X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/0ff5badea3e5156e9795cf5724f3dffed49ca7b5..7ba4d34a63e54829b1a0d1f5fab102747429e8ff:/paper.tex diff --git a/paper.tex b/paper.tex index 523716f..f1247fe 100644 --- a/paper.tex +++ b/paper.tex @@ -77,10 +77,10 @@ %\itshape{\journalnamelc}\footnotemark[2]} \author{Charles Emile Ramamonjisoa\affil{1}, - David Laiymani\affil{1}, - Arnaud Giersch\affil{1}, - Lilia Ziane Khodja\affil{2} and - Raphaël Couturier\affil{1} + Lilia Ziane Khodja\affil{2}, + David Laiymani\affil{1}, + Raphaël Couturier\affil{1} and + Arnaud Giersch\affil{1} } \address{ @@ -107,13 +107,13 @@ %% help developers to better tune their applications for a given multi-core %% architecture. -%% In this paper we focus our attention on the simulation of iterative algorithms to solve sparse linear systems on large clusters. We study the behavior of the widely used GMRES algorithm and two different variants of the Multisplitting algorithms: one using synchronous iterations and another one with asynchronous iterations. +%% In this paper we focus our attention on the simulation of iterative algorithms to solve sparse linear systems on large clusters. We study the behavior of the widely used GMRES algorithm and two different variants of the Multisplitting algorithms: one using synchronous iterations and another one with asynchronous iterations. %% For each algorithm we have simulated %% different architecture parameters to evaluate their influence on the overall -%% execution time. +%% execution time. %% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm. -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 accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications. +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 accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications. In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm. @@ -244,60 +244,109 @@ magnitude. To our knowledge, there is no study on this problematic. \section{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. +In the scope of this paper, the Simgrid +toolkitSimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}, +an open source framework actively developped by its community, has been choosen +to simulate the behavior of the solvers algorithms in different grid +computational configurations. Simgrid pretends to be non-specialized in opposite +to some other simulators which stayed to be very specific oriented-application. +One of the well-known Simgrid advantage is its SMPI (Simulated MPI). SMPI +purpose is to execute by simulation in a similar way as in real life, an MPI +distributed application and to get accurate results with the detailed resources +consumption. Several studies have demonstrated the accuracy of the simulation +compared with execution on real physical architectures. In addition of SMPI, +Simgrid provides other API which can be convienent for different distrbuted +applications: computational grid applications, High Performance Computing (HPC), +P2P but also clouds applications. In this paper we use the SMPI API. It +implements about \np[\%]{80} of the MPI 2.0 standard and allows minor +modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward} +(see Section~\ref{sec:04.02}). + + + Provided as an input to the simulator, at least $3$ XML files describe the + computational grid resources: number of clusters in the grid, number of + processors/cores in each cluster, detailed description of the intra and inter + networks and the list of the hosts in each cluster (see the details in Section~\ref{sec:expe}). Simgrid uses a fluid model to simulate the program execution. + This gives several simulation modes which produce accurate + results~\cite{bedaride+degomme+genaud+al.2013.toward, + velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode + really executes the computation but "intercepts" the communications (running + time is then evaluated according to the parameters of the simulated platform). + It is also possible for SimGrid/SMPI to only keep duration of large + computations by skipping them. Moreover the application can be run "in vitro" + by sharing some in-memory structures between the simulated processes and + thus allowing the use of very large data scale. + + +The choice of Simgrid/SMPI as a simulator tool in this study has been emphasized +by the results obtained by several studies to validate, in real environments, +the behavior of different network models simulated in +Simgrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline +the comparison between real MPI executions and SimGrid/SMPI +ones\cite{guermouche+renard.2010.first, clauss+stillwell+genaud+al.2011.single, +bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of +SimGrid simulations. + + + + + + +% 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} @@ -321,7 +370,7 @@ A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L, \end{equation} where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}. -\begin{figure}[t] +\begin{figure}[htpb] %\begin{algorithm}[t] %\caption{Block