+
\documentclass[conference]{IEEEtran}
\usepackage[T1]{fontenc}
\begin{document}
-\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
+\title{Simulation of Asynchronous Iterative Algorithms Using SimGrid}
\author{%
\IEEEauthorblockN{%
Charles Emile Ramamonjisoa\IEEEauthorrefmark{1},
+ Lilia Ziane Khodja\IEEEauthorrefmark{2},
David Laiymani\IEEEauthorrefmark{1},
- Arnaud Giersch\IEEEauthorrefmark{1},
- Lilia Ziane Khodja\IEEEauthorrefmark{2} and
+ Arnaud Giersch\IEEEauthorrefmark{1} and
Raphaël Couturier\IEEEauthorrefmark{1}
}
\IEEEauthorblockA{\IEEEauthorrefmark{1}%
\maketitle
-\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
-capacity. One solution is to run the program in a virtual environment
-simulating a real interconnected computers architecture. The results are
-convincing and useful solutions are obtained with far fewer resources
-than in a real platform. However, challenges remain for the convergence
-and efficiency of a class of algorithms that concern us here, namely
-numerical parallel iterative algorithms executed in asynchronous mode,
-especially in a large scale level. Actually, such algorithm requires a
-balance and a compromise between computation and communication time
-during the execution. Two important factors determine the success of the
-experimentation: the convergence of the iterative algorithm on a large
-scale and the execution time reduction in asynchronous mode. Once again,
-from the current work, a simulated environment like SimGrid provides
-accurate results which are difficult or even impossible to obtain in a
-physical platform by exploiting the flexibility of the simulator on the
-computing units clusters and the network structure design. Our
-experimental outputs showed a saving of up to \np[\%]{40} for the algorithm
-execution time in asynchronous mode compared to the synchronous one with
-a residual precision up to \np{E-11}. Such successful results open
-perspectives on experimentations for running the algorithm on a
-simulated large scale growing environment and with larger problem size.
-
-\LZK{Long\ldots}
+
+Synchronous iterative algorithms are often less scalable than asynchronous
+iterative ones. Performing large scale experiments with different kind of
+network parameters is not easy because with supercomputers such parameters are
+fixed. So one solution consists in using simulations first in order to analyze
+what parameters could influence or not the behaviors of an algorithm. In this
+paper, we show that it is interesting to use SimGrid to simulate the behaviors
+of asynchronous iterative algorithms. For that, we compare the behaviour of a
+synchronous GMRES algorithm with an asynchronous multisplitting one with
+simulations in which we choose some parameters. Both codes are real MPI
+codes. Simulations allow us to see when the multisplitting algorithm can be more
+efficient than the GMRES one to solve a 3D Poisson problem.
+
% no keywords for IEEE conferences
% Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
increasing complexity of these requested applications combined with a continuous increase of their sizes lead to write
distributed and parallel algorithms requiring significant hardware resources (grid computing, clusters, broadband
network, etc.) but also a non-negligible CPU execution time. We consider in this paper a class of highly efficient
-parallel algorithms called \emph{numerical iterative algorithms} executed in a distributed environment. As their name
+parallel algorithms called \emph{iterative algorithms} executed in a distributed environment. As their name
suggests, these algorithms solve a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
$X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods
demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.
-Parallelization of such algorithms generally involve the division of the problem into several \emph{blocks} that will
-be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new
-iteration starts and until the approximate solution is reached. These parallel computations can be performed either in
-\emph{synchronous} mode where a new iteration begins only when all nodes communications are completed,
-or in \emph{asynchronous} mode where processors can continue independently with few or no synchronization points. For
-instance in the \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model~\cite{bcvc06:ij}, local
-computations do not need to wait for required data. Processors can then perform their iterations with the data present
-at that time. Even if the number of iterations required before the convergence is generally greater than for the
-synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to
-synchronizations especially in a grid computing context (see~\cite{Bahi07} for more details).
