X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/9793696f84bb746d5af3969fb0bc9105f9cb3a99..ca1429f05161a13a6c9cc1eb4a62dcb8217c06d2:/paper.tex?ds=sidebyside diff --git a/paper.tex b/paper.tex index fa447f6..ab8f9ab 100644 --- a/paper.tex +++ b/paper.tex @@ -24,6 +24,8 @@ % Extension pour les liens intra-documents (tagged PDF) % et l'affichage correct des URL (commande \url{http://example.com}) %\usepackage{hyperref} +\usepackage{multirow} + \usepackage{url} \DeclareUrlCommand\email{\urlstyle{same}} @@ -94,27 +96,26 @@ 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 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} @@ -150,10 +151,10 @@ 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 @@ -166,7 +167,7 @@ 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. @@ -174,16 +175,16 @@ 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. @@ -196,20 +197,20 @@ 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}. @@ -219,7 +220,7 @@ 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 @@ -227,17 +228,17 @@ 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. @@ -318,9 +319,9 @@ 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{figure}[htpb] %\begin{algorithm}[t] %\caption{Block Jacobi two-stage multisplitting method} \begin{algorithmic}[1] @@ -358,7 +359,7 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde \end{equation} The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}). -\begin{figure}[t] +\begin{figure}[htbp] %\begin{algorithm}[t] %\caption{Krylov two-stage method using block Jacobi multisplitting} \begin{algorithmic}[1] @@ -387,37 +388,31 @@ 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: \begin{itemize} - \item hostfile: hosts description file. + \item hostfile: hosts description file, \item platform: file describing the platform architecture: clusters (CPU power, -\dots{}), intra cluster network description, inter cluster network (bandwidth bw, -latency lat, \dots{}). +\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: @@ -425,8 +420,8 @@ 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, @@ -435,19 +430,18 @@ 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: \begin{equation} \frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega @@ -479,9 +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 @@ -489,14 +483,11 @@ results comparison and analysis. In the scope of this study, we retain on the one hand the algorithm execution mode (synchronous and asynchronous) and on the other hand the execution time and the number of iterations to reach the convergence. \\ -\textbf{Step 4 }: Set up the different grid testbed environments that will be -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 +\textbf{Step 4}: Set up the different grid testbed environments that will be +simulated in the simulator tool to run the program. The following architectures +have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number represents the number of clusters in the grid and the second number represents -the number of hosts (processors/cores) in each cluster. The network has been -designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a -latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links -(resp. inter-clusters backbone links). \\ +the number of hosts (processors/cores) in each cluster. \\ \textbf{Step 5}: Conduct an extensive and comprehensive testings within these configurations by varying the key parameters, especially @@ -505,8 +496,7 @@ 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 @@ -519,10 +509,10 @@ 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 @@ -533,168 +523,148 @@ 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. -\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode} - -In the scope of this paper, our first objective is to analyze when the Krylov -Multisplitting method has better performance than the classical GMRES -method. With a synchronous iterative method, better performance means a -smaller number of iterations and execution time before reaching the convergence. -For a systematic study, the experiments should figure out that, for various -grid parameters values, the simulator will confirm the targeted outcomes, -particularly for poor and slow networks, focusing on the impact on the -communication performance on the chosen class of algorithm. -The following paragraphs present the test conditions, the output results -and our comments.\\ +\subsection{Comparison between GMRES and two-stage multisplitting algorithms in synchronous mode} +In the scope of this paper, our first objective is to analyze when the synchronous Krylov two-stage method has better performance than the classical GMRES method. With a synchronous iterative method, better performance means a smaller number of iterations and execution time before reaching the convergence. - -\subsubsection{Execution of the algorithms on various computational grid -architectures and scaling up the input matrix size} -\ \\ -% environment +Table~\ref{tab:01} summarizes the parameters used in the different simulations: the grid architectures, the network of inter-clusters backbone links and the matrix sizes of the 3D Poisson problem. However, for all simulations we fix the network parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency $lat$=8$\times$10$^{-6}$. In what follows, we will present the test conditions, the output results and our comments. \begin{table} [ht!] \begin{center} -\begin{tabular}{r c } - \hline - Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline - Network & N2 : bw=1Gbits/s - lat=5.10$^{-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 - \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?} -\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}. Idem dans le texte, les figures, etc.