Jacobi two-stage multisplitting method} \begin{algorithmic}[1] @@ -359,7 +408,7 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde \end{equation} The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}). -\begin{figure}[t] +\begin{figure}[htbp] %\begin{algorithm}[t] %\caption{Krylov two-stage method using block Jacobi multisplitting} \begin{algorithmic}[1] @@ -407,10 +456,10 @@ in which several clusters are geographically distant, so there are intra and inter-cluster communications. In the following, these parameters are described: \begin{itemize} - \item hostfile: hosts description file. + \item hostfile: hosts description file, \item platform: file describing the platform architecture: clusters (CPU power, \dots{}), intra cluster network description, inter cluster network (bandwidth $bw$, -latency $lat$, \dots{}). +latency $lat$, \dots{}), \item archi : grid computational description (number of clusters, number of nodes/processors in each cluster). \end{itemize} @@ -442,8 +491,6 @@ In this section, experiments for both multisplitting algorithms are reported. Fi \subsection{The 3D Poisson problem} \label{3dpoisson} - - We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form: \begin{equation} \frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega @@ -462,7 +509,13 @@ where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ \end{equation} until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid. -In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic block of the problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries. +In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ +sub-problems such that $L$ is the number of clusters and $p$ is the number of +processors in each cluster. We apply the three-dimensional partitioning instead +of the row-by-row one in order to reduce the size of the data shared at the +sub-problems boundaries. In this case, each processor is in charge of +parallelepipedic block of the problem and has at most six neighbors in the same +cluster or in distant clusters with which it shares data at boundaries. \subsection{Study setup and simulation methodology} @@ -485,16 +538,11 @@ results comparison and analysis. In the scope of this study, we retain on the one hand the algorithm execution mode (synchronous and asynchronous) and on the other hand the execution time and the number of iterations to reach the convergence. \\ -\textbf{Step 4 }: Set up the different grid testbed environments that will be +\textbf{Step 4}: Set up the different grid testbed environments that will be simulated in the simulator tool to run the program. The following architectures have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number represents the number of clusters in the grid and the second number represents -the number of hosts (processors/cores) in each cluster. The network has been -designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a -latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links -(resp. inter-clusters backbone links). \\ - -\LZK{Il me semble que le bw et lat des deux réseaux varient dans les expés d'une simu à l'autre. On vire la dernière phrase?} +the number of hosts (processors/cores) in each cluster. \\ \textbf{Step 5}: Conduct an extensive and comprehensive testings within these configurations by varying the key parameters, especially @@ -525,262 +573,178 @@ the program output results: Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster and between distant clusters. This parameter is application dependent. - In a grid environment, it is common to distinguish, on the one hand, the - "intra-network" which refers to the links between nodes within a cluster and - on the other hand, the "inter-network" which is the backbone link between - clusters. In practice, these two networks have different speeds. - The intra-network generally works like a high speed local network with a - high bandwidth and very low latency. In opposite, the inter-network connects - clusters sometime via heterogeneous networks components through internet with - a lower speed. The network between distant clusters might be a bottleneck - for the global performance of the application. - -\subsection{Comparison of GMRES and Krylov two-stage algorithms in synchronous mode} - -In the scope of this paper, our first objective is to analyze when the Krylov -two-stage 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 Multisplitting method better performance compared to classical GMRES, particularly on poor and slow networks. -\LZK{Pas du tout claire la dernière phrase (For a systematic...)!!} -\RCE { Reformule autrement} - -In what follows, we will present the test conditions, the output results and our comments.