X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/4eeb6e06e742a4b5d9adcaae60633c22d9e5cb5b..74d1e243756c3cce524e2fb4f987836574784199:/hpcc.tex?ds=inline diff --git a/hpcc.tex b/hpcc.tex index 8772c9c..5ae025f 100644 --- a/hpcc.tex +++ b/hpcc.tex @@ -72,6 +72,7 @@ \RC{Ordre des auteurs pas définitif.} \begin{abstract} +\AG{L'abstract est AMHA incompréhensible et ne donne pas envie de lire la suite.} In recent years, the scalability of large-scale implementation in a distributed environment of algorithms becoming more and more complex has always been hampered by the limits of physical computing resources @@ -151,11 +152,11 @@ approach of the simulation of AIAC algorithms using a simulation tool (i.e. the SimGrid toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of asynchronous mode algorithms by comparing their performance with the synchronous mode. More precisely, we had implemented a program for solving large -non-symmetric linear system of equations by numerical method GMRES (Generalized -Minimal Residual) []\AG[]{[]?}.\LZK{Problème traité dans le papier est symétrique ou asymétrique? (Poisson 3D symétrique?)} We show, that with minor modifications of the +linear system of equations by numerical method GMRES (Generalized +Minimal Residual) \cite{ref1}. We show, that with minor modifications of the initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a real AIAC application on different computing architectures. The simulated -results we obtained are in line with real results exposed in ??\AG[]{??}. +results we obtained are in line with real results exposed in ??\AG[]{ref?}. SimGrid had allowed us to launch the application from a modest computing infrastructure by simulating different distributed architectures composed by clusters nodes interconnected by variable speed networks. With selected @@ -165,6 +166,9 @@ in the simulated environment, the experimental results have demonstrated not only the algorithm convergence within a reasonable time compared with the physical environment performance, but also a time saving of up to \np[\%]{40} in asynchronous mode. +\AG{Il faudrait revoir la phrase précédente (couper en deux?). Là, on peut + avoir l'impression que le gain de \np[\%]{40} est entre une exécution réelle + et une exécution simulée!} This article is structured as follows: after this introduction, the next section will give a brief description of iterative asynchronous model. Then, the simulation framework SimGrid is presented with the settings to create various @@ -187,7 +191,9 @@ times generated by synchronizations are very penalizing. One way to overcome thi \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model. Here, local computations do not need to wait for required data. Processors can then perform their iterations with the data present at that time. Figure~\ref{fig:aiac} illustrates this model where the gray blocks represent the computation phases, the white spaces the idle -times and the arrows the communications. With this algorithmic model, the number of iterations required before the +times and the arrows the communications. +\AG{There are no ``white spaces'' on the figure.} +With this algorithmic model, the number of iterations required before the convergence is generally greater than for the two former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially in a grid computing context. @@ -252,8 +258,11 @@ with their computing power, the interconnection links with their bandwidth and latency, and the routing strategy. The simulated running time of the application is computed according to these properties. +%%% TODO: add some words+refs about SimGrid's accuracy and scalability.} + \AG{Faut-il ajouter quelque-chose ?} -\CER{Comme tu as décrit la plateforme d'exécution, on peut ajouter éventuellement le fichier XML contenant des hosts dans les clusters formant la grille} +\CER{Comme tu as décrit la plateforme d'exécution, on peut ajouter éventuellement le fichier XML contenant des hosts dans les clusters formant la grille + \AG{Bof.}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Simulation of the multisplitting method} @@ -322,19 +331,7 @@ is solved independently by a cluster and communications are required to update t \label{algo:01} \end{figure} -Algorithm on Figure~\ref{algo:01} shows the main key points of the -multisplitting method to solve a large sparse linear system. This algorithm is -based on an outer-inner iteration method where the parallel synchronous GMRES -method is used to solve the inner iteration. It is executed in parallel by each -cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors -with the subscript $l$ represent the local data for cluster $l$, while -$\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and -$\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with -neighboring clusters. At every outer iteration $k$, asynchronous communications -are performed between processors of the local cluster and those of distant -clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in -Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are -exchanged by message passing using MPI non-blocking communication routines. +Algorithm on Figure~\ref{algo:01} shows the main key points of the multisplitting method to solve a large sparse linear system. This algorithm is based on an outer-inner iteration method where the parallel synchronous GMRES method is used to solve the inner iteration. It is executed in parallel by each cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors with the subscript $l$ represent the local data for cluster $l$, while $\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with neighboring clusters. At every outer iteration $k$, asynchronous communications are performed between processors of the local cluster and those