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
+of asynchronous iterative algorithms. For that, we compare the behavior 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
+simulations which let us easily choose some parameters. Both codes are real MPI
+codes and simulations allow us to see when the asynchronous multisplitting algorithm can be more
efficient than the GMRES one to solve a 3D Poisson problem.
$X_{0}$ 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 (synchronous or asynchronous) applications 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.
-
-<<<<<<< HEAD
-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 involves 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 AIAC 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 AIAC 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 AIAC
-application on different computing architectures.
+real asynchronous iterative application. {\bf The contribution of the present
+ paper can be summarized 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) method
+\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
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.
+faster than GMRES with two distant clusters. In this way, we present an original solution to optimize the use of a simulation
+tool to run efficiently an asynchronous iterative parallel algorithm in a grid architecture
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.
->>>>>>> 6785b9ef58de0db67c33ca901c7813f3dfdc76e0
+
\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
+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. 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}
+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}
-\RC{Je serais partant de virer AIAC et laisser asynchronous algorithms... à voir}
%% It is very challenging to develop efficient applications for large scale,
%% heterogeneous and distributed platforms such as computing grids. Researchers and
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 algorithms comes from the fact it is necessary to run the algorithm
+asynchronous iterative 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,
+nodes, inter and intra clusters bandwidth and latency, etc.) and of the
+algorithm (number of splittings with the multisplitting algorithm), the
+multisplitting code will obtain the solution more or less quickly. Of 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.
+asynchronous iterative algorithms before being able to run real experiments.
framework to study the behavior of large-scale distributed systems. As its name
says, it emanates from the grid computing community, but is nowadays used to
study grids, clouds, HPC or peer-to-peer systems. The early versions of SimGrid
-date from 1999, but it's still actively developed and distributed as an open
-source software. Today, it's one of the major generic tools in the field of
+date from 1999, but it is still actively developed and distributed as an open
+source software. Today, it is one of the major generic tools in the field of
simulation for large-scale distributed systems.
SimGrid provides several programming interfaces: MSG to simulate Concurrent
interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
languages. SMPI is the interface that has been used for the work exposed in
this paper. The SMPI interface implements about \np[\%]{80} of the MPI 2.0
-standard~\cite{bedaride:hal-00919507}, and supports applications written in C or
-Fortran, with little or no modifications.
+standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports
+applications written in C or Fortran, with little or no modifications.
-Within SimGrid, the execution of a distributed application is simulated on a
-single machine. The application code is really executed, but some operations
+Within SimGrid, the execution of a distributed application is simulated by a
+single process. The application code is really executed, but some operations
like the communications are intercepted, and their running time is computed
according to the characteristics of the simulated execution platform. The
description of this target platform is given as an input for the execution, by
the mean of an XML file. It describes the properties of the platform, such as
the computing nodes 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.
+their bandwidth and latency, and the routing strategy. The scheduling of the
+simulated processes, as well as the simulated running time of the application is
+computed according to these properties.
To compute the durations of the operations in the simulated world, and to take
into account resource sharing (e.g. bandwidth sharing between competing
communications), SimGrid uses a fluid model. This allows to run relatively fast
simulations, while still keeping accurate
-results~\cite{bedaride:hal-00919507,tomacs13}. Moreover, depending on the
+results~\cite{bedaride+degomme+genaud+al.2013.toward,
+ velho+schnorr+casanova+al.2013.validity}. Moreover, depending on the
simulated application, SimGrid/SMPI allows to skip long lasting computations and
to only take their duration into account. When the real computations cannot be
skipped, but the results have no importance for the simulation results, there is
several simulated processes, and thus to reduce the whole memory consumption.
These two techniques can help to run simulations at a very large scale.
+The validity of simulations with SimGrid has been asserted by several studies.
+See, for example, \cite{velho+schnorr+casanova+al.2013.validity} and articles
+referenced therein for the validity of the network models. Comparisons between
+real execution of MPI applications on the one hand, and their simulation with
+SMPI on the other hand, are presented in~\cite{guermouche+renard.2010.first,
+ clauss+stillwell+genaud+al.2011.single,
+ bedaride+degomme+genaud+al.2013.toward}. All these works conclude that
+SimGrid is able to simulate pretty accurately the real behavior of the
+applications.