\\ - -%\subsubsection{Execution of the algorithms on various computational grid architectures and scaling up the input matrix size} -\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes} -\ \\ -% environment + In a grid environment, it is common to distinguish, on one hand, the + \textit{intra-network} which refers to the links between nodes within a + cluster and on the other hand, the \textit{inter-network} which is the + backbone link between clusters. In practice, these two networks have + different speeds. The intra-network generally works like a high speed + local network with a high bandwidth and very low latency. In opposite, the + inter-network connects clusters sometime via heterogeneous networks components + through internet with a lower speed. The network between distant clusters + might be a bottleneck for the global performance of the application. + + +\subsection{Comparison between GMRES and two-stage multisplitting algorithms in +synchronous mode} +In the scope of this paper, our first objective is to analyze +when the synchronous Krylov two-stage 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. + +Table~\ref{tab:01} summarizes the parameters used in the different simulations: +the grid architectures (i.e. the number of clusters and the number of nodes per +cluster), the network of inter-clusters backbone links and the matrix sizes of +the 3D Poisson problem. However, for all simulations we fix the network +parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency +$lat=8\mu$s. In what follows, we will present the test conditions, the output +results and our comments. \begin{table} [ht!] \begin{center} -\begin{tabular}{ll } - \hline - Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline - \multirow{2}{*}{Network} & Inter (N2): $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline - & Intra (N1): $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\ - \multirow{2}{*}{Matrix size} & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline - & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =170 $\times$ 170 $\times$ 170 \\ \hline - \end{tabular} -\caption{Test conditions: various grid configurations with the matrix sizes 150$^3$ or 170$^3$} -\LZK{Ce sont les caractéristiques du réseau intra ou inter clusters? Ce n'est pas précisé...} -\RCE{oui c est precise} +\begin{tabular}{ll} +\hline +Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ +\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\ + & $N2$: $bw$=1Gbs, $lat=50\mu$s \\ +\multirow{2}{*}{Matrix size} & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\ + & $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline +\end{tabular} +\caption{Parameters for the different simulations} \label{tab:01} \end{center} \end{table} - -In this section, we analyze the simulations conducted on various grid configurations presented in Table~\ref{tab:01}. Figure~\ref{fig:01} shows, for all grid configurations and a given matrix size, a non-variation in the number of iterations for the classical GMRES algorithm, which is not the case of the Krylov two-stage algorithm. -%% First, the results in Figure~\ref{fig:01} -%% show for all grid configurations the non-variation of the number of iterations of -%% classical GMRES for a given input matrix size; it is not the case for the -%% multisplitting method. -\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...} -\RC{Les légendes ne sont pas explicites...} -\RCE{Corrige} - -\begin{figure} [ht!] - \begin{center} - \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf} - \end{center} - \caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$ -\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}} -\LZK{Pour quelle taille du problème sont calculés les nombres d'itérations? Que représente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?} -\RCE {Corrige} - \label{fig:01} -\end{figure} - -The execution times between the two algorithms is significant with different -grid architectures, even with the same number of processors (for example, 2 $\times$ 16 -and 4 $\times 8$). We can observe a better 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 32 (grid 2 $\times$ 16) to 64 processors/cores (grid 8 $\times$ 8). Note that even with a grid 8 $\times$ 8 having the maximum number of clusters, the execution time of the multisplitting method is in average 32\% less compared to GMRES. -\RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?} -\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exécution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?