X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/da679beb2e5ea3a7a07a474d5a751de50de54f8b..d08d0574c6782fe3a320861ccd6ec60a2ab6025e:/hpcc.tex?ds=inline diff --git a/hpcc.tex b/hpcc.tex index 84f5278..29d00a1 100644 --- a/hpcc.tex +++ b/hpcc.tex @@ -41,17 +41,18 @@ \algnewcommand\Output{\item[\algorithmicoutput]} \newcommand{\MI}{\mathit{MaxIter}} +\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}} \begin{document} -\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid} +\title{Simulation of Asynchronous Iterative Algorithms Using SimGrid} \author{% \IEEEauthorblockN{% Charles Emile Ramamonjisoa\IEEEauthorrefmark{1}, + Lilia Ziane Khodja\IEEEauthorrefmark{2}, David Laiymani\IEEEauthorrefmark{1}, - Arnaud Giersch\IEEEauthorrefmark{1}, - Lilia Ziane Khodja\IEEEauthorrefmark{2} and + Arnaud Giersch\IEEEauthorrefmark{1} and Raphaël Couturier\IEEEauthorrefmark{1} } \IEEEauthorblockA{\IEEEauthorrefmark{1}% @@ -70,32 +71,20 @@ \maketitle -\RC{Ordre des auteurs pas définitif.} \begin{abstract} -\AG{L'abstract est AMHA incompréhensible et ne donne pas envie de lire la suite.} -In recent years, the scalability of large-scale implementation in a -distributed environment of algorithms becoming more and more complex has -always been hampered by the limits of physical computing resources -capacity. One solution is to run the program in a virtual environment -simulating a real interconnected computers architecture. The results are -convincing and useful solutions are obtained with far fewer resources -than in a real platform. However, challenges remain for the convergence -and efficiency of a class of algorithms that concern us here, namely -numerical parallel iterative algorithms executed in asynchronous mode, -especially in a large scale level. Actually, such algorithm requires a -balance and a compromise between computation and communication time -during the execution. Two important factors determine the success of the -experimentation: the convergence of the iterative algorithm on a large -scale and the execution time reduction in asynchronous mode. Once again, -from the current work, a simulated environment like SimGrid provides -accurate results which are difficult or even impossible to obtain in a -physical platform by exploiting the flexibility of the simulator on the -computing units clusters and the network structure design. Our -experimental outputs showed a saving of up to \np[\%]{40} for the algorithm -execution time in asynchronous mode compared to the synchronous one with -a residual precision up to \np{E-11}. Such successful results open -perspectives on experimentations for running the algorithm on a -simulated large scale growing environment and with larger problem size. + +Synchronous iterative algorithms are often less scalable than asynchronous +iterative ones. Performing large scale experiments with different kind of +network parameters is not easy because with supercomputers such parameters are +fixed. So one solution consists in using simulations first in order to analyze +what parameters could influence or not the behaviors of an algorithm. In this +paper, we show that it is interesting to use SimGrid to simulate the behaviors +of asynchronous iterative algorithms. For that, we compare the behaviour of a +synchronous GMRES algorithm with an asynchronous multisplitting one with +simulations in which we choose some parameters. Both codes are real MPI +codes. Simulations allow us to see when the multisplitting algorithm can be more +efficient than the GMRES one to solve a 3D Poisson problem. + % no keywords for IEEE conferences % Keywords: Algorithm distributed iterative asynchronous simulation SimGrid @@ -108,7 +97,7 @@ problems raised by researchers on various scientific disciplines but also by in increasing complexity of these requested applications combined with a continuous increase of their sizes lead to write distributed and parallel algorithms requiring significant hardware resources (grid computing, clusters, broadband network, etc.) but also a non-negligible CPU execution time. We consider in this paper a class of highly efficient -parallel algorithms called \emph{numerical iterative algorithms} executed in a distributed environment. As their name +parallel algorithms called \emph{iterative algorithms} executed in a distributed environment. As their name suggests, these algorithms solve a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value $X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}. @@ -124,57 +113,60 @@ at that time. Even if the number of iterations required before the convergence i synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially in a grid computing context (see~\cite{Bahi07} for more details). -Parallel numerical applications (synchronous or asynchronous) may have