X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/8a89c7d0f5ad8152c74c9cc31938b78e12b06a96..2f78f080350308e2f46d8eff8d66a8e127fee583:/paper.tex?ds=sidebyside diff --git a/paper.tex b/paper.tex index e6bf766..f9434f9 100644 --- a/paper.tex +++ b/paper.tex @@ -94,31 +94,26 @@ 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 developers 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. -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 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. -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. +\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} @@ -154,10 +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 @@ -170,7 +165,7 @@ 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 and of the application data can -lead to very different numbers of iterations to reach the converge and so to +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. @@ -178,16 +173,16 @@ platform scenarios. The {\bf main contribution of this paper} is to show that the use of a simulation tool (i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real parallel applications (i.e. large linear system solvers) can help developers to -better tune their application for a given multi-core architecture. To show the +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) +multisplitting algorithm with the GMRES (Generalized Minimal RESidual) solver~\cite{saad86} in synchronous mode. The simulation results allow us to -determine which method to choose given a specified multi-core architecture. +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 multisplitting method is more efficient than GMRES for large +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. @@ -200,20 +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. \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}. @@ -223,7 +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 @@ -231,17 +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. @@ -322,7 +317,7 @@ where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L, \label{eq:03} \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 ({\it Generalized Minimal RESidual})~\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}. +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{algorithm}[t] @@ -391,26 +386,20 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini \subsection{Simulation of the two-stage methods using SimGrid toolkit} \label{sec:04.02} -One of our objectives when simulating the application in Simgrid is, as in real +One of our objectives when simulating the application in SimGrid is, as in real life, to get accurate results (solutions of the problem) but also to ensure the test reproducibility under the same conditions. According to our experience, -very few modifications are required to adapt a MPI program for the Simgrid +very few modifications are required to adapt a MPI program for the SimGrid simulator using SMPI (Simulator MPI). The first modification is to include SMPI -libraries and related header files (smpi.h). The second modification is to +libraries and related header files (\verb+smpi.h+). The second modification is to suppress all global variables by replacing them with local variables or using a -Simgrid selector called "runtime automatic switching" +SimGrid selector called "runtime automatic switching" (smpi/privatize\_global\_variables). Indeed, global variables can generate side effects on runtime between the threads running in the same process and generated by -Simgrid to simulate the grid environment. - -%\RC{On vire cette phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The -%last modification on the MPI program pointed out for some cases, the review of -%the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which -%might cause an infinite loop. - +SimGrid to simulate the grid environment. -\paragraph{Simgrid Simulator parameters} -\ \\ \noindent Before running a Simgrid benchmark, many parameters for the +\paragraph{Parameters of the simulation in SimGrid} +\ \\ \noindent Before running a SimGrid benchmark, many parameters for the computation platform must be defined. For our experiments, we consider platforms in which several clusters are geographically distant, so there are intra and inter-cluster communications. In the following, these parameters are described: @@ -418,10 +407,10 @@ inter-cluster communications. In the following, these parameters are described: \begin{itemize} \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{}). +\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$, +latency $lat$, \dots{}). \item archi : grid computational description (number of clusters, number of -nodes/processors for each cluster). +nodes/processors in each cluster). \end{itemize} \noindent In addition, the following arguments are given to the programs at runtime: @@ -429,8 +418,8 @@ In addition, the following arguments are given to the programs at runtime: \begin{itemize} \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$, \item inner precision $\TOLG$ and outer precision $\TOLM$, - \item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively, - \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments, + \item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}), + \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones, \item matrix off-diagonal value is fixed to $-1.0$, \item number of vectors in matrix $S$ (i.e. value of $s$), \item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method, @@ -439,17 +428,18 @@ In addition, the following arguments are given to the programs at runtime: \item execution mode: synchronous or asynchronous. \end{itemize} -It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine. +It should also be noticed that both solvers have been executed with the SimGrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine. %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% -\section{Experimental Results} +\section{Experimental results} \label{sec:expe} -In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described. +In this section, experiments for both multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described. \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: @@ -483,9 +473,9 @@ have been chosen for the study in this paper. \\ \textbf{Step 2}: Collect the software materials needed for the experimentation. In our case, we have two variants algorithms for the resolution of the -3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting -method. In addition, the Simgrid simulator has been chosen to simulate the -behaviors of the distributed applications. Simgrid is running in a virtual +3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting +method. In addition, the SimGrid simulator has been chosen to simulate the +behaviors of the distributed applications. SimGrid is running in a virtual machine on a simple laptop. \\ \textbf{Step 3}: Fix the criteria which will be used for the future @@ -494,10 +484,10 @@ 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 -simulated in the simulator tool to run the program. The following architecture -has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. The