} -\RCE{Remarquez que meme avec une grille 8x8, le multi est toujours plus performant} - -\subsubsection{Simulations for two different inter-clusters network speeds \\} - -\begin{table} [ht!] +\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\} + +In this section, we analyze the simulations conducted on various grid +configurations and for different sizes of the 3D Poisson problem. The parameters +of the network between clusters is fixed to $N2$ (see +Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and +a given matrix size of 170$^3$ elements, a non-variation in the number of +iterations for the classical GMRES algorithm, which is not the case of the +Krylov two-stage algorithm. In fact, with multisplitting algorithms, the number +of splitting (in our case, it is equal to the number of clusters) influences on the +convergence speed. The higher the number of splitting is, the slower the +convergence of the algorithm is (see the output results obtained from +configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs. +8$\times$8). + +The execution times between both algorithms is significant with different grid +architectures. The synchronous Krylov two-stage algorithm presents better +performances than the GMRES algorithm, even for a high number of clusters (about +$32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can +observe a better sensitivity of the Krylov two-stage algorithm (compared to the +GMRES one) when scaling up the number of the processors in the computational +grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is +about $40\%$ better on $64$ processors (grid of 8$\times$8) than $32$ processors +(grid of 2$\times$16). + +\begin{figure}[ht] \begin{center} -\begin{tabular}{ll} - \hline - Grid architecture & 2$\times$16, 4$\times$8\\ %\hline - \multirow{2}{*}{Inter Network} & N1: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline - & N2: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\ - Matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline - \end{tabular} -\caption{Test conditions: grid configurations 2$\times$16 and 4$\times$8 with networks N1 vs. N2} -\label{tab:02} +\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf} \end{center} -\end{table} - -In this section, the experiments compare the behavior of the algorithms running on a -speeder inter-cluster network (N2) and also on a less performant network (N1) respectively defined in the test conditions Table~\ref{tab:02}. \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 4 $\times$ 16 or 8 $\times$ 8: the reduction factor is around $2$. 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\%. - - +\caption{Various grid configurations with two matrix sizes: $150^3$ and $170^3$} +\label{fig:01} +\end{figure} -%\begin{wrapfigure}{l}{100mm} -\begin{figure} [ht!] +\subsubsection{Simulations for two different inter-clusters network speeds\\} +In Figure~\ref{fig:02} we present the execution times of both algorithms to +solve a 3D Poisson problem of size $150^3$ on two different simulated network +$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from +this figure that the Krylov two-stage algorithm is sensitive to the number of +clusters (i.e. it is better to have a small number of clusters). However, we can +notice an interesting behavior of the Krylov two-stage algorithm. It is less +sensitive to bad network bandwidth and latency for the inter-clusters links than +the GMRES algorithms. This means that the multisplitting methods are more +efficient for distributed systems with high latency networks. + +\begin{figure}[ht] \centering \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf} -\caption{Various grid configurations with networks N1 vs N2 -\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}} -\RCE{Corrige} +\caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$} +%\LZK{CE, remplacer les ``,'' des décimales par un ``.''} +%\RCE{ok} \label{fig:02} \end{figure} -%\end{wrapfigure} +\subsubsection{Network latency impacts on performances\\} +Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm. -\subsubsection{Network latency impacts on performance} -\ \\ -\begin{table} [ht!] -\centering -\begin{tabular}{r c } - \hline - Grid Architecture & 2 $\times$ 16\\ %\hline - \multirow{2}{*}{Inter Network N1} & $bw$=1Gbs, \\ %\hline - & $lat$= From 8$\times$10$^{-6}$ to $6.10^{-5}$ second \\ - Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline - \end{tabular} -\caption{Test conditions: network latency impacts} -\label{tab:03} -\end{table} - -\begin{figure} [ht!] +\begin{figure}[ht] \centering \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf} -\caption{Network latency impacts on execution time -\AG{\np{E-6}}} +\caption{Network latency impacts on performances} \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 which means that the GMRES seems tolerate more the network latency variation with a less rate increase of the execution time. However, the execution time factor between the two algorithms varies from 2.2 to 1.5 times with a network latency decreasing from $8.10^{-6}$ to $6.10^{-5}$. +\subsubsection{Network bandwidth impacts on performances\\} -\RC{Les 2 précédentes phrases me semblent en contradiction....} -\RCE{Reformule} +Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of +$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to +solve a 3D Poisson problem of size $150^3$. The results of increasing the +network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for +both algorithms by reducing the execution times. However, the Krylov two-stage +algorithm presents a better performance gain in the considered bandwidth +interval with a gain of $40\%$ compared to only about $24\%$ for the classical +GMRES algorithm. -\subsubsection{Network bandwidth impacts on performance} -\ \\ -\begin{table} [ht!] -\centering -\begin{tabular}{r c } - \hline - Grid Architecture & 2 $\times$ 16\\ %\hline -\multirow{2}{*}{Inter Network N1} & $bw$=From 1Gbs to 10 Gbs \\ %\hline - & $lat$= 5.10$^{-5}$ second \\ - Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\ - \end{tabular} -\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}} -\RCE{C est le bw} -\label{tab:04} -\end{table} - - -\begin{figure} [ht!] +\begin{figure}[ht] \centering \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf} -\caption{Network bandwith impacts on execution time -\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.} -\RCE{Corrige} +\caption{Network bandwith impacts on performances} \label{fig:04} \end{figure} -The results of increasing the network bandwidth show the improvement of the -performance for both algorithms by reducing the execution time (see -Figure~\ref{fig:04}). However, in this case, the Krylov multisplitting method -presents a better performance in the considered bandwidth interval with a gain -of $40\%$ which is only around $24\%$ for the classical GMRES. - -\subsubsection{Input matrix size impacts on performance} -\ \\ -\begin{table} [ht!] -\centering -\begin{tabular}{r c } - \hline - Grid Architecture & 4 $\times$ 8\\ %\hline - Inter Network & $bw$=1Gbs - $lat$=5.10$^{-5}$ \\ - Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 40$^{3}$ to 200$^{3}$\\ \hline - \end{tabular} -\caption{Test conditions: Input matrix size impacts} -\label{tab:05} -\end{table} - - -\begin{figure} [ht!] +\subsubsection{Matrix size impacts on performances\\} + +In these experiments, the matrix size of the 3D Poisson problem is varied from +$50^3$ to $190^3$ elements. The simulated computational grid is composed of $4$ +clusters of $8$ processors each interconnected by the network $N2$ (see +Table~\ref{tab:01}). As shown in Figure~\ref{fig:05}, the execution +times for both algorithms increase with increased matrix sizes. For all problem +sizes, the GMRES algorithm is always slower than the Krylov two-stage algorithm. +Moreover, for this benchmark, it seems that the greater the problem size is, the +bigger the ratio between execution times of both algorithms is. We can also +observe that for some problem sizes, the convergence (and thus the execution +time) of the Krylov two-stage algorithm varies quite a lot. +%This is due to the 3D partitioning of the 3D matrix of the Poisson problem. +These findings may help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up. + +\begin{figure}[ht] \centering \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf} -\caption{Problem size impacts on execution time} +\caption{Problem size impacts on performances} \label{fig:05} \end{figure} -In these experiments, the input matrix size has been set from $N_{x} = N_{y} -= N_{z} = 40$ to $200$ side elements that is from $40^{3} = 64.000$ to $200^{3} -= 8,000,000$ points. Obviously, as shown in Figure~\ref{fig:05}, the execution -time for both algorithms increases when the input matrix size also increases. -But the interesting results are: -\begin{enumerate} - \item the important increase ($10$ times) of the number of iterations needed to - reach the convergence for the classical GMRES algorithm particularly, 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} - \RCE{Le nombre d'iterations augmente de 10 fois, cela surtout a partir de N=150} - -\item the classical GMRES execution time is almost the double for $N_{x}=140$ - compared with the Krylov multisplitting method. -\end{enumerate} - -These findings may help a lot end users to setup the best and the optimal -targeted environment for the application deployment when focusing on the problem -size scale up. It should be noticed that the same test has been done with the -grid 2 $\times$ 16 leading to the same conclusion. - -\subsubsection{CPU Power impacts on performance} +\subsubsection{CPU power impacts on performances\\} -\begin{table} [ht!] -\centering -\begin{tabular}{r c } - \hline - Grid architecture & 2 $\times$ 16\\ %\hline - Inter Network & N2 : $bw$=1Gbs - $lat$=5.10$^{-5}$ \\ %\hline - Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ - CPU Power & From 3 to 19 GFlops \\ \hline - \end{tabular} -\caption{Test conditions: CPU Power impacts} -\label{tab:06} -\end{table} +Using the SimGrid simulator flexibility, we have tried to determine the impact +of the CPU power of the processors in the different clusters on performances of +both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The +simulation is conducted on a grid of $2\times16$ processors interconnected by +the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size +$150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance +gain, about $95\%$ for both algorithms, after improving the CPU power of +processors. -\begin{figure} [ht!] +\begin{figure}[ht] \centering \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf} -\caption{CPU