-
-Parallel numerical applications (synchronous or asynchronous) may have different
-configuration and deployment requirements. Quantifying their resource
-allocation policies and application scheduling algorithms in grid computing
-environments under varying load, CPU power and network speeds is very costly,
-very labor intensive and very time
-consuming~\cite{Calheiros:2011:CTM:1951445.1951450}. The case of AIAC
-algorithms is even more problematic since they are very sensible to the
-execution environment context. For instance, variations in the network bandwidth
-(intra and inter-clusters), in the number and the power of nodes, in the number
-of clusters\dots{} can lead to very different number of iterations and so to
-very different execution times. Then, it appears that the use of simulation
-tools to explore various platform scenarios and to run large numbers of
-experiments quickly can be very promising. In this way, the use of a simulation
-environment to execute parallel iterative algorithms found some interests in
-reducing the highly cost of access to computing resources: (1) for the
-applications development life cycle and in code debugging (2) and in production
-to get results in a reasonable execution time with a simulated infrastructure
-not accessible with physical resources. Indeed, the launch of distributed
-iterative asynchronous algorithms to solve a given problem on a large-scale
-simulated environment challenges to find optimal configurations giving the best
-results with a lowest residual error and in the best of execution time.
-
-To our knowledge, there is no existing work on the large-scale simulation of a
-real AIAC application. The aim of this paper is twofold. First we give a first
-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
-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[]{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. In the simulated environment, after setting appropriate
-network and cluster parameters like the network bandwidth, latency or the processors power,
-the experimental results have demonstrated a asynchronous execution time saving up to \np[\%]{40} in
-compared to the synchronous 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!}
-\CER{La phrase a été modifiée}
-
-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
-distributed architectures. The algorithm of the multisplitting method based on GMRES \LZK{??? GMRES n'utilise pas la méthode de multisplitting! Sinon ne doit on pas expliquer le choix d'une méthode de multisplitting?} \CER{La phrase a été corrigée} written with MPI primitives and
-its adaptation to SimGrid with SMPI (Simulated MPI) is detailed in the next section. At last, the experiments results
-carried out will be presented before some concluding remarks and future works.
+Parallelization of such algorithms generally involve the division of the problem
+into several \emph{blocks} that will be solved in parallel on multiple
+processing units. The latter will communicate each intermediate results before a
+new iteration starts and until the approximate solution is reached. These
+parallel computations can be performed either in \emph{synchronous} mode where a
+new iteration begins only when all nodes communications are completed, or in
+\emph{asynchronous} mode where processors can continue independently with no
+synchronization points~\cite{bcvc06:ij}. In this case, local computations do not
+need to wait for required data. Processors can then perform their iterations
+with the data present at that time. Even if the number of iterations required
+before the convergence is generally greater than for the synchronous case,
+asynchronous iterative algorithms can significantly reduce overall execution
+times by suppressing idle times due to synchronizations especially in a grid
+computing context (see~\cite{Bahi07} for more details).
+
+Parallel applications based on a (synchronous or asynchronous) iteration model
+may have different configuration and deployment requirements. Quantifying their
+resource allocation policies and application scheduling algorithms in grid
+computing environments under varying load, CPU power and network speeds is very
+costly, very labor intensive and very time
+consuming~\cite{Calheiros:2011:CTM:1951445.1951450}. The case of asynchronous
+iterative algorithms is even more problematic since they are very sensible to
+the execution environment context. For instance, variations in the network
+bandwidth (intra and inter-clusters), in the number and the power of nodes, in
+the number of clusters\dots{} can lead to very different number of iterations
+and so to very different execution times. Then, it appears that the use of
+simulation tools to explore various platform scenarios and to run large numbers
+of experiments quickly can be very promising. In this way, the use of a
+simulation environment to execute parallel iterative algorithms found some
+interests in reducing the highly cost of access to computing resources: (1) for
+the applications development life cycle and in code debugging (2) and in
+production to get results in a reasonable execution time with a simulated
+infrastructure not accessible with physical resources. Indeed, the launch of
+distributed iterative asynchronous algorithms to solve a given problem on a
+large-scale simulated environment challenges to find optimal configurations
+giving the best results with a lowest residual error and in the best of
+execution time.
