-
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-
-
\usepackage[T1]{fontenc}
-\usepackage{ucs}
-%\usepackage[utf8x]{inputenc}
-\usepackage{lmodern}
-\usepackage{color}
-%% Jolis entetes %%
-\usepackage[Glenn]{fncychap}
-%\usepackage{amsmath}
+\usepackage[utf8]{inputenc}
+\usepackage{amsfonts,amssymb}
+\usepackage{amsmath}
+%\usepackage{algorithm}
+\usepackage{algpseudocode}
%\usepackage{amsthm}
-%\usepackage{amsfonts}
-%\usepackage{graphicx}
-%\usepackage{xspace}
-% Definition des marges
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-\setpapersize[portrait]{A4}
-\usepackage[francais]{babel}
-% Extension pour les graphiques EPS
-%\usepackage[dvips]{graphicx}
-\usepackage[pdftex,final]{graphicx}
+\usepackage{graphicx}
+\usepackage[american]{babel}
% Extension pour les liens intra-documents (tagged PDF)
% et l'affichage correct des URL (commande \url{http://example.com})
-\usepackage{hyperref}
+%\usepackage{hyperref}
+
+\usepackage{url}
+\DeclareUrlCommand\email{\urlstyle{same}}
-\ifCLASSINFOpdf
- \usepackage[pdftex]{graphicx}
- \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
-\else
-\fi
+\usepackage[autolanguage,np]{numprint}
+\AtBeginDocument{%
+ \renewcommand*\npunitcommand[1]{\text{#1}}
+ \npthousandthpartsep{}}
+\usepackage{xspace}
+\usepackage[textsize=footnotesize]{todonotes}
+\newcommand{\AG}[2][inline]{%
+ \todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
+\newcommand{\DL}[2][inline]{%
+ \todo[color=yellow!50,#1]{\sffamily\textbf{DL:} #2}\xspace}
+\newcommand{\LZK}[2][inline]{%
+ \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+\newcommand{\RC}[2][inline]{%
+ \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
-% correct bad hyphenation here
-\hyphenation{op-tical net-works semi-conduc-tor}
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+\newcommand{\MI}{\mathit{MaxIter}}
\begin{document}
-%
-% paper title
-% can use linebreaks \\ within to get better formatting as desired
-\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
+\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
-% author names and affiliations
-% use a multiple column layout for up to three different
-% affiliations
-\author{\IEEEauthorblockN{Raphaël Couturier and Arnaud Giersch and David Laiymani and Charles-Emile Ramamonjisoa}
-\IEEEauthorblockA{Femto-ST Institute - DISC Department\\
-Université de Franche-Comté\\
-Belfort\\
-Email: raphael.couturier@univ-fcomte.fr}
-%\and
-%\IEEEauthorblockN{Arnaud Giersch}
-%\IEEEauthorblockA{Twentieth Century Fox\\
-%Springfield, USA\\
-%Email: homer@thesimpsons.com}
-%\and
-%\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
-%\IEEEauthorblockA{Starfleet Academy\\
-%San Francisco, California 96678-2391\\
-%Telephone: (800) 555--1212\\
-%Fax: (888) 555--1212
+\author{%
+ \IEEEauthorblockN{%
+ Charles Emile Ramamonjisoa\IEEEauthorrefmark{1},
+ David Laiymani\IEEEauthorrefmark{1},
+ Arnaud Giersch\IEEEauthorrefmark{1},
+ Lilia Ziane Khodja\IEEEauthorrefmark{2} and
+ Raphaël Couturier\IEEEauthorrefmark{1}
+ }
+ \IEEEauthorblockA{\IEEEauthorrefmark{1}%
+ Femto-ST Institute -- DISC Department\\
+ Université de Franche-Comté,
+ IUT de Belfort-Montbéliard\\
+ 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
+ Email: \email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}
+ }
+ \IEEEauthorblockA{\IEEEauthorrefmark{2}%
+ Inria Bordeaux Sud-Ouest\\
+ 200 avenue de la Vieille Tour, 33405 Talence cedex, France \\
+ Email: \email{lilia.ziane@inria.fr}
+ }
}
-
-
-% make the title area
\maketitle
-
+\RC{Ordre des auteurs pas définitif.}
\begin{abstract}
-%\boldmath
-The abstract goes here.
+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.
+
+% no keywords for IEEE conferences
+% Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
\end{abstract}
-% IEEEtran.cls defaults to using nonbold math in the Abstract.
