X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/e139a43b147ddae98f95e987560caadfa4c845c8..56dc4d55704617d8f459826573f8bd2beeb5b5b3:/paper.tex?ds=sidebyside diff --git a/paper.tex b/paper.tex index 0d24694..7fd9704 100644 --- a/paper.tex +++ b/paper.tex @@ -70,8 +70,8 @@ -\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\affil{1}, @@ -94,27 +94,26 @@ analysis of simulated grid-enabled numerical iterative algorithms} Email:~\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 -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 main contribution of this paper is to show that the use of a -simulation tool (here we have decided to use the SimGrid toolkit) can really -help developpers to better tune their applications for a given multi-core -architecture. - -In particular we focus our attention on two parallel iterative algorithms based -on the Multisplitting algorithm and we compare them to the GMRES algorithm. -These algorithms are used to solve linear systems. Two different variants of -the Multisplitting are studied: one using synchronoous 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. The obtain simulated results confirm the real results -previously obtained on different real multi-core architectures and also confirm -the efficiency of the asynchronous multisplitting algorithm compared to the -synchronous GMRES method. +\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 +%% 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 main contribution of this paper is to show that the use of a +%% simulation tool (here we have decided to use the SimGrid toolkit) can really +%% 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. +%% 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 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. + +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. \end{abstract} @@ -136,7 +135,7 @@ 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 @@ -150,10 +149,10 @@ task cannot begin a new iteration while it has not received data dependencies from its neighbors. We say that the iteration computation follows a \textit{synchronous} scheme. In the asynchronous scheme a task can compute a new iteration without having to wait for the data dependencies coming from its -neighbors. Both communication and computations are \textit{asynchronous} +neighbors. Both communications and computations are \textit{asynchronous} inducing that there is no more idle time, due to synchronizations, between two iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks -that we detail in section~\ref{sec:asynchro} but even if the number of +that we detail in Section~\ref{sec:asynchro} but even if the number of iterations required to converge is generally greater than for the synchronous case, it appears that the asynchronous iterative scheme can significantly reduce overall execution times by suppressing idle times due to @@ -165,34 +164,30 @@ 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. - -\LZK{Pas trop convainquant comme argument pour valider l'approche de simulation. \\On peut dire par exemple: on a pu simuler différents algos itératifs à large échelle (le plus connu GMRES et deux variantes de multisplitting) et la simulation nous a permis (sans avoir le vrai matériel) de déterminer quelle serait la meilleure solution pour une telle configuration de l'archi ou vice versa.\\A revoir...} - -The obtained results on different -simulated multi-core architectures confirm the real results previously obtained -on non simulated architectures. - -\LZK{Il n y a pas dans la partie expé cette comparaison et confirmation des résultats entre la simulation et l'exécution réelle des algos sur les vrais clusters.\\ Sinon on pourrait ajouter dans la partie expé une référence vers le journal supercomput de krylov multi pour confirmer que cette méthode est meilleure que GMRES sur les clusters large échelle.} - -We also confirm the efficiency of the -asynchronous multisplitting algorithm compared to the synchronous GMRES. - -\LZK{P.S.: Pour tout le papier, le principal objectif n'est pas de faire des comparaisons entre des méthodes itératives!!\\Sinon, les deux algorithmes Krylov multisplitting synchrone et multisplitting asynchrone sont plus efficaces que GMRES sur des clusters à large échelle.\\Et préciser, si c'est vraiment le cas, que le multisplitting asynchrone est plus efficace et adapté aux clusters distants par rapport aux deux autres algos (je n'ai pas encore lu la partie expé)} - -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 convergence 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 applications 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 for a given 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 Krylov 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 @@ -200,22 +195,20 @@ asynchronous iterative application. This paper is organized as follows. Section~\ref{sec:asynchro} presents the iteration model we use and more particularly the asynchronous scheme. In -section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented. +Section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented. Section~\ref{sec:04} details the different solvers that we use. Finally our -experimental results are presented in section~\ref{sec:expe} followed by some +experimental results are presented in Section~\ref{sec:expe} followed by some concluding remarks and perspectives. -\LZK{Proposition d'un titre pour le papier: Grid-enabled simulation of large-scale linear iterative solvers.