+
+
+To our knowledge, there is no existing work on the large-scale simulation of a
+real asynchronous iterative application. {\bf The contribution of the present
+ paper can be summarised in two main points}. First we give a first approach
+of the simulation of asynchronous iterative algorithms using a simulation tool
+(i.e. the SimGrid toolkit~\cite{SimGrid}). Second, we confirm the
+effectiveness of the asynchronous multisplitting algorithm by comparing its
+performance with the synchronous GMRES (Generalized Minimal Residual)
+\cite{ref1}. Both these codes can be used to solve large linear systems. In
+this paper, we focus on a 3D Poisson problem. We show, that with minor
+modifications of the initial MPI code, the SimGrid toolkit allows us to perform
+a test campaign of a real asynchronous iterative application on different
+computing architectures.
+% The simulated 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. Parameters of the
+network platforms are the bandwidth and the latency of inter cluster
+network. Parameters on the cluster's architecture are the number of machines and
+the computation power of a machine. Simulations show that the asynchronous
+multisplitting algorithm can solve the 3D Poisson problem approximately twice
+faster than GMRES with two distant clusters.
+
+
+
+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
+distributed architectures. Then, the multisplitting method is presented, it is
+based on GMRES to solve each block obtained of the splitting. This code is
+written with MPI primitives and its adaptation to SimGrid with SMPI (Simulated
+MPI) is detailed in the next section. At last, the simulation results carried
+out will be presented before some concluding remarks and future works.
+
\section{Motivations and scientific context}
-As exposed in the introduction, parallel iterative methods are now widely used in many scientific domains. They can be
-classified in three main classes depending on how iterations and communications are managed (for more details readers
-can refer to~\cite{bcvc06:ij}). In the \textit{Synchronous Iterations~-- Synchronous Communications (SISC)} model data
-are exchanged at the end of each iteration. All the processors must begin the same iteration at the same time and
-important idle times on processors are generated. The \textit{Synchronous Iterations~-- Asynchronous Communications
-(SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously
-i.e. without stopping current computations. This technique allows to partially overlap communications by computations
-but unfortunately, the overlapping is only partial and important idle times remain. It is clear that, in a grid
-computing context, where the number of computational nodes is large, heterogeneous and widely distributed, the idle
-times generated by synchronizations are very penalizing. One way to overcome this problem is to use the
-\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.
-\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.\LZK{Répétition par rapport à l'intro}
+As exposed in the introduction, parallel iterative methods are now widely used
+in many scientific domains. They can be classified in three main classes
+depending on how iterations and communications are managed (for more details
+readers can refer to~\cite{bcvc06:ij}). In the synchronous iterations model,
+data are exchanged at the end of each iteration. All the processors must begin
+the same iteration at the same time and important idle times on processors are
+generated. It is possible to use asynchronous communications, in this case, the
+model can be compared to the previous one except that data required on another
+processor are sent asynchronously i.e. without stopping current computations.
+This technique allows to partially overlap communications by computations but
+unfortunately, the overlapping is only partial and important idle times remain.
+It is clear that, in a grid computing context, where the number of computational
+nodes is large, heterogeneous and widely distributed, the idle times generated
+by synchronizations are very penalizing. One way to overcome this problem is to
+use the asynchronous iterations 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. 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},
+asynchronous iterative algorithms can significantly reduce overall execution
+times by suppressing idle times due to synchronizations especially in a grid
+computing context.