}} +\begin{tabular}{ll} +\hline +Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ +\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\ + & $N2$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ +\multirow{2}{*}{Matrix size} & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\ + & $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline +\end{tabular} +\caption{Parameters for the different simulations} \label{tab:01} \end{center} \end{table} +\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\} +In this section, we analyze the simulations conducted on various grid +configurations and for different sizes of the 3D Poisson problem. The parameters +of the network between clusters is fixed to $N2$ (see +Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and a +given matrix size 170$^3$ elements, a non-variation in the number of iterations +for the classical GMRES algorithm, which is not the case of the Krylov two-stage +algorithm. In fact, with multisplitting algorithms, the number of splitting (in +our case, it is the number of clusters) influences on the convergence speed. The +higher the number of splitting is, the slower the convergence of the algorithm +is (see the output results obtained from configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs. 8$\times$8). +The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16). +\begin{figure}[t] +\begin{center} +\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf} +\end{center} +\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$} +\label{fig:01} +\end{figure} -In this section, we analyze the performance of algorithms running on various -grid configurations (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01} -show for all grid configurations the non-variation of the number of iterations of -classical GMRES for a given input matrix size; it is not the case for the -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...} +\subsubsection{Simulations for two different inter-clusters network speeds\\} +In this section, the experiments compare the behavior of the algorithms running on a +speeder inter-cluster network (N2) and also on a less performant network (N1) respectively defined in the test conditions Table~\ref{tab:02}. +%\RC{Il faut définir cela avant...} +Figure~\ref{fig:02} shows that end users will reduce the execution time +for both algorithms when using a grid architecture like 4 $\times$ 16 or 8 $\times$ 8: the reduction factor is around $2$. The results depict also that when +the network speed drops down (variation of 12.5\%), the difference between the two Multisplitting algorithms execution times can reach more than 25\%. -\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} -\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}} - \label{fig:01} +\begin{figure}[t] +\centering +\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf} +\caption{Various grid configurations with networks $N1$ vs. $N2$} +\label{fig:02} \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 -(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?} -\subsubsection{Running on two different inter-clusters network speeds \\} -\begin{table} [ht!] -\begin{center} -\begin{tabular}{r c } - \hline - Grid Architecture & 2x16, 4x8\\ %\hline - Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline - - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\ - Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline - \end{tabular} -\caption{Test conditions: grid 2x16 and 4x8 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...} -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\%. -%\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 -\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}} -\label{fig:02} -\end{figure} -%\end{wrapfigure} -\subsubsection{Network latency impacts on performance} -\ \\ + + + + + + + + + + + + +\subsubsection{Network latency impacts on performance\\} + \begin{table} [ht!] \centering \begin{tabular}{r c } \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 + Grid Architecture & 2 $\times$ 16\\ %\hline + \multirow{2}{*}{Inter Network N1} & $bw$=1Gbs, \\ %\hline + & $lat$= From 8$\times$10$^{-6}$ to $6.10^{-5}$ second \\ + Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline \end{tabular} \caption{Test conditions: network latency impacts} \label{tab:03} \end{table} - - -\begin{figure} [ht!] +\begin{figure} [htbp] \centering \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf} -\caption{Network latency impacts on execution time -\AG{\np{E-6}}} +\caption{Network latency impacts on execution time} +%\AG{\np{E-6}}} \label{fig:03} \end{figure} +In Table~\ref{tab:03}, parameters for the influence of the network latency are +reported. According to the results of Figure~\ref{fig:03}, a degradation of the +network latency from $8.10^{-6}$ to $6.10^{-5}$ implies an absolute time +increase of more than $75\%$ (resp. $82\%$) of the execution for the classical +GMRES (resp. Krylov multisplitting) algorithm. The execution time factor +between the two algorithms varies from 2.2 to 1.5 times with a network latency +decreasing from $8.10^{-6}$ to $6.10^{-5}$ second. -According to the results of Figure~\ref{fig:03}, a degradation of the network -latency from $8.10^{-6}$ to $6.10^{-5}$ implies an absolute time increase of -more than $75\%$ (resp. $82\%$) of the execution for the classical GMRES -(resp. Krylov multisplitting) algorithm. In addition, it appears that the -Krylov multisplitting method tolerates more the network latency variation with a -less rate increase of the execution time.\RC{Les 2 précédentes phrases me - semblent en contradiction....} Consequently, in the worst case ($lat=6.10^{-5 -}$), the execution time for GMRES is almost the double than the time of the -Krylov multisplitting, even though, the performance was on the same order of -magnitude with a latency of $8.10^{-6}$. -\subsubsection{Network bandwidth impacts on performance} -\ \\ +\subsubsection{Network bandwidth impacts on performance\\} + \begin{table} [ht!] \centering \begin{tabular}{r c } \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 \\ + Grid Architecture & 2 $\times$ 16\\ %\hline +\multirow{2}{*}{Inter Network N1} & $bw$=From 1Gbs to 10 Gbs \\ %\hline + & $lat$= 5.10$^{-5}$ second \\ + Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\ \end{tabular} -\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}} +\caption{Test conditions: Network bandwidth impacts} +% \RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau} +%\RCE{C est le bw} \label{tab:04} \end{table} -\begin{figure} [ht!] +\begin{figure} [htbp] \centering \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf} -\caption{Network bandwith impacts on execution time -\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.