-% This preserves the distinction between vectors and scalars. However,
-% if the conference you are submitting to favors bold math in the abstract,
-% then you can use LaTeX's standard command \boldmath at the very start
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-% math in the abstract anyway.
-
-% no keywords
-
-
-
-
-% For peer review papers, you can put extra information on the cover
-% page as needed:
-% \ifCLASSOPTIONpeerreview
-% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
-% \fi
-%
-% For peerreview papers, this IEEEtran command inserts a page break and
-% creates the second title. It will be ignored for other modes.
-\IEEEpeerreviewmaketitle
-
-
\section{Introduction}
-Présenter un bref état de l'art sur la simulation d'algos parallèles. Présenter rapidement les algos itératifs asynchrones et leurs avantages. Parler de leurs inconvénients en particulier la difficulté de déploiement à grande échelle donc il serait bien de simuler. Dire qu'à notre connaissance il n'existe pas de simulation de ce type d'algo.
-Présenter les travaux et les résultats obtenus. Annoncer le plan.
+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
+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
+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
+non-symmetric linear system of equations by numerical method GMRES (Generalized
+Minimal Residual) []\AG[]{[]?}. 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[]{??}.
+SimGrid had allowed us to launch the application from a modest computing
+infrastructure by simulating different distributed architectures composed by
+clusters nodes interconnected by variable speed networks. With selected
+parameters on the network platforms (bandwidth, latency of inter cluster
+network) and on the clusters architecture (number, capacity calculation power)
+in the simulated environment, the experimental results have demonstrated not
+only the algorithm convergence within a reasonable time compared with the
+physical environment performance, but also a time saving of up to \np[\%]{40} in
+asynchronous mode.
+
+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 used by GMRES 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.
-\section{The asynchronous iteration model}
+\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. 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.
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[width=8cm]{AIAC.pdf}
+ \caption{The Asynchronous Iterations~-- Asynchronous Communications 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{}
+
+\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.
+
+
-Décrire le modèle asynchrone. Je m'en charge (DL)
\section{SimGrid}
-Décrire SimGrid (Arnaud)
+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's still actively developed and distributed as an open
+source software. Today, it's 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
+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.
+
+With 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 to be simulated 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 node
+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.
+
+\AG{Faut-il ajouter quelque-chose ?}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Simulation of 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*}
+ \left(\begin{array}{ccc}
+ A_{11} & \cdots & A_{1L} \\
+ \vdots & \ddots & \vdots\\
+ A_{L1} & \cdots & A_{LL}
+ \end{array} \right)
+ \times
+ \left(\begin{array}{c}
+ X_1 \\
+ \vdots\\
+ X_L
+ \end{array} \right)
+ =
+ \left(\begin{array}{c}
+ B_1 \\
+ \vdots\\
+ B_L
+ \end{array} \right)
+\end{equation*}
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$.
+
+The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
+\begin{equation}
+ \label{eq:4.1}
+ \left\{
+ \begin{array}{l}
+ A_{ll}X_l = Y_l \text{, such that}\\
+ Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
+ \end{array}
+ \right.
+\end{equation}
+is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers.
+
+\begin{figure}[!t]
+ %%% IEEE instructions forbid to use an algorithm environment here, use figure
+ %%% instead
+\begin{algorithmic}[1]
+\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
+\Output $X_l$ (solution sub-vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$
+\State Set the initial guess $x^0$
+\For {$k=0,1,2,\ldots$ until the global convergence}
+\State Restart outer iteration with $x^0=x^k$
+\State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
+\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters
+\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
+\EndFor
+
+\Statex
+
+\Function {InnerSolver}{$x^0$, $k$}
+\State Compute local right-hand side $Y_l$:
+ \begin{equation*}
+ Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0
+ \end{equation*}
+\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method
+\State \Return $X_l^k$
+\EndFunction
+\end{algorithmic}
+\caption{A multisplitting solver with GMRES method}
+\label{algo:01}
+\end{figure}
+
+Algorithm on Figure~\ref{algo:01} shows the main key points of the
+multisplitting method to solve a large sparse linear system. This algorithm is
+based on an outer-inner iteration method where the parallel synchronous GMRES
+method is used to solve the inner iteration. It is executed in parallel by each
+cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors
+with the subscript $l$ represent the local data for cluster $l$, while
+$\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and
+$\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with
+neighboring clusters. At every outer iteration $k$, asynchronous communications
+are performed between processors of the local cluster and those of distant
+clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in
+Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are
+exchanged by message passing using MPI non-blocking communication routines.