of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines. \begin{figure}[!t] \centering @@ -352,7 +349,7 @@ Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around the virtual ring from a master processor to another until the global convergence is achieved. So starting from the cluster with rank 1, each master processor $i$ sets the token to \textit{True} if the local convergence is achieved or to -\text\it{False} otherwise, and sends it to master processor $i+1$. Finally, the +\textit{False} otherwise, and sends it to master processor $i+1$. Finally, the global convergence is detected when the master of cluster 1 receives from the master of cluster $L$ a token set to \textit{True}. In this case, the master of cluster 1 broadcasts a stop message to masters of other clusters. In this work, @@ -361,13 +358,13 @@ condition is satisfied \begin{equation*} (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon) \end{equation*} -where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$. +where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the +tolerance threshold of the error computed between two successive local solution +$X_l^k$ and $X_l^{k+1}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code -debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between the six neighbors of each point (left,right,front,behind,top,down) in a cubic partitionned submatrix within a cluster or between clusters, \CER{J'ai rajouté quelques précisions mais serait-il nécessaire de décrire a ce niveau la discrétisation 3D ?} -\LZK{Non ce n'est pas nécessaire. A ce niveau, on décrit l'algorithme général de multisplitting. Donc, je pense qu'il est préférable de ne pas préciser le schéma de communication qui peut changer selon le type de problème. \\ {\bf Par exemple: Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters}} -the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. In synchronous +debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. In synchronous mode, the execution of the program raised no particular issue but in asynchronous mode, the review of the sequence of MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions and with the addition of the primitive MPI\_Test was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm. \CER{On voulait en fait montrer la simplicité de l'adaptation de l'algo a SimGrid. Les problèmes rencontrés décrits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais préféré avoir un peu plus de détails sur l'adaptation de la version async} @@ -377,9 +374,7 @@ global variables have been moved to local variables for each subroutine. In fact shared memory used by threads simulating each computing unit in the SimGrid architecture. Second, the alignment of certain types of variables such as ``long int'' had also to be reviewed. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid. In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real -environment. We have tested in synchronous mode with a simulated platform starting from a modest 2 or 3 clusters grid to a larger configuration like simulating -Grid5000 with more than 1500 hosts with 5000 cores~\cite{bolze2006grid}. - +environment. We have successfully executed the code in synchronous mode using GMRES algorithm compared with a multisplitting method in asynchrnous mode after few modification. \section{Experimental results} @@ -396,27 +391,58 @@ study that the results depend on the following parameters: \textit{external} precision are critical. They allow to ensure not only the convergence of the algorithm but also to get the main objective of the experimentation of the simulation in having an execution time in asynchronous - less than in synchronous mode (i.e. speed-up less than 1). + less than in synchronous mode. The ratio between the execution time of asynchronous compared to the synchronous mode is defined as the "relative gain". So, our objective running the algorithm in SimGrid is to obtain a relative gain greater than 1. \end{itemize} -\LZK{Propositions pour changer le terme ``speedup'': acceleration ratio ou relative gain} - -A priori, obtaining a speedup less than 1 would be difficult in a local area +\LZK{Propositions pour remplacer le terme ``speedup'': acceleration ratio ou relative gain} +\CER{C'est fait. En conséquence, les tableaux et les commentaires ont été aussi modifiés} +A priori, obtaining a relative gain greater than 1 would be difficult in a local area network configuration where the synchronous mode will take advantage on the rapid exchange of information on such high-speed links. Thus, the methodology adopted was to launch the application on clustered network. In this last configuration, degrading the inter-cluster network performance will -\textit{penalize} the synchronous mode allowing to get a speedup lower than 1. -This action simulates the case of clusters linked with long distance network +\textit{penalize} the synchronous mode allowing to get a relative gain greater than 1. +This action simulates the case of distant clusters linked with long distance network like Internet. +In this paper, we solve the 3D Poisson problem whose the mathematical model is +\begin{equation} +\left\{ +\begin{array}{l} +\nabla^2 u = f \text{~in~} \Omega \\ +u =0 \text{~on~} \Gamma =\partial\Omega +\end{array} +\right. +\label{eq:02} +\end{equation} +where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. The general iteration scheme of our multisplitting method in a 3D domain using a seven point stencil could be written as +\begin{equation} +\begin{array}{ll} +u^{k+1}(x,y,z)= & u^k(x,y,z) - \frac{1}{6}\times\\ + & (u^k(x-1,y,z) + u^k(x+1,y,z) + \\ + & u^k(x,y-1,z) + u^k(x,y+1,z) + \\ + & u^k(x,y,z-1) + u^k(x,y,z+1)), +\end{array} +\label{eq:03} +\end{equation} +where the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite. + +The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries. + +\begin{figure}[!t] +\centering + \includegraphics[width=80mm,keepaspectratio]{partition} +\caption{Example of the 3D data partitioning between two clusters of processors.