\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{AIAC.pdf}
- \caption{The Asynchronous Iterations~-- Asynchronous Communications model}
+ \caption{The asynchronous iterations model}
\label{fig:aiac}
\end{figure}
-It is very challenging to develop efficient applications for large scale,
-heterogeneous and distributed platforms such as computing grids. Researchers and
-engineers have to develop techniques for maximizing application performance of
-these multi-cluster platforms, by redesigning the applications and/or by using
-novel algorithms that can account for the composite and heterogeneous nature of
-the platform. Unfortunately, the deployment of such applications on these very
-large scale systems is very costly, labor intensive and time consuming. In this
-context, it appears that the use of simulation tools to explore various platform
-scenarios at will and to run enormous numbers of experiments quickly can be very
-promising. Several works\dots{}
+%% It is very challenging to develop efficient applications for large scale,
+%% heterogeneous and distributed platforms such as computing grids. Researchers and
+%% engineers have to develop techniques for maximizing application performance of
+%% these multi-cluster platforms, by redesigning the applications and/or by using
+%% novel algorithms that can account for the composite and heterogeneous nature of
+%% the platform. Unfortunately, the deployment of such applications on these very
+%% large scale systems is very costly, labor intensive and time consuming. In this
+%% context, it appears that the use of simulation tools to explore various platform
+%% scenarios at will and to run enormous numbers of experiments quickly can be very
+%% promising. Several works\dots{}
+
+%% \AG{Several works\dots{} what?\\
+% Le paragraphe suivant se trouve déjà dans l'intro ?}
+In the context of asynchronous algorithms, the number of iterations to reach the
+convergence depends on the delay of messages. With synchronous iterations, the
+number of iterations is exactly the same than in the sequential mode (if the
+parallelization process does not change the algorithm). So the difficulty with
+asynchronous iteratie algorithms comes from the fact it is necessary to run the algorithm
+with real data. In fact, from an execution to another the order of messages will
+change and the number of iterations to reach the convergence will also change.
+According to all the parameters of the platform (number of nodes, power of
+nodes, inter and intra clusrters bandwith and latency, ....) and of the
+algorithm (number of splitting with the multisplitting algorithm), the
+multisplitting code will obtain the solution more or less quickly. Or course,
+the GMRES method also depends of the same parameters. As it is difficult to have
+access to many clusters, grids or supercomputers with many different network
+parameters, it is interesting to be able to simulate the behaviors of
+asynchronous iterative algoritms before being able to runs real experiments.
+
-\AG{Several works\dots{} what?\\
- Le paragraphe suivant se trouve déjà dans l'intro ?}
-In the context of AIAC algorithms, the use of simulation tools is even more
-relevant. Indeed, this class of applications is very sensible to the execution
-environment context. For instance, variations in the network bandwidth (intra
-and inter-clusters), in the number and the power of nodes, in the number of
-clusters\dots{} can lead to very different number of iterations and so to very
-different execution times.
tolerance threshold of the error computed between two successive local solution
$X_\ell^k$ and $X_\ell^{k+1}$.
+
+
+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. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
+\begin{equation}
+\begin{array}{l}
+u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
+%u(x,y,z)= & \frac{1}{6}\times [u(x-1,y,z) + u(x+1,y,z) + \\
+ % & u(x,y-1,z) + u(x,y+1,z) + \\
+ % & u(x,y,z-1) + u(x,y,z+1) - \\ & h^2f(x,y,z)],
+\end{array}
+\label{eq:03}
+\end{equation}
+where $h$ is the distance between two adjacent elements in the spatial discretization scheme and 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}
+
+
+
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 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}
-\CER{Le problème majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a été le plantage en SMPI de Waitall après un Isend et Irecv. J'avais proposé un workaround en utilisant un MPI\_wait séparé pour chaque échange a la place d'un waitall unique pour TOUTES les échanges, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais après, tu as modifié le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de détection de la convergence et du coup, l'échange global avec waitall a aussi fonctionné.}
+%\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}
+%\CER{Le problème majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a été le plantage en SMPI de Waitall après un Isend et Irecv. J'avais proposé un workaround en utilisant un MPI\_wait séparé pour chaque échange a la place d'un waitall unique pour TOUTES les échanges, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais après, tu as modifié le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de détection de la convergence et du coup, l'échange global avec waitall a aussi fonctionné.}
Note here that the use of SMPI functions optimizer for memory footprint and CPU usage is not recommended knowing that one wants to get real results by simulation.
-As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. First, all declared
-global variables have been moved to local variables for each subroutine. In fact, global variables generate side effects arising from the concurrent access of
-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.