+
+\begin{figure}[!t]
+\centering
+ \includegraphics[width=60mm,keepaspectratio]{clustering}
+\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
+\label{fig:4.1}
+\end{figure}
+
+The global convergence of the asynchronous multisplitting solver is detected
+when the clusters of processors have all converged locally. We implemented the
+global convergence detection process as follows. On each cluster a master
+processor is designated (for example the processor with rank 1) and masters of
+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 $i$
+sets the token to \textit{True} if the local convergence is achieved or to
+\text\it{False} otherwise, and sends it to master processor $i+1$. Finally, the
+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,
+the local convergence on each cluster $l$ is detected when the following
+condition is satisfied
+\begin{equation*}
+ (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)
+\end{equation*}
+where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SIMGRI\LZK[]{SimGrid} unless some code
+debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between the six neighbors of each point in a submatrix within a cluster or
+between clusters, \LZK{Il faut expliquer pourquoi 6 points voisins (7-point stencil problem)}
+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.
+\LZK{Peut-être mettre plus de précisions sur les difficultés rencontrées dans la version async et les adaptaions effectuées pour SimGrid}
+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. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
+In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real
+environment. We have tested in synchronous mode with a simulated platform starting from a modest 2 or 3 clusters grid to a larger configuration like simulating
+Grid5000 with more than 1500 hosts with 5000 cores~\cite{bolze2006grid}. Once the code debugging and adaptation were complete, the next section shows our methodology and experimental results.\LZK{Dernière phrase peut être supprimée}
-\section{Simulation of the multi-splitting method}
-Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
\section{Experimental results}
-\section{Conclusion}
+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
+study that the results depend on the following parameters:
+\begin{itemize}
+\item At the network level, we found that the most critical values are the
+ bandwidth (bw) and the network latency (lat).
+\item Hosts 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
+ \textit{external} precision are critical. They allow to ensure not only the
+ convergence of the algorithm but also to get the main objective of the
+ experimentation of the simulation in having an execution time in asynchronous
+ less than in synchronous mode (i.e. speed-up less than 1).
+\end{itemize}
+\LZK{Propositions pour changer le terme ``speedup'': acceleration ratio ou relative gain}
+
+A priori, obtaining a speedup less than 1 would be difficult in a local area
+network configuration where the synchronous mode will take advantage on the
+rapid exchange of information on such high-speed links. Thus, the methodology
+adopted was to launch the application on clustered network. In this last
+configuration, degrading the inter-cluster network performance will
+\textit{penalize} the synchronous mode allowing to get a speedup lower than 1.
+This action simulates the case of clusters linked with long distance network
+like Internet.
+
+As a first step, the algorithm was run on a network consisting of two clusters
+containing 50 hosts each, totaling 100 hosts. Various combinations of the above
+factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a
+matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from
+$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
+\text{\np{5211000}}$ entries.
+\LZK{Donner le type et la description du problème traité (problème symétrique Poisson 3D) et préciser peut être aussi qu'on a utilisé un partitionnement 3D}
+
+% 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
+ \renewcommand{\arraystretch}{1.3}%
+ \begin{tabular}{|>{\bfseries}r%
+ |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
+ \end{tabular}}
+
+\begin{table}[!t]
+ \centering
+ \caption{2 clusters, each with 50 nodes}
+ \label{tab.cluster.2x50}
+
+ \begin{mytable}{6}
+ \hline
+ bw
+ & 5 & 5 & 5 & 5 & 5 & 50 \\
+ \hline
+ lat
+ & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ \hline
+ power
+ & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 \\
+ \hline
+ size
+ & 62 & 62 & 62 & 100 & 100 & 110 \\
+ \hline
+ Prec/Eprec
+ & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\
+ \hline
+ speedup
+ & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\
+ \hline
+ \end{mytable}
+
+ \smallskip
+
+ \begin{mytable}{6}
+ \hline
+ bw
+ & 50 & 50 & 50 & 50 & 10 & 10 \\
+ \hline
+ lat
+ & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01 \\
+ \hline
+ power
+ & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 \\
+ \hline
+ size
+ & 120 & 130 & 140 & 150 & 171 & 171 \\
+ \hline
+ Prec/Eprec
+ & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5} & \np{E-5} \\
+ \hline
+ speedup
+ & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778 \\
+ \hline
+ \end{mytable}
+\end{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
+speedups less than 1 with a matrix size from 62 to 100 elements.