+
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation of the multisplitting method}
+
+\subsection{The multisplitting method}
%Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping
\begin{equation*}
\begin{figure}[!t]
\centering
\includegraphics[width=60mm,keepaspectratio]{clustering}
-\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
+\caption{Example of three distant clusters of processors.}
\label{fig:4.1}
\end{figure}
all clusters are interconnected by a virtual unidirectional ring network (see
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 $i$
+is achieved. So starting from the cluster with rank 1, each master processor $\ell$
sets the token to \textit{True} if the local convergence is achieved or to
-\textit{False} otherwise, and sends it to master processor $i+1$. Finally, the
+\textit{False} otherwise, and sends it to master processor $\ell+1$. 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,
\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
+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 differences 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}{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)),
+\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 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.
+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.
\end{figure}
+\subsection{Simulation of the multisplitting method using SimGrid and SMPI}
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.}
+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. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method, the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
+and with the addition of the primitive \texttt{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.}
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.
+environment. We have successfully executed the code for the synchronous GMRES algorithm compared with our asynchronous multisplitting algorithm after few modifications.
\item Hosts processors power (GFlops) can also influence on the results.
\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,
- 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}
+ algorithm but also to get the main objective in getting an execution time with the asynchronous multisplitting less than with synchronous GMRES.
+ \end{itemize}
+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.
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
+area network configuration where the synchronous GMRES method will take advantage on the
rapid exchange of information on such high-speed links. Thus, the methodology
adopted was to launch the application on a clustered network. In this
configuration, degrading the inter-cluster network performance will penalize the
simulates the case of distant clusters linked with long distance network as in grid computing context.
-% 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 $N=N_x = N_y = N_z = \text{62}$ 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
\begin{table}[!t]
\centering
- \caption{2 clusters, each with 50 nodes}
+ \caption{Relative gain of the multisplitting algorithm compared to GMRES for
+ different configurations with 2 clusters, each one composed of 50 nodes. Latency = $20$ms}
\label{tab.cluster.2x50}
- \begin{mytable}{6}
- \hline
- bandwidth (Mbits/s)
- & 5 & 5 & 5 & 5 & 5 & 50 \\
+ \begin{mytable}{5}
\hline
- latency (ms)
- & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ bandwidth (Mbit/s)
+ & 5 & 5 & 5 & 5 & 5 \\
\hline
+ % latency (ms)
+ % & 20 & 20 & 20 & 20 & 20 \\
+ %\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 $(N)$
+ & $62^3$ & $62^3$ & $62^3$ & $100^3$ & $100^3$ \\
\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}
- \hline
- bandwidth (Mbits/s)
- & 50 & 50 & 50 & 50 \\ % & 10 & 10 \\
+ \begin{mytable}{5}
\hline
- latency (ms)
- & 0.02 & 0.02 & 0.02 & 0.02 \\ % & 0.03 & 0.01 \\
+ bandwidth (Mbit/s)
+ & 50 & 50 & 50 & 50 & 50 \\ % & 10 & 10 \\
\hline
+ %latency (ms)
+ %& 20 & 20 & 20 & 20 & 20 \\ % & 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 $(N)$
+ & $110^3$ & $120^3$ & $130^3$ & $140^3$ & $150^3$ \\ % & 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}
+%\RC{Du coup la latence est toujours la même, pourquoi la mettre dans la table?}
+
%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
%relative gains greater than 1 with a matrix size from 62 to 100 elements.
-\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}
+%\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}
%\begin{table}[!t]
% \centering
% \caption{3 clusters, each with 33 nodes}
\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;
+\item PLATFORM: XML file description of the platform architecture with the
+ following characteristics:
+ % two clusters (cluster1 and cluster2) with the following characteristics:
+ \begin{itemize}
+ \item 2 clusters of 50 hosts each;
+ \item Processor unit power: \np[GFlops]{1} or \np[GFlops]{1.5};
+ \item Intra-cluster network bandwidth: \np[Gbit/s]{1.25} and latency: \np[$\mu$s]{50};
+ \item Inter-cluster network bandwidth: \np[Mbit/s]{5} or \np[Mbit/s]{50} and latency: \np[ms]{20};
+ \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
+ clusters> <Number of hosts in cluster1> <Number of hosts in cluster2>;
+\item Maximum numbers of outer and inner iterations;
+\item Outer and inner precisions on the residual error;
+\item Matrix size $N_x$, $N_y$ and $N_z$;
+\item Matrix diagonal value: $6$ (see Equation~(\ref{eq:03}));
+\item Matrix off-diagonal values: $-1$;
+\item Communication mode: asynchronous.
