X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/56dc4d55704617d8f459826573f8bd2beeb5b5b3..2f78f080350308e2f46d8eff8d66a8e127fee583:/paper.tex?ds=inline diff --git a/paper.tex b/paper.tex index 7fd9704..f9434f9 100644 --- a/paper.tex +++ b/paper.tex @@ -317,7 +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] @@ -386,26 +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. +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. - - -\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: @@ -413,10 +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: @@ -424,8 +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, @@ -434,17 +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: @@ -478,9 +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 @@ -489,10 +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). \\ @@ -504,8 +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 @@ -518,10 +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 @@ -532,8 +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. @@ -561,12 +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 + 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} @@ -577,13 +571,13 @@ 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!] @@ -596,12 +590,11 @@ multisplitting method. \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 \\} @@ -609,20 +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 & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \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\%. @@ -631,7 +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} @@ -644,7 +637,7 @@ 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 & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline \end{tabular} @@ -680,7 +673,7 @@ 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 & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\ \end{tabular} @@ -709,7 +702,7 @@ 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 \end{tabular} @@ -741,7 +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} @@ -749,7 +742,7 @@ 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 & $N_{x} = 150 \times 150 \times 150$\\ \hline \end{tabular} @@ -810,7 +803,7 @@ 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$^{-2}$\\