first number +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$50. The first number represents the number of clusters in the grid and the second number represents -the number of hosts (processors/cores) in each cluster. The network has been +the number of hosts (processors/cores) in each cluster. The network has been designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links (resp. inter-clusters backbone links). \\ @@ -509,8 +499,7 @@ input data. \\ \textbf{Step 6} : Collect and analyze the output results. -\subsection{Factors impacting distributed applications performance in -a grid environment} +\subsection{Factors impacting distributed applications performance in a grid environment} When running a distributed application in a computational grid, many factors may have a strong impact on the performance. First of all, the architecture of the @@ -523,10 +512,10 @@ Another important factor impacting the overall performance of the application is the network configuration. Two main network parameters can modify drastically the program output results: \begin{enumerate} -\item the network bandwidth (bw=bits/s) also known as "the data-carrying +\item the network bandwidth ($bw$ in bits/s) also known as "the data-carrying capacity" of the network is defined as the maximum of data that can transit from one point to another in a unit of time. -\item the network latency (lat : microsecond) defined as the delay from the +\item the network latency ($lat$ in microseconds) defined as the delay from the start time to send a simple data from a source to a destination. \end{enumerate} Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster @@ -537,8 +526,8 @@ and between distant clusters. This parameter is application dependent. 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 bandwith and very low latency. In opposite, the inter-network connects - clusters sometime via heterogeneous networks components throuth internet with + 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. @@ -566,12 +555,12 @@ architectures and scaling up the input matrix size} \begin{center} \begin{tabular}{r c } \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 + Grid Architecture & 2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8\\ %\hline + Inter Network N2 & bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\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 matrix size N$_{x}$=N$_{y}$=N$_{z}$=150 or 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} @@ -582,31 +571,30 @@ architectures and scaling up the input matrix size} In this section, we analyze the performance of algorithms running on various -grid configurations (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01} +grid configurations (2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8) and using an inter-network N2 defined in the test conditions in Table~\ref{tab:01}. 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...} +%\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...} \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 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} -The execution times between the two algorithms is significant with different -grid architectures, even with the same number of processors (for example, 2x16 -and 4x8). We can observ the low sensitivity of the Krylov multisplitting method +Secondly, 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 observ the sensitivity of the Krylov multisplitting method (compared with the classical GMRES) when scaling up the number of the processors -in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs -$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?} +in the grid: in average, the reduction of the execution time for GMRES (resp. Multisplitting) algorithm is around $40\%$ (resp. around $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors (grid 8 $\times$ 8) processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?} \subsubsection{Running on two different inter-clusters network speeds \\} @@ -614,20 +602,20 @@ $40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC \begin{center} \begin{tabular}{r c } \hline - Grid Architecture & 2x16, 4x8\\ %\hline - Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline + Grid Architecture & 2 $\times$ 16, 4 $\times$ 8\\ %\hline + Inter Networks & 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} +\caption{Test conditions: grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2} \label{tab:02} \end{center} \end{table} -These experiments compare the behavior of the algorithms running first on a -speed inter-cluster network (N1) and also on a less performant network (N2). \RC{Il faut définir cela avant...} +In this section, the experiments compare the behavior of the algorithms running on a +speeder inter-cluster network (N1) and also on a less performant network (N2) 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 4x16 or 8x8: the reduction is about $2$. The results depict also that when +for both algorithms when using a grid architecture like 4 $\times$ 16 or 8 $\times$ 8: 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\%. @@ -636,7 +624,7 @@ the network speed drops down (variation of 12.5\%), the difference between t \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 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2 \AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}} \label{fig:02} \end{figure} @@ -649,9 +637,9 @@ the network speed drops down (variation of 12.5\%), the difference between t \centering \begin{tabular}{r c } \hline - Grid Architecture & 2x16\\ %\hline + Grid Architecture & 2 $\times$ 16\\ %\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} @@ -685,9 +673,9 @@ magnitude with a latency of $8.10^{-6}$. \centering \begin{tabular}{r c } \hline - Grid Architecture & 2x16\\ %\hline + Grid Architecture & 2 $\times$ 16\\ %\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} @@ -714,9 +702,9 @@ of $40\%$ which is only around $24\%$ for the classical GMRES. \centering \begin{tabular}{r c } \hline - Grid Architecture & 4x8\\ %\hline + Grid Architecture & 4 $\times$ 8\\ %\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} @@ -746,7 +734,7 @@ But the interesting results are: 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 2x16 leading to the same conclusion. +grid 2 $\times$ 16 leading to the same conclusion. \subsubsection{CPU Power impacts on performance} @@ -754,9 +742,9 @@ grid 2x16 leading to the same conclusion. \centering \begin{tabular}{r c } \hline - Grid architecture & 2x16\\ %\hline + Grid architecture & 2 $\times$ 16\\ %\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} @@ -815,11 +803,11 @@ The test conditions are summarized in the table~\ref{tab:07}: \\ \centering \begin{tabular}{r c } \hline - Grid Architecture & 2x50 totaling 100 processors\\ %\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 + 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} @@ -899,7 +887,11 @@ application data can lead to very different numbers of iterations to reach the converge and so to very different execution times. -Our future works... +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.