different -configuration and deployment requirements. Quantifying their resource -allocation policies and application scheduling algorithms in grid computing -environments under varying load, CPU power and network speeds is very costly, -very labor intensive and very time -consuming~\cite{Calheiros:2011:CTM:1951445.1951450}. The case of AIAC -algorithms is even more problematic since they are very sensible to the +Parallel (synchronous or asynchronous) applications may have different +configuration and deployment requirements. Quantifying their resource +allocation policies and application scheduling algorithms in grid computing +environments under varying load, CPU power and network speeds is very costly, +very labor intensive and very time +consuming~\cite{Calheiros:2011:CTM:1951445.1951450}. The case of AIAC +algorithms is even more problematic since they are very sensible to the execution environment context. For instance, variations in the network bandwidth -(intra and inter-clusters), in the number and the power of nodes, in the number -of clusters\dots{} can lead to very different number of iterations and so to -very different execution times. Then, it appears that the use of simulation -tools to explore various platform scenarios and to run large numbers of -experiments quickly can be very promising. In this way, the use of a simulation -environment to execute parallel iterative algorithms found some interests in -reducing the highly cost of access to computing resources: (1) for the -applications development life cycle and in code debugging (2) and in production -to get results in a reasonable execution time with a simulated infrastructure -not accessible with physical resources. Indeed, the launch of distributed -iterative asynchronous algorithms to solve a given problem on a large-scale -simulated environment challenges to find optimal configurations giving the best +(intra and inter-clusters), in the number and the power of nodes, in the number +of clusters\dots{} can lead to very different number of iterations and so to +very different execution times. Then, it appears that the use of simulation +tools to explore various platform scenarios and to run large numbers of +experiments quickly can be very promising. In this way, the use of a simulation +environment to execute parallel iterative algorithms found some interests in +reducing the highly cost of access to computing resources: (1) for the +applications development life cycle and in code debugging (2) and in production +to get results in a reasonable execution time with a simulated infrastructure +not accessible with physical resources. Indeed, the launch of distributed +iterative asynchronous algorithms to solve a given problem on a large-scale +simulated environment challenges to find optimal configurations giving the best results with a lowest residual error and in the best of execution time. -To our knowledge, there is no existing work on the large-scale simulation of a -real AIAC application. The aim of this paper is twofold. First we give a first -approach of the simulation of AIAC algorithms using a simulation tool (i.e. the -SimGrid toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of -asynchronous mode algorithms by comparing their performance with the synchronous -mode. More precisely, we had implemented a program for solving large -linear system of equations by numerical method GMRES (Generalized -Minimal Residual) \cite{ref1}. We show, that with minor modifications of the -initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a -real AIAC application on different computing architectures. The simulated -results we obtained are in line with real results exposed in ??\AG[]{ref?}. -SimGrid had allowed us to launch the application from a modest computing -infrastructure by simulating different distributed architectures composed by -clusters nodes interconnected by variable speed networks. With selected -parameters on the network platforms (bandwidth, latency of inter cluster -network) and on the clusters architecture (number, capacity calculation power) -in the simulated environment, the experimental results have demonstrated not -only the algorithm convergence within a reasonable time compared with the -physical environment performance, but also a time saving of up to \np[\%]{40} in -asynchronous mode. -\AG{Il faudrait revoir la phrase précédente (couper en deux?). Là, on peut - avoir l'impression que le gain de \np[\%]{40} est entre une exécution réelle - et une exécution simulée!