} - \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 +Asynchronous iterative methods have been studied for many years theoretically and practically. Many methods have been considered and convergence results have been proved. These methods can be used to solve, in parallel, fixed point problems (i.e. problems for which the solution is $x^\star =f(x^\star)$. In practice, -asynchronous iterations methods can be used to solve, for example, linear and +asynchronous iteration methods can be used to solve, for example, linear and non-linear systems of equations or optimization problems, interested readers are invited to read~\cite{BT89,bahi07}. @@ -225,7 +218,7 @@ algorithm that supports both the synchronous or the asynchronous iteration model requires very few modifications to be able to be executed in both variants. In practice, only the communications and convergence detection are different. In the synchronous mode, iterations are synchronized whereas in the asynchronous -one, they are not. It should be noticed that non blocking communications can be +one, they are not. It should be noticed that non-blocking communications can be used in both modes. Concerning the convergence detection, synchronous variants can use a global convergence procedure which acts as a global synchronization point. In the asynchronous model, the convergence detection is more tricky as @@ -233,17 +226,17 @@ 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 +for the asynchronous scheme (this number 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 +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 +architectures are often very important. So, prior to deployment 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 +iterative algorithms. The problematic is then to show that the results produced 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. @@ -252,6 +245,57 @@ magnitude. To our knowledge, there is no study on this problematic. 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} @@ -519,10 +563,11 @@ architectures and scaling up the input matrix size} \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 + Input 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 input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}} +\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{table} @@ -545,7 +590,8 @@ multisplitting method. \begin{center} \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf} \end{center} - \caption{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170\RC{idem}} + \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} @@ -566,7 +612,7 @@ $40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC 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 + Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \end{tabular} \caption{Test conditions: grid 2x16 and 4x8 with networks N1 vs N2} \label{tab:02} @@ -578,16 +624,15 @@ speed inter-cluster network (N1) and also on a less performant network (N2). 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\%. -%\RC{c'est pas clair : la différence entre quoi et quoi?} -%\DL{pas clair} -%\RCE{Modifie} + %\begin{wrapfigure}{l}{100mm} \begin{figure} [ht!] \centering \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf} -\caption{Grid 2x16 and 4x8 with networks N1 vs 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} @@ -601,7 +646,7 @@ the network speed drops down (variation of 12.5\%), the difference between t \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 + 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} @@ -612,7 +657,8 @@ the network speed drops down (variation of 12.5\%), the difference between t \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} @@ -636,7 +682,7 @@ magnitude with a latency of $8.10^{-6}$. \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 \\ + 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}} \label{tab:04} @@ -646,7 +692,8 @@ magnitude with a latency of $8.10^{-6}$. \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} @@ -664,7 +711,7 @@ of $40\%$ which is only around $24\%$ for the classical GMRES. \hline Grid Architecture & 4x8\\ %\hline Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ - Input matrix size & N$_{x}$ = From 40 to 200\\ \hline + Input matrix size & $N_{x}$ = From 40 to 200\\ \hline \end{tabular} \caption{Test conditions: Input matrix size impacts} \label{tab:05} @@ -704,7 +751,7 @@ grid 2x16 leading to the same conclusion. \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 + Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline \end{tabular} \caption{Test conditions: CPU Power impacts} \label{tab:06} @@ -722,10 +769,16 @@ 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} -\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}. \subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode} @@ -761,7 +814,7 @@ The test conditions are summarized in the table~\ref{tab:07}: \\ 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 + Input matrix size & $N_{x}$ = From 62 to 150\\ %\hline Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\ \end{tabular} \caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode} @@ -772,7 +825,7 @@ 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 -Figure~\ref{fig:07} are reported the best grid configurations allowing +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 @@ -787,7 +840,7 @@ geographically distant clusters through the internet. \end{tabular}} -\begin{figure}[!t] +\begin{table}[!t] \centering %\begin{table} % \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES} @@ -815,12 +868,38 @@ geographically distant clusters through the internet. \end{mytable} %\end{table} \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES} - \label{fig:07} -\end{figure} + \label{tab:08} +\end{table} \section{Conclusion} -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 +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 +magnitude) with the works presented in~\cite{couturier15}. Simulated results +also confirm the efficiency of the asynchronous multisplitting +algorithm compared to the synchronous GMRES especially in case of +geographically distant clusters. + +These results are important since it is very time consuming to find optimal +configuration and deployment requirements for a given 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. This problematic is even more difficult for the asynchronous +scheme where a small 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 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 +make the real computation. So we plan to focus our research on that problematic. + %\section*{Acknowledgment} @@ -830,6 +909,7 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX- \bibliographystyle{wileyj} \bibliography{biblio} + \end{document} %%% Local Variables: