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\begin{document}
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-\author{\IEEEauthorblockN{Raphaël Couturier and Arnaud Giersch and David Laiymani and Charles-Emile Ramamonjisoa}
+\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\\
\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 iterative executed
+in a distributed environment. As their name suggests, these algorithm
+solves a given problem that might be NP- complete complex 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. Generally, to reduce the complexity and the
+execution time, the problem is divided into several "pieces" that will
+be solved in parallel on multiple processing units. The latter will
+communicate each intermediate results before a new iteration starts
+until the approximate solution is reached. These distributed parallel
+computations can be performed either in "synchronous" communication mode
+where a new iteration begin only when all nodes communications are
+completed, either "asynchronous" mode where processors can continue
+independently without or few synchronization points. Despite the
+effectiveness of iterative approach, a major drawback of the method is
+the requirement of huge resources in terms of computing capacity,
+storage and high speed communication network. Indeed, limited physical
+resources are blocking factors for large-scale deployment of parallel
+algorithms.
+
+In recent years, 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. According our knowledge, no testing of large-scale
+simulation of the class of algorithm solving to achieve real results has
+been undertaken to date. We had in the scope of this work implemented a
+program for solving large non-symmetric linear system of equations by
+numerical method GMRES (Generalized Minimal Residual ) in the simulation
+environment Simgrid . The simulated platform 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 addition, it has been
+permitted to show the effectiveness of asynchronous mode algorithm by
+comparing its performance with the synchronous mode time. 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 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 will be presented with the
+settings to create various distributed architectures. The algorithm of
+the multi -splitting method used by GMRES written with MPI primitives
+and its adaptation to Simgrid with SMPI (Simulation MPI ) will be in the
+next section . At last, the experiments results carried out will be
+presented before the conclusion which we will announce the opening of
+our future work after the results.
\section{The asynchronous iteration model}
Décrire SimGrid (Arnaud)
-\section{Simulation of the multi-splitting method}
-Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
+
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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 $y$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi partitioning to solve this linear system on a large scale platform composed of $L$ clusters of processors. In this case, we apply a row-by-row splitting without overlapping
+\[
+\left(\begin{array}{ccc}
+A_{11} & \cdots & A_{1L} \\
+\vdots & \ddots & \vdots\\
+A_{L1} & \cdots & A_{LL}
+\end{array} \right)
+\times
+\left(\begin{array}{c}
+X_1 \\
+\vdots\\
+X_L
+\end{array} \right)
+=
+\left(\begin{array}{c}
+Y_1 \\
+\vdots\\
+Y_L
+\end{array} \right)\]
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,i\in\{1,\ldots,L\}$ $A_{li}$ is a rectangular block of $A$ of size $n_l\times n_i$, $X_l$ and $Y_l$ are sub-vectors of $x$ and $y$, respectively, each of size $n_l$ and $\sum_{l} n_l=\sum_{i} n_i=n$.
+
+The multisplitting method proceeds by iteration to solve in parallel the linear system by $L$ clusters of processors, in such a way each sub-system
+\begin{equation}
+\left\{
+\begin{array}{l}
+A_{ll}X_l = Y_l \mbox{,~such that}\\
+Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+\end{array}
+\right.
+\label{eq:4.1}
+\end{equation}
+is solved independently by a cluster and communication are required to update the right-hand side sub-vectors $Y_l$, such that the sub-vectors $X_i$ 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 GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers.
+
+\begin{algorithm}
+\caption{A multisplitting solver with inner iteration GMRES method}
+\begin{algorithmic}[1]
+\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
+\Output $X_l$ (local solution vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$, $x^0$
+\State Initialize the shared vector $\hat{x}=x^0$
+\For {$k=1,2,3,\ldots$ until the global convergence}
+\State $x^0=\hat{x}$
+\State Inner iteration solver: \Call{InnerSolver}{$x^0$, $k$}
+\State Exchange the local solution ${X}_l^k$ with the neighboring clusters and copy the shared vector elements in $\hat{x}$
+\EndFor
+
+\Statex
+
+\Function {InnerSolver}{$x^0$, $k$}
+\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
+\State Solving the local splitting $A_{ll}X_l^k=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the local initial guess
+\State \Return $X_l^k$
+\EndFunction
+\end{algorithmic}
+\label{algo:01}
+\end{algorithm}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+
+
+
\section{Experimental results}
\centering
\caption{2 clusters X 50 nodes}
\label{tab.cluster.2x50}
- \includegraphics[width=209pt]{img-1.eps}
+ \includegraphics[width=209pt]{img1.jpg}
\end{table}
\begin{table}
\centering
- \caption{3 clusters X 33 n\oe{}uds}
+ \caption{3 clusters X 33 nodes}
\label{tab.cluster.3x33}
- \includegraphics[width=209pt]{img-1.eps}
+ \includegraphics[width=209pt]{img2.jpg}
\end{table}
\begin{table}
\centering
- \caption{3 clusters X 67 noeuds}
+ \caption{3 clusters X 67 nodes}
\label{tab.cluster.3x67}
- \includegraphics[width=128pt]{img-2.eps}
+% \includegraphics[width=160pt]{img3.jpg}
+ \includegraphics[scale=0.5]{img3.jpg}
\end{table}
\paragraph*{Interpretations and comments}
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