} - -This article is structured as follows: after this introduction, the next section will give a brief description of -iterative asynchronous model. Then, the simulation framework SimGrid is presented with the settings to create various -distributed architectures. The algorithm of the multisplitting method used by GMRES written with MPI primitives and -its adaptation to SimGrid with SMPI (Simulated MPI) is detailed in the next section. At last, the experiments results -carried out will be presented before some concluding remarks and future works. +To our knowledge, there is no existing work on the large-scale simulation of a +real AIAC application. {\bf The contribution of the present paper can be + summarised in two main points}. First we give a first approach of the +simulation of AIAC algorithms using a simulation tool (i.e. the SimGrid +toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of the +asynchronous multisplitting algorithm by comparing its performance with the +synchronous GMRES (Generalized Minimal Residual) \cite{ref1}. Both these codes +can be used to solve large linear systems. In this paper, we focus on a 3D +Poisson problem. We show, that with minor modifications of the initial MPI +code, the SimGrid toolkit allows us to perform a test campaign of a real AIAC +application on different computing architectures. +% The simulated results we +%obtained are in line with real results exposed in ??\AG[]{ref?}. +SimGrid had allowed us to launch the application from a modest computing +infrastructure by simulating different distributed architectures composed by +clusters nodes interconnected by variable speed networks. Parameters of the +network platforms are the bandwidth and the latency of inter cluster +network. Parameters on the cluster's architecture are the number of machines and +the computation power of a machine. Simulations show that the asynchronous +multisplitting algorithm can solve the 3D Poisson problem approximately twice +faster than GMRES with two distant clusters. + + + +This article is structured as follows: after this introduction, the next section +will give a brief description of iterative asynchronous model. Then, the +simulation framework SimGrid is presented with the settings to create various +distributed architectures. Then, the multisplitting method is presented, it is +based on GMRES to solve each block obtained of the splitting. This code is +written with MPI primitives and its adaptation to SimGrid with SMPI (Simulated +MPI) is detailed in the next section. At last, the simulation results carried +out will be presented before some concluding remarks and future works. \section{Motivations and scientific context} @@ -196,7 +188,7 @@ times and the arrows the communications. With this algorithmic model, the number of iterations required before the convergence is generally greater than for the two former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially -in a grid computing context. +in a grid computing context.\LZK{Répétition par rapport à l'intro} \begin{figure}[!t] \centering @@ -254,7 +246,7 @@ like the 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 the mean of an XML file. It describes the properties of the platform, such as -the computing node with their computing power, the interconnection links with +the computing nodes with their computing power, the interconnection links with their bandwidth and latency, and the routing strategy. The simulated running time of the application is computed according to these properties. @@ -293,51 +285,75 @@ Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, whe B_L \end{array} \right) \end{equation*} -in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$. +in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ +are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$, $A_{\ell + m}$ is a rectangular block of $A$ of size $n_\ell\times n_m$, $X_\ell$ and +$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each, +and $\sum_{\ell} n_\ell=\sum_{m} n_m=n$. The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system \begin{equation} \label{eq:4.1} \left\{ \begin{array}{l} - A_{ll}X_l = Y_l \text{, such that}\\ - Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m + A_{\ell\ell}X_\ell = Y_\ell \text{, such that}\\ + Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\ m\neq \ell}}^{L}A_{\ell m}X_m \end{array} \right. \end{equation} -is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers. +is solved independently by a cluster and communications are required to update +the right-hand side sub-vector $Y_\ell$, such that the sub-vectors $X_m$ +represent the data dependencies between the clusters. As each sub-system +(\ref{eq:4.1}) is solved in parallel by a cluster of processors, our +multisplitting method uses an iterative method as an inner solver which is +easier to parallelize and more scalable than a direct method. In this work, we +use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most +used iterative method by many researchers. \begin{figure}[!t] %%% IEEE instructions forbid to use an algorithm environment here, use figure %%% instead \begin{algorithmic}[1] -\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector) -\Output $X_l$ (solution sub-vector)\vspace{0.2cm} -\State Load $A_l$, $B_l$ +\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector) +\Output $X_\ell$ (solution sub-vector)\medskip + +\State Load $A_\ell$, $B_\ell$ \State Set the initial guess $x^0$ \For {$k=0,1,2,\ldots$ until the global convergence} \State Restart outer iteration with $x^0=x^k$ \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$} -\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters -\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$ +\State\label{algo:01:send} Send shared elements of $X_\ell^{k+1}$ to neighboring clusters +\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq \ell}$ \EndFor \Statex \Function {InnerSolver}{$x^0$, $k$} -\State Compute local right-hand side $Y_l$: +\State Compute local right-hand side $Y_\ell$: \begin{equation*} - Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0 + Y_\ell = B_\ell - \sum\nolimits^L_{\substack{m=1\\ m\neq \ell}}A_{\ell m}X_m^0 \end{equation*} -\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method -\State \Return $X_l^k$ +\State Solving sub-system $A_{\ell\ell}X_\ell^k=Y_\ell$ with the parallel GMRES method +\State \Return $X_\ell^k$ \EndFunction \end{algorithmic} \caption{A multisplitting solver with GMRES method} \label{algo:01} \end{figure} -Algorithm on Figure~\ref{algo:01} shows the main key points of the multisplitting method to solve a large sparse linear system. This algorithm is based on an outer-inner iteration method where the parallel synchronous GMRES method is used to solve the inner iteration. It is executed in parallel by each cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors with the subscript $l$ represent the local data for cluster $l$, while $\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with neighboring clusters. At every outer iteration $k$, asynchronous communications are performed between processors of the local cluster and those of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines. +Algorithm on Figure~\ref{algo:01} shows the main key points of the +multisplitting method to solve a large sparse linear system. This algorithm is +based on an outer-inner iteration method where the parallel synchronous GMRES +method is used to solve the inner iteration. It is executed in parallel by each +cluster of processors. For all $\ell,m\in\{1,\ldots,L\}$, the matrices and +vectors with the subscript $\ell$ represent the local data for cluster $\ell$, +while $\{A_{\ell m}\}_{m\neq \ell}$ are off-diagonal matrices of sparse matrix +$A$ and $\{X_m\}_{m\neq \ell}$ contain vector elements of solution $x$ shared +with neighboring clusters. At every outer iteration $k$, asynchronous +communications are performed between processors of the local cluster and those +of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in +Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are +exchanged by message passing using MPI non-blocking communication routines. \begin{figure}[!t] \centering @@ -359,14 +375,14 @@ sets the token to \textit{True} if the local convergence is achieved or to global convergence is detected when the master of cluster 1 receives from the master of cluster $L$ a token set to \textit{True}. In this case, the master of cluster 1 broadcasts a stop message to masters of other clusters. In this work, -the local convergence on each cluster $l$ is detected when the following +the local convergence on each cluster $\ell$ is detected when the following condition is satisfied \begin{equation*} - (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon) + (k\leq \MI) \text{ or } (\|X_\ell^k - X_\ell^{k+1}\|_{\infty}\leq\epsilon) \end{equation*} where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution -$X_l^k$ and $X_l^{k+1}$. +$X_\ell^k$ and $X_\ell^{k+1}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code @@ -378,9 +394,12 @@ Note here that the use of SMPI functions optimizer for memory footprint and CPU As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. First, all declared global variables have been moved to local variables for each subroutine. In fact, global variables generate side effects arising from the concurrent access of shared memory used by threads simulating each computing unit in the SimGrid architecture. Second, the alignment of certain types of variables such as ``long int'' had -also to be reviewed. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid. +also to be reviewed. +\AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.} + Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid. In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real -environment. We have successfully executed the code in synchronous mode using GMRES algorithm compared with a multisplitting method in asynchrnous mode after few modification. +environment. We have successfully executed the code in synchronous mode using parallel GMRES algorithm compared with our multisplitting algorithm in asynchronous mode after few modifications. + \section{Experimental results} @@ -390,26 +409,34 @@ parameters and the program arguments allows us to compare outputs from the code study that the results depend on the following parameters: \begin{itemize} \item At the network level, we found that the most critical values are the - bandwidth (bw) and the network latency (lat). + bandwidth and the network latency. \item Hosts power (GFlops) can also influence on the results. \item Finally, when submitting job batches for execution, the arguments values - passed to the program like the maximum number of iterations or the - \textit{external} precision are critical. They allow to ensure not only the - convergence of the algorithm but also to get the main objective of the - experimentation of the simulation in having an execution time in asynchronous - less than in synchronous mode. The ratio between the execution time of asynchronous compared to the synchronous mode is defined as the "relative gain". So, our objective running the algorithm in SimGrid is to obtain a relative gain greater than 1. + passed to the program like the maximum number of iterations or the external + precision are critical. They allow to ensure not only the convergence of the + algorithm but also to get the main objective of the experimentation of the + simulation in having an execution time in asynchronous less than in + synchronous mode. The ratio between the execution time of asynchronous + compared to the synchronous mode is defined as the \emph{relative gain}. So, + our objective running the algorithm in SimGrid is to obtain a relative gain + greater than 1. + \AG{$t_\text{async} / t_\text{sync} > 1$, l'objectif est donc que ça dure plus + longtemps (que ça aille moins vite) en asynchrone qu'en synchrone ? + Ce n'est pas plutôt l'inverse ?} \end{itemize} -\LZK{Propositions pour remplacer le terme ``speedup'': acceleration ratio ou relative gain} -\CER{C'est fait. En conséquence, les tableaux et les commentaires ont été aussi modifiés} -A priori, obtaining a relative gain greater than 1 would be difficult in a local area -network configuration where the synchronous mode will take advantage on the + +A priori, obtaining a relative gain greater than 1 would be difficult in a local +area network configuration where the synchronous mode will take advantage on the rapid exchange of information on such high-speed links. Thus, the methodology adopted was to launch the application on clustered network. In this last -configuration, degrading the inter-cluster network performance will -\textit{penalize} the synchronous mode allowing to get a relative gain greater than 1. -This action simulates the case of distant clusters linked with long distance network -like Internet. - +configuration, degrading the inter-cluster network performance will penalize the +synchronous mode allowing to get a relative gain greater than 1. This action +simulates the case of distant clusters linked with long distance network like +Internet. + +\AG{Cette partie sur le poisson 3D + % on sait donc que ce n'est pas une plie ou une sole (/me fatigué) + n'est pas à sa place. Elle devrait être placée plus tôt.} In this paper, we solve the 3D Poisson problem whose the mathematical model is \begin{equation} \left\{ @@ -444,10 +471,11 @@ The parallel solving of the 3D Poisson problem with our multisplitting method re As a first step, the algorithm was run on a network consisting of two clusters containing 50 hosts each, totaling 100 hosts. Various combinations of the above -factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a +factors have provided the results shown in Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = -\text{\np{5211000}}$ entries. +\text{\np{5000211}}$ entries. +\AG{Expliquer comment lire les tableaux.} % use the same column width for the following three tables \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}} @@ -464,10 +492,10 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = \begin{mytable}{6} \hline - bw + bandwidth & 5 & 5 & 5 & 5 & 5 & 50 \\ \hline - lat + latency & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\ \hline power @@ -477,21 +505,22 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = & 62 & 62 & 62 & 100 & 100 & 110 \\ \hline Prec/Eprec - & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\ + & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\ + \hline \hline Relative gain & 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 \\ \hline \end{mytable} - \smallskip + \bigskip \begin{mytable}{6} \hline - bw + bandwidth & 50 & 50 & 50 & 50 & 10 & 10 \\ \hline - lat + latency & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01 \\ \hline power @@ -503,6 +532,7 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = Prec/Eprec & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5} & \np{E-5} \\ \hline + \hline Relative gain & 2.51 & 2.58 & 2.55 & 2.54 & 1.59 & 1.29 \\ \hline @@ -522,10 +552,10 @@ relative gains greater than 1 with a matrix size from 62 to 100 elements. \begin{mytable}{6} \hline - bw + bandwidth & 10 & 5 & 4 & 3 & 2 & 6 \\ \hline - lat + latency & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\ \hline power @@ -537,8 +567,9 @@ relative gains greater than 1 with a matrix size from 62 to 100 elements. Prec/Eprec & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\ \hline + \hline Relative gain - & 1.003 & 1,01 & 1,08 & 0.19 & 1.28 & 1.01 \\ + & 1.003 & 1.01 & 1.08 & 1.19 & 1.28 & 1.01 \\ \hline \end{mytable} \end{table} @@ -554,9 +585,9 @@ Table~\ref{tab.cluster.3x67}. \begin{mytable}{1} \hline - bw & 1 \\ + bandwidth & 1 \\ \hline - lat & 0.02 \\ + latency & 0.02 \\ \hline power & 1 \\ \hline @@ -564,6 +595,7 @@ Table~\ref{tab.cluster.3x67}. \hline Prec/Eprec & \np{E-5} \\ \hline + \hline Relative gain & 1.11 \\ \hline \end{mytable} @@ -573,11 +605,12 @@ Note that the program was run with the following parameters: \paragraph*{SMPI parameters} +~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).} \begin{itemize} - \item HOSTFILE: Hosts file description. - \item PLATFORM: file description of the platform architecture : clusters (CPU power, -\dots{}), intra cluster network description, inter cluster network (bandwidth bw, -lat latency, \dots{}). +\item HOSTFILE: Hosts file description. +\item PLATFORM: file description of the platform architecture : clusters (CPU + power, \dots{}), intra cluster network description, inter cluster network + (bandwidth, latency, \dots{}). \end{itemize} @@ -589,7 +622,7 @@ lat latency, \dots{}). \item Internal and external precisions; \item Matrix size $N_x$, $N_y$ and $N_z$; \item Matrix diagonal value: \np{6.0}; - \item Matrix Off-diagonal value: \np{-1.0}; + \item Matrix off-diagonal value: \np{-1.0}; \item Execution Mode: synchronous or asynchronous. \end{itemize} @@ -611,7 +644,7 @@ increasing the matrix size up to 100 elements, it was necessary to increase the CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a convergence of the algorithm with the same order of asynchronous mode efficiency. Maintaining such a system power but this time, increasing network throughput inter cluster up to -\np[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5 is obtained with +\np[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5\AG[]{2.5 ?} is obtained with high external precision of \np{E-11} for a matrix size from 110 to 150 side elements. @@ -622,14 +655,17 @@ matrix size of 62 elements, equality between the performance of the two modes (synchronous and asynchronous) is achieved with an inter cluster of \np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency greater than 1.2 with a matrix size of 100 points, it was necessary to degrade the inter cluster network bandwidth from 5 to \np[Mbit/s]{2}. +\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ??? + Quelle est la perte de perfs en faisant ça ?} A last attempt was made for a configuration of three clusters but more powerful with 200 nodes in total. The convergence with a relative gain around 1.1 was obtained with a bandwidth of \np[Mbit/s]{1} as shown in Table~\ref{tab.cluster.3x67}. -\LZK{Dans le papier, on compare les deux versions synchrone et asycnhrone du multisplitting. Y a t il des résultats pour comparer gmres parallèle classique avec multisplitting asynchrone? Ca permettra de montrer l'intérêt du multisplitting asynchrone sur des clusters distants} -\CER{En fait, les résultats ont été obtenus en comparant les temps d'exécution entre l'algo classique GMRES en mode synchrone avec le multisplitting en mode asynchrone, le tout sur un environnement de clusters distants} +\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...} +\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)} +\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??} \section{Conclusion} The experimental results on executing a parallel iterative algorithm in @@ -690,3 +726,5 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX- % LocalWords: Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua % LocalWords: Fortran GFlops priori Mbit de du fcomte multisplitting scalable % LocalWords: SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib +% LocalWords: intra durations nonsingular Waitall discretization discretized +% LocalWords: InnerSolver Isend Irecv