\usepackage{graphicx}
\usepackage[american]{babel}
% Extension pour les liens intra-documents (tagged PDF)
-% et l'affichage correct des URL (commande \url{http://example.com})
+% et l'affichage correct des UR (commande \url{http://example.com})
%\usepackage{hyperref}
\usepackage{url}
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
+fixed. So, one solution consists in using simulations first in order to analyze
+what parameters could influence or not the behavior of an algorithm. In this
+paper, we show that it is interesting to use SimGrid to simulate the behavior
+of asynchronous iterative algorithms. For that, we compare the behavior of a
synchronous GMRES algorithm with an asynchronous multisplitting one with
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
\section{Introduction}
-Parallel computing and high performance computing (HPC) are becoming more and more imperative for solving various
-problems raised by researchers on various scientific disciplines but also by industrial in the field. Indeed, the
+Parallel computing and high performance computing (HPC) are becoming more and more imperative to solve various
+problems raised by researchers on various scientific disciplines but also by industrialists in the field. Indeed, the
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
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
+parallel computations can be performed either in a \emph{synchronous} mode, where a
+new iteration begins only when all nodes communications are completed, or in an
\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,
+with the data present at that time. Even if the number of required iterations
+before the convergence is generally greater than in 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
+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
+computing environments under varying load, CPU power and network speeds are 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
+iterative algorithms is even more problematic since they are very sensitive 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
+of experiments quickly can be very promising.
+
+Thus, using a simulation environment to execute parallel iterative algorithms can prove to be very interesting to reduce 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.
+infrastructure not accessible with physical resources. Indeed, to find optimal configurations
+giving the best results with a lowest residual error and in the best
+execution time is very challenging for large scale distributed iterative asynchronous algorithms
To our knowledge, there is no existing work on the large-scale simulation of a
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
+efficiency of the asynchronous multisplitting algorithm by comparing its
+performances 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
+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
+SimGrid has 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.
+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
This article is structured as follows: after this introduction, the next section
-will give a brief description of iterative asynchronous model. Then, the
+will give a brief description of the 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
+based on GMRES to solve each block obtained from 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 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
+As described 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 and useless idle times used for synchronization 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.
+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 communications to be partially overlapped by computations
+but unfortunately, the overlapping is only partial and useless idle times used for synchronization 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
%% \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
+convergence depends on the delay of the 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 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
+asynchronous iterative algorithms comes from the fact that it is necessary to run the algorithm
+with real data. Indeed, from one execution to the other 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, etc.) and of the
+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
+the GMRES method also depends on 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.
+parameters, it is interesting to be able to simulate the behavior of
+asynchronous iterative algorithms before being able to run real experiments.
\section{SimGrid}
-SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid} is a simulation
-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 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~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
+is a simulation framework to study the behavior of large-scale distributed
+systems. As its name suggests, 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 back 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
Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
run real applications written in MPI~\cite{MPI}. Apart from the native C
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
+languages. SMPI is the interface that has been used for the work described 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 (cf Section IV - paragraph B).
-Within SimGrid, the execution of a distributed application is simulated on a
-single machine. The application code is really executed, but some operations
-like the communications are intercepted, and their running time is computed
+Within SimGrid, the execution of a distributed application is simulated by a
+single process. The application code is really executed, but some operations,
+like 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
+means 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
+are 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
+communications), SimGrid uses a fluid model. This allows users 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
-also the possibility to share dynamically allocated data structures between
+skipped, but the results are unimportant for the simulation results, it is
+also possible to share dynamically allocated data structures between
several simulated processes, and thus to reduce the whole memory consumption.
-These two techniques can help to run simulations at a very large scale.
+These two techniques can help to run simulations on 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}
\label{algo:01}
\end{figure}
-Algorithm on Figure~\ref{algo:01} shows the main key points of the
+The algorithm in 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
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 $\ell$
+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 $\ell+1$. 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,
+cluster 1 broadcasts a stop message to the masters of other clusters. In this work,
the local convergence on each cluster $\ell$ is detected when the following
condition is satisfied
\begin{equation*}
- (k\leq \MI) \text{ or } (\|X_\ell^k - X_\ell^{k+1}\|_{\infty}\leq\epsilon)
+ (k=\MI) \text{ or } (\|X_\ell^k - X_\ell^{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
-$X_\ell^k$ and $X_\ell^{k+1}$.
+$X_\ell^k$ and $X_\ell^{k+1}$. It should be noted that with asynchronous iterative algorithms, we cannot use a classical norm (which would require to synchronize all processors), such as $\|X_\ell^k - X_\ell^{k+1}\|_{2}$ for example. Nevertheless, in our experiments, we check that the final result is correct, for this we compute the precision with $max_i | A*x-b |_i$.
-In this paper, we solve the 3D Poisson problem whose the mathematical model is
+In this paper, we solve the 3D Poisson problem whose mathematical model is
\begin{equation}
\left\{
\begin{array}{l}
\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 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
+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 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),
\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.
+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 one 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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-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. 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
+We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid. Only, some code
+debugging has been required. 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. 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
+As mentioned, upon this adaptation, the algorithm is executed as in real life in the simulated environment after the following minor changes. The scope of all declared
+global variables have been moved to local subroutines. 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?}
\begin{itemize}
\item At the network level, we found that the most critical values are the
bandwidth and the network latency.
-\item Hosts processors power (GFlops) can also influence on the results.
+\item Host processor power (GFlops) can also influence 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 with the asynchronous multisplitting less than with synchronous GMRES.
adopted was to launch the application on a clustered network. In this
configuration, degrading the inter-cluster network performance will 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 as in grid computing context.
+simulates the case of distant clusters linked with long distance networks as in grid computing context.
-Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
+Both codes were simulated on a two clusters based network with 50 hosts each, totalling 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). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one.
+\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is on 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
\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 whith the following characteristics: %two clusters (cluster1 and cluster2) with the following characteristics :
+\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};
\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
+the results, have given a relative gain of more than 2.5, showing the effectiveness of the
asynchronous multisplitting compared to GMRES with two distant clusters.
With these settings, Table~\ref{tab.cluster.2x50} shows
-that after setting the bandwidth of the inter cluster network to \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
+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
+stable even if the residual error precision varies from \np{E-5} to \np{E-9}. By
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 is obtained with
+CPU power by \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency. Maintaining a relative gain of $2.5$ and such processor power but increasing network throughput inter cluster up to \np[Mbit/s]{50}, is obtained with
high external precision of \np{E-11} for a matrix size from $110^3$ to $150^3$ side
-elements.
+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
%\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
\section{Conclusion}
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
+In this work, we show that SimGrid is one of efficient simulation tool that has enabled us to
reach the following two objectives:
\begin{enumerate}
\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 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.
+Our results have shown that with two distant clusters, the asynchronous multisplitting method is faster by \np[\%]{40} compared to the synchronous GMRES method
+which is not negligible for solving complex practical problems with ever increasing size.
Several studies have already addressed the performance execution time of
this class of algorithm. The work presented in this paper has
% 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
