X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/1162f45db29e22394d4564bea71aef816419cba2..17320d7072dfa57c74d77096d2bc06929517f19c:/paper.tex

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--- a/paper.tex
+++ b/paper.tex
@@ -1,4 +1,17 @@
-\documentclass[conference]{IEEEtran}
+\documentclass[times]{cpeauth}
+
+\usepackage{moreverb}
+
+%\usepackage[dvips,colorlinks,bookmarksopen,bookmarksnumbered,citecolor=red,urlcolor=red]{hyperref}
+
+%\newcommand\BibTeX{{\rmfamily B\kern-.05em \textsc{i\kern-.025em b}\kern-.08em
+%T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
+
+\def\volumeyear{2015}
+
+\usepackage{graphicx}
+\usepackage{wrapfig}
+\usepackage{grffile}
 
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
@@ -23,6 +36,7 @@
 
 \usepackage{xspace}
 \usepackage[textsize=footnotesize]{todonotes}
+
 \newcommand{\AG}[2][inline]{%
   \todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
@@ -38,7 +52,10 @@
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
-\newcommand{\MI}{\mathit{MaxIter}}
+\newcommand{\TOLG}{\mathit{tol_{gmres}}}
+\newcommand{\MIG}{\mathit{maxit_{gmres}}}
+\newcommand{\TOLM}{\mathit{tol_{multi}}}
+\newcommand{\MIM}{\mathit{maxit_{multi}}}
 
 \usepackage{array}
 \usepackage{color, colortbl}
@@ -48,35 +65,37 @@
 \newcolumntype{g}{>{\columncolor{Gray}}c}
 \definecolor{Gray}{gray}{0.9}
 
+
+
 \begin{document}
 \RCE{Titre a confirmer.}
 \title{Comparative performance analysis of simulated grid-enabled numerical iterative algorithms}
+%\itshape{\journalnamelc}\footnotemark[2]}
 
-\author{%
-  \IEEEauthorblockN{%
-    Charles Emile Ramamonjisoa and
+\author{    Charles Emile Ramamonjisoa and
     David Laiymani and
     Arnaud Giersch and
     Lilia Ziane Khodja and
     Raphaël Couturier
-  }
-  \IEEEauthorblockA{%
+}
+
+\address{
+	\centering    
     Femto-ST Institute - DISC Department\\
     Université de Franche-Comté\\
     Belfort\\
     Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
-  }
 }
 
-\maketitle
+%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
 
 \begin{abstract}
 ABSTRACT
+\end{abstract}
 
+\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
 
-Keywords : Algorithm distributed iterative asynchronous simulation simgrid performance
-
-\end{abstract}
+\maketitle
 
 \section{Introduction}
 
@@ -84,17 +103,506 @@ Keywords : Algorithm distributed iterative asynchronous simulation simgrid perfo
 
 \section{SimGrid}
 
-\section{Simulation of the multisplitting method}
-
-\section{Experiments and results}
-
-\paragraph*{1. Study setup and methodology}
-
-\paragraph*{2. Factors impacting distributed applications performance in a grid environment}
-
-\paragraph*{3. Comparing GMRES and Multisplitting algorithms in synchronous mode}
-
-\paragraph*{4. Comparing GMRES in synchronous mode and Multisplitting algorithms in asynchronous mode}
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Two-stage splitting methods}
+\label{sec:04}
+\subsection{Multisplitting methods for sparse linear systems}
+\label{sec:04.01}
+Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$
+\begin{equation}
+Ax=b,
+\label{eq:01}
+\end{equation}
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
+\label{eq:02}
+\end{equation}
+where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$~\cite{O'leary85,White86}, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix. The iterations of the multisplitting methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system   
+\begin{equation}
+M_\ell y_\ell = c_\ell^k,\mbox{~such that~} c_\ell^k = N_\ell x^k + b,
+\label{eq:03}
+\end{equation}
+then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01})
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y_\ell.
+\label{eq:04}
+\end{equation}
+The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{O'leary85,bahi97,Bai99,bahi07}. %It is dependent on the condition  
+%\begin{equation}
+%\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1,
+%\label{eq:05}
+%\end{equation}
+%where $\rho$ is the spectral radius of the square matrix. 
+The multisplitting methods are convergent: 
+\begin{itemize}
+\item if $A^{-1}>0$ and the splittings of matrix $A$ are weak regular (i.e. $M^{-1}\geq 0$ and $M^{-1}N\geq 0$) when the iterations are synchronous, or
+\item if $A$ is M-matrix and its splittings are regular (i.e. $M^{-1}\geq 0$ and $N\geq 0$) when the iterations are asynchronous.
+\end{itemize}
+The solutions of the different linear sub-systems~(\ref{eq:03}) arising from the multisplitting of matrix $A$ can be either computed exactly with a direct method or approximated with an iterative method. In the latter case, the multisplitting methods are called {\it inner-outer iterative methods} or {\it two-stage multisplitting methods}. This kind of methods uses two nested iterations: the outer iteration and the inner iteration (that of the iterative method).
+
+In this paper we are focused on two-stage multisplitting methods, in their both versions synchronous and asynchronous, where the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} is used as an inner iteration. Furthermore, our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. weighting matrices $E_\ell$ have only zero and one factors). In this case, the iteration of the multisplitting method presented by (\ref{eq:03}) and~(\ref{eq:04}) can be rewritten in the following form 
+\begin{equation}
+A_{\ell\ell} x_\ell^{k+1} = b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m},\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
+\label{eq:05}
+\end{equation}
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. In each outer iteration $k$ until the convergence, each sub-system arising from the block Jacobi multisplitting
+\begin{equation}
+A_{\ell\ell} x_\ell = c_\ell,
+\label{eq:06}
+\end{equation}
+is solved iteratively using GMRES method and independently from other sub-systems by a cluster of processors. The right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. Algorithm~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:06}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold of GMRES respectively.  
+
+\begin{algorithm}[t]
+\caption{Block Jacobi two-stage multisplitting method}
+\begin{algorithmic}[1]
+  \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+  \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+  \State Set the initial guess $x^0$
+  \For {$k=1,2,3,\ldots$ until convergence}
+    \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+    \State $x^k_\ell=Solve(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\label{solve}
+    \State Send $x_\ell^k$ to neighboring clusters\label{send}
+    \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters\label{recv}
+  \EndFor
+\end{algorithmic}
+\label{alg:01}
+\end{algorithm}
+
+Multisplitting methods are more advantageous for large distributed computing platforms composed of hundreds or even thousands of processors interconnected by high latency networks. In this context, the parallel asynchronous model is preferred to the synchronous one to reduce overall execution times of the algorithms, even if it generally requires more iterations to converge. The asynchronous model allows the communications to be overlapped by computations which suppresses the idle times resulting from the synchronizations. So in asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Algorithm~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged
+\begin{equation}
+k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
+\label{eq:07}
+\end{equation}    
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold of the two-stage algorithm. The procedure of the convergence detection is implemented as follows. All clusters are interconnected by a virtual unidirectional ring network around which a Boolean token circulates from a cluster to another.  
+
+
+
+
+
+
+
+\subsection{Simulation of two-stage methods using SimGrid framework}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Experimental, Results and Comments}
+
+
+\textbf{V.1. Setup study and Methodology}
+
+To conduct our study, we have put in place the following methodology 
+which can be reused with any grid-enabled applications.
+
+\textbf{Step 1} : Choose with the end users the class of algorithms or 
+the application to be tested. Numerical parallel iterative algorithms 
+have been chosen for the study in the paper. 
+
+\textbf{Step 2} : Collect the software materials needed for the 
+experimentation. In our case, we have three variants algorithms for the 
+resolution of three 3D-Poisson problem: (1) using the classical GMRES alias Algo-1 in this 
+paper, (2) using the multisplitting method alias Algo-2 and (3) an 
+enhanced version of the multisplitting method as Algo-3. In addition, 
+SIMGRID simulator has been chosen to simulate the behaviors of the 
+distributed applications. SIMGRID is running on the Mesocentre 
+datacenter in Franche-Comte University $[$10$]$ but also in a virtual 
+machine on a laptop.
+
+\textbf{Step 3} : Fix the criteria which will be used for the future 
+results comparison and analysis. In the scope of this study, we retain 
+in one hand the algorithm execution mode (synchronous and asynchronous) 
+and in the other hand the execution time and the number of iterations of 
+the application before obtaining the convergence.
+
+\textbf{Step 4 }: Setup up the different grid testbeds environment 
+which will be simulated in the simulator tool to run the program. The 
+following architecture has been configured in Simgrid : 2x16 - that is a 
+grid containing 2 clusters with 16 hosts (processors/cores) each -, 4x8, 
+4x16, 8x8 and 2x50. The network has been designed to operate with a 
+bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8E-6 
+microseconds (resp. 5E-5) for the intra-clusters links (resp. 
+inter-clusters backbone links).
+
+\textbf{Step 5}: Process an extensive and comprehensive testings 
+within these configurations in varying the key parameters, especially 
+the CPU power capacity, the network parameters and also the size of the 
+input matrix. Note that some parameters should be invariant to allow the 
+comparison like some program input arguments.
+
+\textbf{Step 6} : Collect and analyze the output results.
+
+\textbf{ V.2. Factors impacting distributed applications performance in 
+a grid environment}
+
+From our previous experience on running distributed application in a 
+computational grid, many factors are identified to have an impact on the 
+program behavior and performance on this specific environment. Mainly, 
+first of all, the architecture of the grid itself can obviously 
+influence the performance results of the program. The performance gain 
+might be important theoretically when the number of clusters and/or the 
+number of nodes (processors/cores) in each individual cluster increase. 
+
+Another important factor impacting the overall performance of the 
+application is the network configuration. Two main network parameters 
+can modify drastically the program output results : (i) the network 
+bandwidth (bw=bits/s) also known as "the data-carrying capacity" 
+$[$13$]$ of the network is defined as the maximum of data that can pass 
+from one point to another in a unit of time. (ii) the network latency 
+(lat : microsecond) defined as the delay from the start time to send the 
+data from a source and the final time the destination have finished to 
+receive it. Upon the network characteristics, another impacting factor 
+is the application dependent volume of data exchanged between the nodes 
+in the cluster and between distant clusters. Large volume of data can be 
+transferred in transit between the clusters and nodes during the code 
+execution. 
+
+ In a grid environment, it is common to distinguish in one hand, the 
+"\,intra-network" which refers to the links between nodes within a 
+cluster and in the other hand, the "\,inter-network" which is the 
+backbone link between clusters. By design, these two networks perform 
+with different speed. The intra-network generally works like a high 
+speed local network with a high bandwith and very low latency. In 
+opposite, the inter-network connects clusters sometime via heterogeneous 
+networks components thru internet with a lower speed. The network 
+between distant clusters might be a bottleneck for the global 
+performance of the application. 
+
+\textbf{V.3 Comparing GMRES and Multisplitting algorithms in 
+synchronous mode}
+
+In the scope of this paper, our first objective is to demonstrate the 
+Algo-2 (Multisplitting method) shows a better performance in grid 
+architecture compared with Algo-1 (Classical GMRES) both running in 
+\textbf{\textit{synchronous mode}}. Better algorithm performance 
+should mean a less number of iterations output and a less execution time 
+before reaching the convergence. For a systematic study, the experiments 
+should figure out that, for various grid parameters values, the 
+simulator will confirm the targeted outcomes, particularly for poor and 
+slow networks, focusing on the impact on the communication performance 
+on the chosen class of algorithm $[$12$]$.
+
+The following paragraphs present the test conditions, the output results 
+and our comments.
+
+
+\textit{3.a Executing the algorithms on various computational grid 
+architecture scaling up the input matrix size}
+\\
+
+% environment
+\begin{footnotesize}
+\begin{tabular}{r c }
+ \hline  
+ Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
+ Network & N2 : bw=1Gbs-lat=5E-05 \\ %\hline
+ Input matrix size & N$_{x}$ =150 x 150 x 150 and\\ %\hline
+ - & N$_{x}$ =170 x 170 x 170    \\ \hline
+ \end{tabular}
+\end{footnotesize}
+
+
+ Table 1 : Clusters x Nodes with NX=150 or NX=170
+
+\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
+
+
+The results in figure 1 show the non-variation of the number of 
+iterations of classical GMRES for a given input matrix size; it is not 
+the case for the multisplitting method. 
+
+%\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
+\centering
+\includegraphics[width=60mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\caption{Cluster x Nodes NX=150 and NX=170} 
+%\label{overflow}}
+\end{figure}
+%\end{wrapfigure}
+
+Unless the 8x8 cluster, the time 
+execution difference between the two algorithms is important when 
+comparing between different grid architectures, even with the same number of 
+processors (like 2x16 and 4x8 = 32 processors for example). The 
+experiment concludes the low sensitivity of the multisplitting method 
+(compared with the classical GMRES) when scaling up to higher input 
+matrix size. 
+
+\textit{3.b Running on various computational grid architecture}
+
+% environment
+\begin{footnotesize}
+\begin{tabular}{r c }
+ \hline  
+ Grid & 2x16, 4x8\\ %\hline
+ Network & N1 : bw=10Gbs-lat=8E-06 \\ %\hline
+ - & N2 : bw=1Gbs-lat=5E-05 \\
+ Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline \\
+ \end{tabular}
+\end{footnotesize}
+
+%Table 2 : Clusters x Nodes - Networks N1 x N2
+%\RCE{idem pour tous les tableaux de donnees}
+
+
+%\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
+\centering
+\includegraphics[width=60mm]{cluster_x_nodes_n1_x_n2.pdf}
+\caption{Cluster x Nodes N1 x N2}
+%\label{overflow}}
+\end{figure}
+%\end{wrapfigure}
+
+The experiments compare the behavior of the algorithms running first on 
+speed inter- cluster network (N1) and a less performant network (N2). 
+The figure 2 shows that end users will gain to reduce the execution time 
+for both algorithms in using a grid architecture like 4x16 or 8x8: the 
+performance was increased in a factor of 2. The results depict also that 
+when the network speed drops down, the difference between the execution 
+times can reach more than 25\%. 
+
+\textit{\\\\\\\\\\\\\\\\\\3.c Network latency impacts on performance}
+
+% environment
+\begin{footnotesize}
+\begin{tabular}{r c }
+ \hline  
+ Grid & 2x16\\ %\hline
+ Network & N1 : bw=1Gbs \\ %\hline
+ Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline\\
+ \end{tabular}
+\end{footnotesize}
+
+Table 3 : Network latency impact
+
+
+\begin{figure} [ht!]
+\centering
+\includegraphics[width=60mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impact on execution time}
+%\label{overflow}}
+\end{figure}
+
+
+According the results in table and figure 3, degradation of the network 
+latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time 
+increase more than 75\% (resp. 82\%) of the execution for the classical 
+GMRES (resp. multisplitting) algorithm. In addition, it appears that the 
+multisplitting method tolerates more the network latency variation with 
+a less rate increase. Consequently, in the worst case (lat=6.10$^{-5
+}$), the execution time for GMRES is almost the double of the time for 
+the multisplitting, even though, the performance was on the same order 
+of magnitude with a latency of 8.10$^{-6}$. 
+
+\textit{3.d Network bandwidth impacts on performance}
+
+% environment
+\begin{footnotesize}
+\begin{tabular}{r c }
+ \hline  
+ Grid & 2x16\\ %\hline
+ Network & N1 : bw=1Gbs - lat=5E-05 \\ %\hline
+ Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline
+ \end{tabular}
+\end{footnotesize}
+
+Table 4 : Network bandwidth impact
+
+\begin{figure} [ht!]
+\centering
+\includegraphics[width=60mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impact on execution time}
+%\label{overflow}
+\end{figure}
+
+
+
+The results of increasing the network bandwidth depict the improvement 
+of the performance by reducing the execution time for both of the two 
+algorithms. However, and again in this case, the multisplitting method 
+presents a better performance in the considered bandwidth interval with 
+a gain of 40\% which is only around 24\% for classical GMRES.
+
+\textit{3.e Input matrix size impacts on performance}
+
+% environment
+\begin{footnotesize}
+\begin{tabular}{r c }
+ \hline  
+ Grid & 4x8\\ %\hline
+ Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
+ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
+ \end{tabular}
+\end{footnotesize}
+
+Table 5 : Input matrix size impact
+
+\begin{figure} [ht!]
+\centering
+\includegraphics[width=60mm]{pb_size_impact_on_execution_time.pdf}
+\caption{Pb size impact on execution time}
+%\label{overflow}}
+\end{figure}
+
+In this experimentation, the input matrix size has been set from 
+Nx=Ny=Nz=40 to 200 side elements that is from 40$^{3}$ = 64.000 to 
+200$^{3}$ = 8.000.000 points. Obviously, as shown in the figure 5, 
+the execution time for the algorithms convergence increases with the 
+input matrix size. But the interesting result here direct on (i) the 
+drastic increase (300 times) of the number of iterations needed before 
+the convergence for the classical GMRES algorithm when the matrix size 
+go beyond Nx=150; (ii) the classical GMRES execution time also almost 
+the double from Nx=140 compared with the convergence time of the 
+multisplitting method. These findings may help a lot end users to setup 
+the best and the optimal targeted environment for the application 
+deployment when focusing on the problem size scale up. Note that the 
+same test has been done with the grid 2x16 getting the same conclusion.
+
+\textit{3.f CPU Power impact on performance}
+
+% environment
+\begin{footnotesize}
+\begin{tabular}{r c }
+ \hline  
+ Grid & 2x16\\ %\hline
+ Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
+ Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
+ \end{tabular}
+\end{footnotesize}
+
+Table 6 : CPU Power impact
+
+\begin{figure} [ht!]
+\centering
+\includegraphics[width=60mm]{cpu_power_impact_on_execution_time.pdf}
+\caption{CPU Power impact on execution time}
+%\label{overflow}}
+\end{figure}
+
+Using the SIMGRID simulator flexibility, we have tried to determine the 
+impact on the algorithms performance in varying the CPU power of the 
+clusters nodes from 1 to 19 GFlops. The outputs depicted in the figure 6 
+confirm the performance gain, around 95\% for both of the two methods, 
+after adding more powerful CPU. Note that the execution time axis in the 
+figure is in logarithmic scale.
+
+ \textbf{V.4 Comparing GMRES in native synchronous mode and 
+Multisplitting algorithms in asynchronous mode}
+
+The previous paragraphs put in evidence the interests to simulate the 
+behavior of the application before any deployment in a real environment. 
+We have focused the study on analyzing the performance in varying the 
+key factors impacting the results. In the same line, the study compares 
+the performance of the two proposed methods in \textbf{synchronous mode
+}. In this section, with the same previous methodology, the goal is to 
+demonstrate the efficiency of the multisplitting method in \textbf{
+asynchronous mode} compare with the classical GMRES staying in the 
+synchronous mode.
+
+Note that the interest of using the asynchronous mode for data exchange 
+is mainly, in opposite of the synchronous mode, the non-wait aspects of 
+the current computation after a communication operation like sending 
+some data between nodes. Each processor can continue their local 
+calculation without waiting for the end of the communication. Thus, the 
+asynchronous may theoretically reduce the overall execution time and can 
+improve the algorithm performance.
+
+As stated supra, SIMGRID simulator tool has been used to prove the 
+efficiency of the multisplitting in asynchronous mode and to find the 
+best combination of the grid resources (CPU, Network, input matrix size, 
+\ldots ) to get the highest "\,relative gain" in comparison with the 
+classical GMRES time. 
+
+
+The test conditions are summarized in the table below : 
+
+% environment
+\begin{footnotesize}
+\begin{tabular}{r c }
+ \hline  
+ Grid & 2x50 totaling 100 processors\\ %\hline
+ Processors & 1 GFlops to 1.5 GFlops\\
+   Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
+   Inter-Network & bw=5 Mbits - lat=2E-02\\
+ Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
+ Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline
+ \end{tabular}
+\end{footnotesize}
+
+Again, comprehensive and extensive tests have been conducted varying the 
+CPU power and the network parameters (bandwidth and latency) in the 
+simulator tool with different problem size. The relative gains greater 
+than 1 between the two algorithms have been captured after each step of 
+the test. Table I below has recorded the best grid configurations 
+allowing a multiplitting method time more than 2.5 times lower than 
+classical GMRES execution and convergence time. The finding thru this 
+experimentation is the tolerance of the multisplitting method under a 
+low speed network that we encounter usually with distant clusters thru the 
+internet.
+
+% 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{Relative gain of the multisplitting algorithm compared with 
+the classical GMRES}
+  \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}
 
 \section{Conclusion}
 CONCLUSION
@@ -106,12 +614,8 @@ CONCLUSION
 The authors would like to thank\dots{}
 
 
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