-\AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.}
-\CER{Ce problème fait partie des modifications que j'ai dû faire dans l'adaptation du programme MPI vers SMPI. IL découle de la différence de la taille des mots en mémoire : en 32 bits, pour les variables declarees en long int, on garde dans les instructions de sortie (printf, sprintf, ...) le format \%lu sinon en 64 bits, on le substitue par \%llu.}
- Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
+As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. The scope of all declared
+global variables have been moved to local to subroutine. Indeed, global variables generate side effects arising from the concurrent access of
+shared memory used by threads simulating each computing unit in the SimGrid architecture.
+%Second, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
+%\AG{compilation or run-time error?}
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 successfully executed the code in synchronous mode using parallel GMRES algorithm compared with our multisplitting algorithm in asynchronous mode after few modifications.
-\section{Experimental results}
+\section{Simulation results}
When the \textit{real} application runs in the simulation environment and produces the expected results, varying the input
parameters and the program arguments allows us to compare outputs from the code execution. We have noticed from this
\item Finally, when submitting job batches for execution, the arguments values
passed to the program like the maximum number of iterations or the precision are critical. They allow us to ensure not only the convergence of the
algorithm but also to get the main objective in getting an execution time in asynchronous communication less than in
- synchronous mode. The ratio between the execution time of synchronous
- compared to the asynchronous mode ($t_\text{sync} / t_\text{async}$) is defined as the \emph{relative gain}. So,
+ synchronous mode. The ratio between the simulated execution time of synchronous GMRES algorithm
+ compared to the asynchronous multisplitting algorithm ($t_\text{GMRES} / t_\text{Multisplitting}$) is defined as the \emph{relative gain}. So,
our objective running the algorithm in SimGrid is to obtain a relative gain
greater than 1.
- \AG{$t_\text{async} / t_\text{sync} > 1$, l'objectif est donc que ça dure plus
- longtemps (que ça aille moins vite) en asynchrone qu'en synchrone ?
- Ce n'est pas plutôt l'inverse ?}
- \CER{J'ai modifie la phrase.}
\end{itemize}
A priori, obtaining a relative gain greater than 1 would be difficult in a local
synchronous mode allowing to get a relative gain greater than 1. This action
simulates the case of distant clusters linked with long distance network as in grid computing context.
-\AG{Cette partie sur le poisson 3D
- % on sait donc que ce n'est pas une plie ou une sole (/me fatigué)
- n'est pas à sa place. Elle devrait être placée plus tôt.}
-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 two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The algorithm convergence with a 3D
-matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
+Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
+factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem ranges from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
-\text{\np{3375000}}$ entries), is obtained in asynchronous in average 2.5 times speeder than the synchronous mode.
-\AG{Expliquer comment lire les tableaux.}
-\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
+\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one.