} +\label{fig:4.2} +\end{figure} + + As a first step, the algorithm was run on a network consisting of two clusters containing 50 hosts each, totaling 100 hosts. Various combinations of the above factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = \text{\np{5211000}}$ entries. -\CER{Voir ma remarque plus si nécessaire de décrire en détail le partitionnement 3D} -\LZK{Je pense qu'il faut donner ici le type du problème traité (Poisson 3D). Le partitionnement 3D permet juste de définir le schéma de dépendances (1 proc a au max 6 voisins dans le cluster local ou dans les clusters distants)} + % use the same column width for the following three tables \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}} \newenvironment{mytable}[1]{% #1: number of columns for data @@ -447,8 +473,8 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = Prec/Eprec & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\ \hline - speedup - & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\ + Relative gain + & 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 \\ \hline \end{mytable} @@ -471,8 +497,8 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = Prec/Eprec & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5} & \np{E-5} \\ \hline - speedup - & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778 \\ + Relative gain + & 2.51 & 2.58 & 2.55 & 2.54 & 1.59 & 1.29 \\ \hline \end{mytable} \end{table} @@ -481,7 +507,7 @@ Then we have changed the network configuration using three clusters containing respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the clusters. In the same way as above, a judicious choice of key parameters has permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the -speedups less than 1 with a matrix size from 62 to 100 elements. +relative gains greater than 1 with a matrix size from 62 to 100 elements. \begin{table}[!t] \centering @@ -505,8 +531,8 @@ speedups less than 1 with a matrix size from 62 to 100 elements. Prec/Eprec & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\ \hline - speedup - & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99 \\ + Relative gain + & 1.003 & 1,01 & 1,08 & 0.19 & 1.28 & 1.01 \\ \hline \end{mytable} \end{table} @@ -532,7 +558,7 @@ Table~\ref{tab.cluster.3x67}. \hline Prec/Eprec & \np{E-5} \\ \hline - speedup & 0.9 \\ + Relative gain & 1.11 \\ \hline \end{mytable} \end{table} @@ -569,7 +595,7 @@ lat latency, \dots{}). After analyzing the outputs, generally, for the configuration with two or three clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50} and~\ref{tab.cluster.3x33}), some combinations of the used parameters affecting -the results have given a speedup less than 1, showing the effectiveness of the +the results have given a relative gain more than 2.5, showing the effectiveness of the asynchronous performance compared to the synchronous mode. In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows @@ -578,29 +604,29 @@ bandwidth, a latency in order of a hundredth of a millisecond and a system power of one GFlops, an efficiency of about \np[\%]{40} in asynchronous mode is obtained for a matrix size of 62 elements. It is noticed that the result remains stable even if we vary the external precision from \np{E-5} to \np{E-9}. By -increasing the problem size up to 100 elements, it was necessary to increase the +increasing the matrix size up to 100 elements, it was necessary to increase the CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a convergence of the algorithm with the same order of asynchronous mode efficiency. Maintaining such a system power but this time, increasing network throughput inter cluster up to -\np[Mbit/s]{50}, the result of efficiency of about \np[\%]{40} is obtained with +\np[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5 is obtained with high external precision of \np{E-11} for a matrix size from 110 to 150 side elements. For the 3 clusters architecture including a total of 100 hosts, Table~\ref{tab.cluster.3x33} shows that it was difficult to have a combination -which gives an efficiency of asynchronous below \np[\%]{80}. Indeed, for a +which gives a relative gain of asynchronous mode more than 1.2. Indeed, for a matrix size of 62 elements, equality between the performance of the two modes (synchronous and asynchronous) is achieved with an inter cluster of -\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency by -\np[\%]{78} with a matrix size of 100 points, it was necessary to degrade the +\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency greater than 1.2 with a matrix size of 100 points, it was necessary to degrade the inter cluster network bandwidth from 5 to \np[Mbit/s]{2}. A last attempt was made for a configuration of three clusters but more powerful -with 200 nodes in total. The convergence with a speedup of \np[\%]{90} was +with 200 nodes in total. The convergence with a relative gain around 1.1 was obtained with a bandwidth of \np[Mbit/s]{1} as shown in Table~\ref{tab.cluster.3x67}. \LZK{Dans le papier, on compare les deux versions synchrone et asycnhrone du multisplitting. Y a t il des résultats pour comparer gmres parallèle classique avec multisplitting asynchrone? Ca permettra de montrer l'intérêt du multisplitting asynchrone sur des clusters distants} +\CER{En fait, les résultats ont été obtenus en comparant les temps d'exécution entre l'algo classique GMRES en mode synchrone avec le multisplitting en mode asynchrone, le tout sur un environnement de clusters distants} \section{Conclusion} The experimental results on executing a parallel iterative algorithm in