\end{itemize}
\paragraph*{Interpretations and comments}
After analyzing the outputs, generally, for the two clusters including one hundred hosts configuration (Tables~\ref{tab.cluster.2x50}), some combinations of parameters affecting
the results have given a relative gain more than 2.5, showing the effectiveness of the
-asynchronous performance compared to the synchronous mode.
+asynchronous multisplitting compared to GMRES with two distant clusters.
With these settings, Table~\ref{tab.cluster.2x50} shows
-that after a deterioration of inter cluster network with a bandwidth of \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
-of one GFlops, an efficiency of about \np[\%]{40} is
-obtained in asynchronous mode for a matrix size of 62 elements. It is noticed that the result remains
+that after setting the bandwidth of the inter cluster network to \np[Mbit/s]{5}, the latency to $20$ millisecond and the processor power
+to one GFlops, an efficiency of about \np[\%]{40} is
+obtained in asynchronous mode for a matrix size of $62^3$ elements. It is noticed that the result remains
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
+increasing the matrix size up to $100^3$ 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
-high external precision of \np{E-11} for a matrix size from 110 to 150 side
+\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 $110^3$ to $150^3$ side
elements.
%For the 3 clusters architecture including a total of 100 hosts,
%(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 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}.
-\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
- Quelle est la perte de perfs en faisant ça ?}
+%\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
+ %Quelle est la perte de perfs en faisant ça ?}
%A last attempt was made for a configuration of three clusters but more powerful
%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}.
-\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
-computers organized with interconnected clusters have been presented.
-Our work has demonstrated that using such a simulation tool allow us to
-reach the following three objectives:
+The simulation of the execution of parallel asynchronous iterative algorithms on large scale clusters has been presented.
+In this work, we show that SimGrid is an efficient simulation tool that allows us to
+reach the following two objectives:
\begin{enumerate}
-\item To have a flexible configurable execution platform resolving the
-hard exercise to access to very limited but so solicited physical
-resources;
-\item to ensure the algorithm convergence with a reasonable time and
-iteration number ;
-\item and finally and more importantly, to find the correct combination
-of the cluster and network specifications permitting to save time in
-executing the algorithm in asynchronous mode.
+\item To have a flexible configurable execution platform that allows us to
+ simulate algorithms for which execution of all parts of
+ the code is necessary. Using simulations before real executions is a nice
+ solution to detect potential scalability problems.
+
+\item To test the combination of the cluster and network specifications permitting to execute an asynchronous algorithm faster than a synchronous one.
\end{enumerate}
-Our results have shown that in certain conditions, asynchronous mode is
-speeder up to \np[\%]{40} than executing the algorithm in synchronous mode
+Our results have shown that with two distant clusters, the asynchronous multisplitting method is faster to \np[\%]{40} compared to the synchronous GMRES method
which is not negligible for solving complex practical problems with more
and more increasing size.
- Several studies have already addressed the performance execution time of
+Several studies have already addressed the performance execution time of
this class of algorithm. The work presented in this paper has
demonstrated an original solution to optimize the use of a simulation
tool to run efficiently an iterative parallel algorithm in asynchronous
mode in a grid architecture.
-\LZK{Perspectives???}
+In future works, we plan to extend our experimentations to larger scale platforms by increasing the number of computing cores and the number of clusters.
+We will also have to increase the size of the input problem which will require the use of a more powerful simulation platform. At last, we expect to compare our simulation results to real execution results on real architectures in order to better experimentally validate our study. Finally, we also plan to study other problems with the multisplitting method and other asynchronous iterative methods.
\section*{Acknowledgment}
This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
-\todo[inline]{The authors would like to thank\dots{}}
+%\todo[inline]{The authors would like to thank\dots{}}
% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% LocalWords: Ouest Vieille Talence cedex scalability experimentations HPC MPI
% LocalWords: Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua
% LocalWords: Fortran GFlops priori Mbit de du fcomte multisplitting scalable
-% LocalWords: SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib
+% LocalWords: SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib Gbit
% LocalWords: intra durations nonsingular Waitall discretization discretized
-% LocalWords: InnerSolver Isend Irecv
+% LocalWords: InnerSolver Isend Irecv parallelization