} +\caption{Network bandwith impacts on execution time} +%\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.} +%\RCE{Corrige} \label{fig:04} \end{figure} @@ -704,55 +674,54 @@ Figure~\ref{fig:04}). However, in this case, the Krylov multisplitting method presents a better performance in the considered bandwidth interval with a gain of $40\%$ which is only around $24\%$ for the classical GMRES. -\subsubsection{Input matrix size impacts on performance} -\ \\ +\subsubsection{Input matrix size impacts on performance\\} + \begin{table} [ht!] \centering \begin{tabular}{r c } \hline - Grid Architecture & 4x8\\ %\hline - Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ - Input matrix size & N$_{x}$ = From 40 to 200\\ \hline + Grid Architecture & 4 $\times$ 8\\ %\hline + Inter Network & $bw$=1Gbs - $lat$=5.10$^{-5}$ \\ + Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 50$^{3}$ to 190$^{3}$\\ \hline \end{tabular} \caption{Test conditions: Input matrix size impacts} \label{tab:05} \end{table} -\begin{figure} [ht!] +\begin{figure} [htbp] \centering \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf} \caption{Problem size impacts on execution time} \label{fig:05} \end{figure} -In these experiments, the input matrix size has been set from $N_{x} = N_{y} -= N_{z} = 40$ to $200$ side elements that is from $40^{3} = 64.000$ to $200^{3} -= 8,000,000$ points. Obviously, as shown in Figure~\ref{fig:05}, the execution -time for both algorithms increases when the input matrix size also increases. -But the interesting results are: -\begin{enumerate} - \item the drastic increase ($10$ times) of the number of iterations needed to - reach the convergence for the classical GMRES algorithm when the matrix size - go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itérations est 10 fois plus long mais la suite de la phrase ne veut rien dire} -\item the classical GMRES execution time is almost the double for $N_{x}=140$ - compared with the Krylov multisplitting method. -\end{enumerate} +In these experiments, the input matrix size has been set from $50^3$ to +$190^3$. Obviously, as shown in Figure~\ref{fig:05}, the execution time for both +algorithms increases when the input matrix size also increases. For all problem +sizes, GMRES is always slower than the Krylov multisplitting. Moreover, for this +benchmark, it seems that the greater the problem size is, the bigger the ratio +between both algorithm execution times is. We can also observ that for some +problem sizes, the Krylov multisplitting convergence varies quite a +lot. Consequently the execution times in that cases also varies. + 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 4 $\times$ 8 leading to the same conclusion. -\subsubsection{CPU Power impacts on performance} +\subsubsection{CPU Power impacts on performance\\} -\begin{table} [ht!] + +\begin{table} [htbp] \centering \begin{tabular}{r c } \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 + Grid architecture & 2 $\times$ 16\\ %\hline + Inter Network & N2 : $bw$=1Gbs - $lat$=5.10$^{-5}$ \\ %\hline + Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ + CPU Power & From 3 to 19 GFlops \\ \hline \end{tabular} \caption{Test conditions: CPU Power impacts} \label{tab:06} @@ -797,25 +766,27 @@ synchronization with the other processors. Thus, the asynchronous may theoretically reduce the overall execution time and can improve the algorithm performance. -\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici} -In this section, Simgrid simulator tool has been successfully used to show -the efficiency of the multisplitting in asynchronous mode and to find the best -combination of the grid resources (CPU, Network, input matrix size, \ldots ) to -get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ / -exec\_time$_{multisplitting}$) in comparison with the classical GMRES time. +In this section, the Simgrid simulator is used to compare the behavior of the +multisplitting in asynchronous mode with GMRES in synchronous mode. Several +benchmarks have been performed with various combination of the grid resources +(CPU, Network, input matrix size, \ldots ). The test conditions are summarized +in Table~\ref{tab:07}. In order to compare the execution times, this table +reports the relative gain between both algorithms. It is defined by the ratio +between the execution time of GMRES and the execution time of the +multisplitting. The ration is greater than one because the asynchronous +multisplitting version is faster than GMRES. -The test conditions are summarized in the table~\ref{tab:07}: \\ -\begin{table} [ht!] +\begin{table} [htbp] \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} @@ -826,7 +797,7 @@ 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 @@ -841,7 +812,7 @@ 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} @@ -857,7 +828,7 @@ geographically distant clusters through the internet. power (GFlops) & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\ \hline - size (N) + size ($N^3$) & 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline Precision @@ -868,10 +839,9 @@ geographically distant clusters through the internet. \hline \end{mytable} %\end{table} - \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES -\AG{C'est un tableau, pas une figure}} - \label{fig:07} -\end{figure} + \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES} + \label{tab:08} +\end{table} \section{Conclusion} @@ -896,7 +866,11 @@ 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.