Power impacts on execution time} +\caption{CPU Power impacts on performances} \label{fig:06} \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 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}. +with a real platform is most of the time not possible or very costly. Moreover +the behavior of both GMRES and Krylov two-stage algorithms is in accordance +with larger real executions on large scale supercomputers~\cite{couturier15}. -\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode} +\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms} The previous paragraphs put in evidence the interests to simulate the behavior of the application before any deployment in a real environment. In this @@ -790,46 +754,38 @@ classical GMRES in \textit{synchronous mode}. The interest of using an asynchronous algorithm is that there is no more synchronization. With geographically distant clusters, this may be essential. -In this case, each processor can compute its iteration freely without any +In this case, each processor can compute its iterations freely without any synchronization with the other processors. Thus, the asynchronous may 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} -\RCE{C est la description du dernier test sync/async avec l'introduction de la notion de relative gain} -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. +In this section, the SimGrid simulator is used to compare the behavior of the +two-stage algorithm in asynchronous mode with GMRES in synchronous mode. +Several benchmarks have been performed with various combinations of the grid +resources (CPU, Network, matrix size, \ldots). The test conditions are +summarized in Table~\ref{tab:02}. -The test conditions are summarized in the table~\ref{tab:07}: \\ -\begin{table} [ht!] +%\LZK{Quelle table repporte les gains relatifs?? Sûrement pas Table II !!} +%\RCE{Table III avec la nouvelle numerotation} + + +\begin{table}[htbp] \centering -\begin{tabular}{r c } +\begin{tabular}{ll} \hline - Grid Architecture & 2 $\times$ 50 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 \\ + Grid architecture & 2$\times$50 totaling 100 processors\\ + Processors Power & 1 GFlops to 1.5 GFlops \\ + \multirow{2}{*}{Network inter-clusters} & $bw$=1.25 Gbits, $lat=50\mu$s \\ + & $bw$=5 Mbits, $lat=20ms$\\ + Matrix size & from $62^3$ to $150^3$\\ + Residual error precision & $10^{-5}$ to $10^{-9}$\\ \hline \\ \end{tabular} -\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode} -\label{tab:07} +\caption{Test conditions: GMRES in synchronous mode vs. Krylov two-stage in asynchronous mode} +\label{tab:02} \end{table} -Again, comprehensive and extensive tests have been conducted with different -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 -Table~\ref{tab:08} 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 through the internet. % use the same column width for the following three tables \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}} @@ -856,7 +812,7 @@ geographically distant clusters through the internet. power (GFlops) & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\ \hline - size (N) + size ($N^3$) & 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline Precision @@ -867,15 +823,24 @@ geographically distant clusters through the internet. \hline \end{mytable} %\end{table} - \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES} - \label{tab:08} + \caption{Relative gains of the two-stage multisplitting algorithm compared with the classical GMRES} + \label{tab:03} \end{table} -\section{Conclusion} +Table~\ref{tab:03} reports the relative gains between both algorithms. It is +defined by the ratio between the execution time of GMRES and the execution time +of the multisplitting. The ratio is greater than one because the asynchronous +multisplitting version is faster than GMRES. In average, the two-stage +multisplitting algorithm 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 through the internet. + +\section{Conclusion} In this paper we have presented the simulation of the execution of three -different parallel solvers on some multi-core architectures. We have show that +different parallel solvers on some multi-core architectures. We have shown that the SimGrid toolkit is an interesting simulation tool that has allowed us to determine which method to choose given a specified multi-core architecture. Moreover the simulated results are in accordance (i.e. with the same order of @@ -897,7 +862,7 @@ converge and so to very different execution times. In future works, we plan to investigate how to simulate the behavior of really large scale applications. For example, if we are interested to simulate the execution of the solvers of this paper with thousand or even dozens of thousands -or core, it is not possible to do that with SimGrid. In fact, this tool will +of cores, it is not possible to do that with SimGrid. In fact, this tool will make the real computation. So we plan to focus our research on that problematic.