+
+\begin{table}[!t]
+ \centering
+ \caption{3 clusters, each with 33 nodes}
+ \label{tab.cluster.3x33}
+
+ \begin{mytable}{6}
+ \hline
+ bw
+ & 10 & 5 & 4 & 3 & 2 & 6 \\
+ \hline
+ lat
+ & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ \hline
+ power
+ & 1 & 1 & 1 & 1 & 1 & 1 \\
+ \hline
+ size
+ & 62 & 100 & 100 & 100 & 100 & 171 \\
+ \hline
+ Prec/Eprec
+ & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
+ \hline
+ speedup
+ & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99 \\
+ \hline
+ \end{mytable}
+\end{table}
+
+In a final step, results of an execution attempt to scale up the three clustered
+configuration but increasing by two hundreds hosts has been recorded in
+Table~\ref{tab.cluster.3x67}.
+
+\begin{table}[!t]
+ \centering
+ \caption{3 clusters, each with 66 nodes}
+ \label{tab.cluster.3x67}
+
+ \begin{mytable}{1}
+ \hline
+ bw & 1 \\
+ \hline
+ lat & 0.02 \\
+ \hline
+ power & 1 \\
+ \hline
+ size & 62 \\
+ \hline
+ Prec/Eprec & \np{E-5} \\
+ \hline
+ speedup & 0.9 \\
+ \hline
+ \end{mytable}
+\end{table}
+
+Note that the program was run with the following parameters:
+
+\paragraph*{SMPI parameters}
+
+\begin{itemize}
+ \item HOSTFILE: Hosts file description.
+ \item PLATFORM: file description of the platform architecture : clusters (CPU power,
+\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
+lat latency, \dots{}).
+\end{itemize}
+
+
+\paragraph*{Arguments of the program}
+
+\begin{itemize}
+ \item Description of the cluster architecture;
+ \item Maximum number of internal and external iterations;
+ \item Internal and external precisions;
+ \item Matrix size $N_x$, $N_y$ and $N_z$;
+ \item Matrix diagonal value: \np{6.0}, \LZK{Off-diagonal values? (-1?)}
+ \item Execution Mode: synchronous or asynchronous.
+\end{itemize}
+
+\paragraph*{Interpretations and comments}
+
+After analyzing the outputs, generally, for the configuration with two or three
+clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50}
+and~\ref{tab.cluster.3x33}), some combinations of the used parameters affecting
+the results have given a speedup less than 1, showing the effectiveness of the
+asynchronous performance compared to the synchronous mode.
+
+In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows
+that with a deterioration of inter cluster network set with \np[Mbit/s]{5} of
+bandwidth, a latency in order of a hundredth of a millisecond and a system power
+of one GFlops, an efficiency of about \np[\%]{40} in asynchronous mode is
+obtained for a matrix size of 62 elements. It is noticed that the result remains
+stable even if we vary the external precision from \np{E-5} to \np{E-9}. By
+increasing the problem size up to 100 elements, it was necessary to increase the
+CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a convergence of the algorithm
+with the same order of asynchronous mode efficiency. Maintaining such a system
+power but this time, increasing network throughput inter cluster up to
+\np[Mbit/s]{50}, the result of efficiency of about \np[\%]{40} is obtained with
+high external precision of \np{E-11} for a matrix size from 110 to 150 side
+elements.
+
+For the 3 clusters architecture including a total of 100 hosts,
+Table~\ref{tab.cluster.3x33} shows that it was difficult to have a combination
+which gives an efficiency of asynchronous below \np[\%]{80}. Indeed, for a
+matrix size of 62 elements, equality between the performance of the two modes
+(synchronous and asynchronous) is achieved with an inter cluster of
+\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency by
+\np[\%]{78} with a matrix size of 100 points, it was necessary to degrade the
+inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
+
+A last attempt was made for a configuration of three clusters but more powerful
+with 200 nodes in total. The convergence with a speedup of \np[\%]{90} was
+obtained with a bandwidth of \np[Mbit/s]{1} as shown in
+Table~\ref{tab.cluster.3x67}.
+
+\LZK{Dans le papier, on compare les deux versions synchrone et asycnhrone du multisplitting. Y a t il des résultats pour comparer gmres parallèle classique avec multisplitting asynchrone? Ca permettra de montrer l'intérêt du multisplitting asynchrone sur des clusters distants}
+\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:
+
+\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.
+\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
+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
+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.
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-
-The authors would like to thank...
-
-
-
-
+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{}}
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-% that's all folks
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