+%\AG{Expliquer comment lire les tableaux.}
+%\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
% 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
\caption{2 clusters, each with 50 nodes}
\label{tab.cluster.2x50}
- \begin{mytable}{6}
+ \begin{mytable}{5}
\hline
- bandwidth (Mbits/s)
- & 5 & 5 & 5 & 5 & 5 & 50 \\
+ bandwidth (Mbit/s)
+ & 5 & 5 & 5 & 5 & 5 \\
\hline
latency (ms)
- & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
\hline
power (GFlops)
- & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 \\
+ & 1 & 1 & 1 & 1.5 & 1.5 \\
\hline
- size
- & 62 & 62 & 62 & 100 & 100 & 110 \\
+ size $(n^3)$
+ & 62 & 62 & 62 & 100 & 100 \\
\hline
Precision
- & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\
+ & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} \\
\hline
\hline
Relative gain
- & 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 \\
+ & 2.52 & 2.55 & 2.52 & 2.57 & 2.54 \\
\hline
\end{mytable}
\bigskip
- \begin{mytable}{6}
+ \begin{mytable}{5}
\hline
- bandwidth (Mbits/s)
- & 50 & 50 & 50 & 50 \\ % & 10 & 10 \\
+ bandwidth (Mbit/s)
+ & 50 & 50 & 50 & 50 & 50 \\ % & 10 & 10 \\
\hline
latency (ms)
- & 0.02 & 0.02 & 0.02 & 0.02 \\ % & 0.03 & 0.01 \\
+ & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\ % & 0.03 & 0.01 \\
\hline
Power (GFlops)
- & 1.5 & 1.5 & 1.5 & 1.5 \\ % & 1 & 1.5 \\
+ & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 \\ % & 1 & 1.5 \\
\hline
- size
- & 120 & 130 & 140 & 150 \\ % & 171 & 171 \\
+ size $(n^3)$
+ & 110 & 120 & 130 & 140 & 150 \\ % & 171 & 171 \\
\hline
Precision
- & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5} & \np{E-5} \\
+ & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5} & \np{E-5} \\
\hline
\hline
Relative gain
- & 2.51 & 2.58 & 2.55 & 2.54 \\ % & 1.59 & 1.29 \\
+ & 2.53 & 2.51 & 2.58 & 2.55 & 2.54 \\ % & 1.59 & 1.29 \\
\hline
\end{mytable}
\end{table}
\paragraph*{SMPI parameters}
-~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).}
-\CER {Précisions ajoutées}
-
\begin{itemize}
\item HOSTFILE: Text file containing the list of the processors units name. Here 100 hosts;
\item PLATFORM: XML file description of the platform architecture : two clusters (cluster1 and cluster2) with the following characteristics :
-
- - Processor unit power : 1.5 GFlops;
-
- - Intracluster network : bandwidth = 1,25 Gbits/s and latency = 5E-05 ms;
-
- - Intercluster network : bandwidth = 5 Mbits/s and latency = 5E-03 ms;
+ \begin{itemize}
+ \item Processor unit power: \np[GFlops]{1.5};
+ \item Intracluster network bandwidth: \np[Gbit/s]{1.25} and latency:
+ \np[$\mu$s]{0.05};
+ \item Intercluster network bandwidth: \np[Mbit/s]{5} and latency:
+ \np[$\mu$s]{5};
+ \end{itemize}
\end{itemize}
\paragraph*{Arguments of the program}
\begin{itemize}
- \item Description of the cluster architecture matching the format <Number of cluster> <Number of hosts in cluster\_1> <Number of hosts in cluster\_2>;
- \item Maximum number of iterations;
- \item Precisions on the residual error;
- \item Matrix size $N_x$, $N_y$ and $N_z$;
- \item Matrix diagonal value: \np{1.0} (See (3));
- \item Matrix off-diagonal value: $-\frac{1}{6}$ (See(3));
- \item Communication mode: Asynchronous.
+\item Description of the cluster architecture matching the format <Number of
+ cluster> <Number of hosts in cluster1> <Number of hosts in cluster2>;
+\item Maximum number of iterations;
+\item Precisions on the residual error;
+\item Matrix size $N_x$, $N_y$ and $N_z$;
+\item Matrix diagonal value: $6$ (See~(\ref{eq:03}));
+\item Matrix off-diagonal value: $-1$;
+\item Communication mode: asynchronous.
\end{itemize}
\paragraph*{Interpretations and comments}
stable even we vary the residual error precision from \np{E-5} to \np{E-9}. By
increasing the matrix size up to 100 elements, it was necessary to increase the
CPU power of \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency. Maintaining such processor power but increasing network throughput inter cluster up to
-\np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5\AG[]{2.5 ?} is obtained with
+\np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5 is obtained with
high external precision of \np{E-11} for a matrix size from 110 to 150 side
elements.
%obtained with a bandwidth of \np[Mbit/s]{1} as shown in
%Table~\ref{tab.cluster.3x67}.
-\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
-\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
-\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
-\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
+%\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
+%\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
+%\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
+%\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
\section{Conclusion}
The experimental results on executing a parallel iterative algorithm in
asynchronous mode on an environment simulating a large scale of virtual
