMRES execution time is almost the double for $N_{x}=140$ compared with the Krylov multisplitting method. \end{enumerate} @@ -663,16 +746,17 @@ grid 2x16 leading to the same conclusion. \subsubsection{CPU Power impacts on performance} -\begin{figure} [ht!] +\begin{table} [ht!] \centering \begin{tabular}{r c } \hline - Grid & 2x16\\ %\hline + Grid architecture & 2x16\\ %\hline Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline \end{tabular} -\caption{CPU Power impacts} -\end{figure} +\caption{Test conditions: CPU Power impacts} +\label{tab:06} +\end{table} \begin{figure} [ht!] \centering @@ -686,17 +770,24 @@ on the algorithms performance in varying the CPU power of the clusters nodes from $1$ to $19$ GFlops. The outputs depicted in Figure~\ref{fig:06} confirm the performance gain, around $95\%$ for both of the two methods, after adding more powerful CPU. +\ \\ +%\DL{il faut une conclusion sur ces tests : ils confirment les résultats déjà +%obtenus en grandeur réelle. Donc c'est une aide précieuse pour les dev. Pas +%besoin de déployer sur une archi réelle} + +To conclude these series of experiments, with SimGrid we have been able to make +many simulations with many parameters variations. Doing all these experiments +with a real platform is most of the time not possible. Moreover the behavior of +both GMRES and Krylov multisplitting methods is in accordance with larger real +executions on large scale supercomputer~\cite{couturier15}. -\DL{il faut une conclusion sur ces tests : ils confirment les résultats déjà -obtenus en grandeur réelle. Donc c'est une aide précieuse pour les dev. Pas -besoin de déployer sur une archi réelle} \subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode} The previous paragraphs put in evidence the interests to simulate the behavior of the application before any deployment in a real environment. In this section, following the same previous methodology, our goal is to compare the -efficiency of the multisplitting method in \textit{ asynchronous mode} with the +efficiency of the multisplitting method in \textit{ asynchronous mode} compared with the classical GMRES in \textit{synchronous mode}. The interest of using an asynchronous algorithm is that there is no more @@ -707,37 +798,39 @@ theoretically reduce the overall execution time and can improve the algorithm performance. \RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici} -As stated before, the Simgrid simulator tool has been successfully used to show +In this section, Simgrid simulator tool has been successfully used to show the efficiency of the multisplitting in asynchronous mode and to find the best combination of the grid resources (CPU, Network, input matrix size, \ldots ) to get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ / exec\_time$_{multisplitting}$) in comparison with the classical GMRES time. -The test conditions are summarized in the table below : \\ +The test conditions are summarized in the table~\ref{tab:07}: \\ -\begin{figure} [ht!] +\begin{table} [ht!] \centering \begin{tabular}{r c } \hline - Grid & 2x50 totaling 100 processors\\ %\hline + Grid Architecture & 2x50 totaling 100 processors\\ %\hline Processors Power & 1 GFlops to 1.5 GFlops\\ Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\ Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\ \end{tabular} -\end{figure} +\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode} +\label{tab:07} +\end{table} Again, comprehensive and extensive tests have been conducted with different -parametes as the CPU power, the network parameters (bandwidth and latency) in -the simulator tool and with different problem size. The relative gains greater -than 1 between the two algorithms have been captured after each step of the -test. In Figure~\ref{table:01} are reported the best grid configurations -allowing the multisplitting method to be more than 2.5 times faster than the +parameters as the CPU power, the network parameters (bandwidth and latency) +and with different problem size. The relative gains greater than $1$ between the +two algorithms have been captured after each step of the test. In +Figure~\ref{fig:07} are reported the best grid configurations allowing +the multisplitting method to be more than $2.5$ times faster than the classical GMRES. These experiments also show the relative tolerance of the multisplitting algorithm when using a low speed network as usually observed with -geographically distant clusters throuth the internet. +geographically distant clusters through the internet. % use the same column width for the following three tables \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}} @@ -775,8 +868,9 @@ geographically distant clusters throuth the internet. \hline \end{mytable} %\end{table} - \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES} - \label{table:01} + \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES +\AG{C'est un tableau, pas une figure}} + \label{fig:07} \end{figure} @@ -784,14 +878,14 @@ geographically distant clusters throuth the internet. CONCLUSION -\section*{Acknowledgment} - +%\section*{Acknowledgment} +\ack This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01). - \bibliographystyle{wileyj} \bibliography{biblio} + \end{document} %%% Local Variables: