corrections keywords

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 4f0fd297961db3ce5c77d8db68d3b5232aa6adc3..8f39683c3c12888c896932a733ab10d210630d9d 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -69,9 +69,8 @@
 
 
 
 
 
 

-\begin{document}
-\RCE{Titre a confirmer.}
-\title{Comparative performance analysis of simulated grid-enabled numerical iterative algorithms}
+\begin{document} \RCE{Titre a confirmer.} \title{Comparative performance
+analysis of simulated grid-enabled numerical iterative algorithms}

 %\itshape{\journalnamelc}\footnotemark[2]}
 
 \author{    Charles Emile Ramamonjisoa and
 %\itshape{\journalnamelc}\footnotemark[2]}
 
 \author{    Charles Emile Ramamonjisoa and


@@ -82,7 +81,7 @@
 }
 
 \address{
 }
 
 \address{

-       \centering    
+       \centering

     Femto-ST Institute - DISC Department\\
     Université de Franche-Comté\\
     Belfort\\
     Femto-ST Institute - DISC Department\\
     Université de Franche-Comté\\
     Belfort\\


@@ -91,70 +90,104 @@
 
 %% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
 
 
 %% 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}
-  The behavior of multicore applications is always a challenge to predict, especially with a new architecture for which no experiment has been performed. With some applications, it is difficult, if not impossible, to build accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications. We have decided to use SimGrid as it enables to benchmark MPI applications.
+\begin{abstract}   The behavior of multicore applications is always a challenge
+to predict, especially with a new architecture for which no experiment has been
+performed. With some applications, it is difficult, if not impossible, to build
+accurate performance models. That is why another solution is to use a simulation
+tool which allows us to change many parameters of the architecture (network
+bandwidth, latency, number of processors) and to simulate the execution of such
+applications. We have decided to use SimGrid as it enables to benchmark MPI
+applications.

 
 In this paper, we focus our attention on two parallel iterative algorithms based
 on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.
 
 In this paper, we focus our attention on two parallel iterative algorithms based
 on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.

-These algorithms  are used to  solve libear  systems. Two different  variantsof the Multisplitting are
-studied: one  using synchronoous  iterations and  another one  with asynchronous
-iterations. For each algorithm we have  tested different parameters to see their
-influence.  We strongly  recommend people  interested  by investing  into a  new
-expensive  hardware  architecture  to   benchmark  their  applications  using  a
-simulation tool before.
+These algorithms  are used to  solve libear  systems. Two different  variantsof
+the Multisplitting are studied: one  using synchronoous  iterations and  another
+one  with asynchronous iterations. For each algorithm we have  tested different
+parameters to see their influence.  We strongly  recommend people  interested
+by investing  into a  new expensive  hardware  architecture  to   benchmark
+their  applications  using  a simulation tool before.
+
+

 
 

-  

 
 

-  

 \end{abstract}
 
 \end{abstract}
 

-\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
+%\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
+%performance} 
+\keywords{Multisplitting algorithms, Synchronous and asynchronous iterations, SimGrid, Simulation, Performance evaluation}

 
 \maketitle
 
 
 \maketitle
 

-\section{Introduction} 
-The use of multi-core architectures for solving large scientific problems seems to  become imperative  in  a  lot  of  cases. 
-Whatever the scale of these architectures (distributed clusters, computational grids, embedded multi-core \ldots) they  are generally 
-well adapted to execute complexe parallel applications operating on a large amount of data.  Unfortunately,  users (industrials or scientists), 
-who need such computational resources, may not have an easy access to such efficient architectures. The cost of using the platform and/or the cost of 
-testing and deploying an application are often very important.  So, in this context it is difficult to optimize a given application for a given 
-architecture. In this way and in order to reduce the access cost to these computing resources it seems very interesting to use a simulation environment. 
-The advantages are numerous: development life cycle, code debugging, ability to obtain results quickly \ldots at the condition that the simulation results are in education with the real ones.
-
-In this paper we focus on a class of highly efficient parallel algorithms called \emph{iterative algorithms}. The
-parallel scheme of iterative methods is quite simple. It generally involves the division of the problem
-into  several  \emph{blocks}  that  will  be  solved  in  parallel  on  multiple
-processing units.  Each processing unit has to
-compute an iteration, to send/receive some data dependencies to/from
-its neighbors and to iterate this process until the convergence of
-the method. Several well-known methods demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.
-In this processing mode a task cannot begin a new iteration while it
-has not received data dependencies from its neighbors. We say that the iteration computation follows a synchronous scheme.
-In the asynchronous scheme a task can compute a new iteration without having to
-wait for the data dependencies coming from its neighbors. Both
-communication and computations are asynchronous inducing that there is
-no more idle times, due to synchronizations, between two
-iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks that we detail in section 2 but even if the number of iterations required to converge is 
-generally  greater  than for the synchronous  case, it appears that the asynchronous  iterative scheme  can significantly  reduce  overall execution
-times by  suppressing idle  times due to  synchronizations~(see \cite{Bahi07} for more details).
-
-Nevertheless, in both cases (synchronous or asynchronous) it is very time consuming to find optimal configuration and deployment requirements 
-for a given application on a given multi-core architecture. Finding good resource allocations policies under varying CPU power, network speeds and 
-loads is very challenging and labor intensive.~\cite{Calheiros:2011:CTM:1951445.1951450}.   This problematic is even more difficult for the asynchronous scheme 
-where variations of the parameters of the execution platform can lead to very different number of iterations required to converge and so to very different execution times.
-In this challenging context we think that the use of a simulation tool can greatly leverage the possibility of testing various platform scenarios.
-
-The main contribution of this paper is to show that the use of a simulation tool (i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real 
-parallel applications (i.e. large linear system solver) can help developers to better tune their application for a given multi-core architecture.
-To show the validity of this approach we first compare the simulated execution of the multisplitting algorithm  with  the  GMRES   (Generalized   Minimal  Residual) solver
-\cite{ref1} in synchronous mode. The obtained results on different simulated multi-core architectures confirm the real results previously obtained on non simulated architectures. 
-We also confirm  the efficiency  of the  asynchronous  multisplitting algorithm  comparing to the synchronous  GMRES. In this way and with a simple computing architecture (a laptop) 
-SimGrid allows us to run a test campaign  of  a  real parallel iterative  applications on  different simulated multi-core architectures. 
-To our knowledge, there is no related work on the large-scale multi-core simulation of a real synchronous and asynchronous iterative application.  
-
-This paper is organized as follows. Section 1 \ref{sec:synchro} presents the iteration model we use and more particularly the asynchronous scheme. 
-In section \ref{sec:simgrid} the SimGrid simulation toolkit is presented. Section \ref{sec:04} details the different solvers that we use. 
-Finally our experimental results are presented in section \ref{\sec:expe} followed by some concluding remarks and perspectives.
+\section{Introduction}  The use of multi-core architectures for solving large
+scientific problems seems to  become imperative  in  a  lot  of  cases.
+Whatever the scale of these architectures (distributed clusters, computational
+grids, embedded multi-core,~\ldots) they  are generally  well adapted to execute
+complex parallel applications operating on a large amount of data.
+Unfortunately,  users (industrials or scientists),  who need such computational
+resources, may not have an easy access to such efficient architectures. The cost
+of using the platform and/or the cost of  testing and deploying an application
+are often very important. So, in this context it is difficult to optimize a
+given application for a given  architecture. In this way and in order to reduce
+the access cost to these computing resources it seems very interesting to use a
+simulation environment.  The advantages are numerous: development life cycle,
+code debugging, ability to obtain results quickly,~\ldots at the condition that
+the simulation results are in education with the real ones.
+
+In this paper we focus on a class of highly efficient parallel algorithms called
+\emph{iterative algorithms}. The parallel scheme of iterative methods is quite
+simple. It generally involves the division of the problem into  several
+\emph{blocks}  that  will  be  solved  in  parallel  on  multiple processing
+units.  Each processing unit has to compute an iteration, to send/receive some
+data dependencies to/from its neighbors and to iterate this process until the
+convergence of the method. Several well-known methods demonstrate the
+convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
+task cannot begin a new iteration while it has not received data dependencies
+from its neighbors. We say that the iteration computation follows a synchronous
+scheme. In the asynchronous scheme a task can compute a new iteration without
+having to wait for the data dependencies coming from its neighbors. Both
+communication and computations are asynchronous inducing that there is no more
+idle times, due to synchronizations, between two iterations~\cite{bcvc06:ij}.
+This model presents some advantages and drawbacks that we detail in
+section~\ref{sec:asynchro} but even if the number of iterations required to
+converge is generally  greater  than for the synchronous  case, it appears that
+the asynchronous  iterative scheme  can significantly  reduce  overall execution
+times by  suppressing idle  times due to  synchronizations~(see~\cite{bahi07}
+for more details).
+
+Nevertheless, in both cases (synchronous or asynchronous) it is very time
+consuming to find optimal configuration and deployment requirements  for a given
+application on a given multi-core architecture. Finding good resource
+allocations policies under varying CPU power, network speeds and  loads is very
+challenging and labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
+problematic is even more difficult for the asynchronous scheme  where variations
+of the parameters of the execution platform can lead to very different number of
+iterations required to converge and so to very different execution times. In
+this challenging context we think that the use of a simulation tool can greatly
+leverage the possibility of testing various platform scenarios.
+
+The main contribution of this paper is to show that the use of a simulation tool
+(i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real  parallel
+applications (i.e. large linear system solvers) can help developers to better
+tune their application for a given multi-core architecture. To show the validity
+of this approach we first compare the simulated execution of the multisplitting
+algorithm  with  the  GMRES   (Generalized   Minimal  Residual)
+solver~\cite{saad86} in synchronous mode. The obtained results on different
+simulated multi-core architectures confirm the real results previously obtained
+on non simulated architectures.  We also confirm  the efficiency  of the
+asynchronous  multisplitting algorithm  comparing to the synchronous  GMRES. In
+this way and with a simple computing architecture (a laptop) SimGrid allows us
+to run a test campaign  of  a  real parallel iterative  applications on
+different simulated multi-core architectures.  To our knowledge, there is no
+related work on the large-scale multi-core simulation of a real synchronous and
+asynchronous iterative application.
+
+This paper is organized as follows. Section~\ref{sec:asynchro} presents the
+iteration model we use and more particularly the asynchronous scheme.  In
+section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented.
+Section~\ref{sec:04} details the different solvers that we use.  Finally our
+experimental results are presented in section~\ref{sec:expe} followed by some
+concluding remarks and perspectives.

 
 
 \section{The asynchronous iteration model}
 
 
 \section{The asynchronous iteration model}


@@ -162,7 +195,7 @@ Finally our experimental results are presented in section \ref{\sec:expe} follow
 
 \section{SimGrid}
  \label{sec:simgrid}
 
 \section{SimGrid}
  \label{sec:simgrid}

- 
+

 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 


@@ -180,12 +213,12 @@ where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side
 x_\ell^{k+1} = A_{\ell\ell}^{-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:02}
 \end{equation}
 x_\ell^{k+1} = A_{\ell\ell}^{-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:02}
 \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. The iterations of these 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 
+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. The iterations of these 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}
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{equation}
 \begin{equation}
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{equation}

-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\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:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{Bru95,bahi07}. 
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\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:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{Bru95,bahi07}.

 
 \begin{figure}[t]
 %\begin{algorithm}[t]
 
 \begin{figure}[t]
 %\begin{algorithm}[t]


@@ -203,28 +236,28 @@ where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are compute
 \end{algorithmic}
 \caption{Block Jacobi two-stage multisplitting method}
 \label{alg:01}
 \end{algorithmic}
 \caption{Block Jacobi two-stage multisplitting method}
 \label{alg:01}

-%\end{algorithm} 
+%\end{algorithm}

 \end{figure}
 
 In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on asynchronous model which allows the communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the 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 Figure~\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:04}
 \end{figure}
 
 In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on asynchronous model which allows the communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the 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 Figure~\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:04}

-\end{equation}    
-where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm. 
+\end{equation}
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.

 
 

-The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration 
+The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration

 \begin{equation}
 S=[x^1,x^2,\ldots,x^s],~s\ll n.
 \label{eq:05}
 \begin{equation}
 S=[x^1,x^2,\ldots,x^s],~s\ll n.
 \label{eq:05}

-\end{equation}   
+\end{equation}

 At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
 \begin{equation}
 \min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
 \label{eq:06}
 At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
 \begin{equation}
 \min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
 \label{eq:06}

-\end{equation} 
+\end{equation}

 The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
 The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).

- 
+

 \begin{figure}[t]
 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{figure}[t]
 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}


@@ -240,7 +273,7 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
        \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
        \State $\tilde{x_\ell}=S_\ell\alpha$
        \State Send $\tilde{x_\ell}$ to neighboring clusters
        \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
        \State $\tilde{x_\ell}=S_\ell\alpha$
        \State Send $\tilde{x_\ell}$ to neighboring clusters

-       \Else 
+       \Else

          \State Send $x_\ell^k$ to neighboring clusters
     \EndIf
     \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
          \State Send $x_\ell^k$ to neighboring clusters
     \EndIf
     \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters


@@ -248,13 +281,13 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \end{algorithmic}
 \caption{Krylov two-stage method using block Jacobi multisplitting}
 \label{alg:02}
 \end{algorithmic}
 \caption{Krylov two-stage method using block Jacobi multisplitting}
 \label{alg:02}

-%\end{algorithm} 
+%\end{algorithm}

 \end{figure}
 
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \label{sec:04.02}
 
 \end{figure}
 
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \label{sec:04.02}
 

-One of our objectives when simulating the application in SIMGRID is, as in real life, to get accurate results (solutions of the problem) but also ensure the test reproducibility under the same conditions. According our experience, very few modifications are required to adapt a MPI program to run in SIMGRID simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine or using a Simgrid selector called "runtime automatic switching" (smpi/privatize\_global\_variables). Indeed, global variables can generate side effects on runtime between the threads running in the same process, generated by the Simgrid to simulate the grid environment.The last modification on the MPI program pointed out for some cases, the review of the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which might cause an infinite loop.     
+One of our objectives when simulating the application in SIMGRID is, as in real life, to get accurate results (solutions of the problem) but also ensure the test reproducibility under the same conditions. According our experience, very few modifications are required to adapt a MPI program to run in SIMGRID simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine or using a Simgrid selector called "runtime automatic switching" (smpi/privatize\_global\_variables). Indeed, global variables can generate side effects on runtime between the threads running in the same process, generated by the Simgrid to simulate the grid environment.The last modification on the MPI program pointed out for some cases, the review of the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which might cause an infinite loop.

 
 
 \paragraph{SIMGRID Simulator parameters}
 
 
 \paragraph{SIMGRID Simulator parameters}


@@ -264,8 +297,8 @@ One of our objectives when simulating the application in SIMGRID is, as in real
        \item plarform: File describing the platform architecture : clusters (CPU power,
 \dots{}), intra cluster network description, inter cluster network (bandwidth bw,
 latency lat, \dots{}).
        \item plarform: File describing the platform architecture : clusters (CPU power,
 \dots{}), intra cluster network description, inter cluster network (bandwidth bw,
 latency lat, \dots{}).

-       \item archi   : Grid computational description (Number of clusters, Number of 
-nodes/processors for each cluster). 
+       \item archi   : Grid computational description (Number of clusters, Number of
+nodes/processors for each cluster).

 \end{itemize}
 
 
 \end{itemize}
 
 


@@ -279,7 +312,7 @@ In addition, the following arguments are given to the programs at runtime:
        \item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 
        \item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 

-At last, note that the two solver algorithms have been executed with the Simgrid selector -cfg=smpi/running\_power which determine the computational power (here 19GFlops) of the simulator host machine.  
+At last, note that the two solver algorithms have been executed with the Simgrid selector -cfg=smpi/running\_power which determine the computational power (here 19GFlops) of the simulator host machine.

 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%


@@ -290,103 +323,103 @@ At last, note that the two solver algorithms have been executed with the Simgrid
 
 \subsection{Setup study and Methodology}
 
 
 \subsection{Setup study and Methodology}
 

-To conduct our study, we have put in place the following methodology 
+To conduct our study, we have put in place the following methodology

 which can be reused for any grid-enabled applications.
 
 which can be reused for 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 
+\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 this paper. \\
 
 have been chosen for the study in this paper. \\
 

-\textbf{Step 2} : Collect the software materials needed for the 
-experimentation. In our case, we have two variants algorithms for the 
-resolution of three 3D-Poisson problem: (1) using the classical GMRES (Algo-1)(2) and the multisplitting method (Algo-2). In addition, SIMGRID simulator has been chosen to simulate the behaviors of the 
+\textbf{Step 2} : Collect the software materials needed for the
+experimentation. In our case, we have two variants algorithms for the
+resolution of three 3D-Poisson problem: (1) using the classical GMRES (Algo-1)(2) and the multisplitting method (Algo-2). 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 but also in a virtual machine on a laptop. \\
 
 distributed applications. SIMGRID is running on the Mesocentre datacenter in Franche-Comte University 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 
+\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. \\
 
 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. 
+\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). \\
 
 inter-clusters backbone links). \\
 

-\textbf{Step 5}: Conduct 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 fixed to be invariant to allow the 
+\textbf{Step 5}: Conduct 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 fixed to be invariant to allow the

 comparison like some program input arguments. \\
 
 \textbf{Step 6} : Collect and analyze the output results.
 
 comparison like some program input arguments. \\
 
 \textbf{Step 6} : Collect and analyze the output results.
 

-\subsection{Factors impacting distributed applications performance in 
+\subsection{Factors impacting distributed applications performance in

 a grid environment}
 
 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" 
-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. 
-
-\subsection{Comparing GMRES and Multisplitting algorithms in 
+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"
+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.
+
+\subsection{Comparing GMRES and Multisplitting algorithms in

 synchronous mode}
 
 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 means 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 
+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 means 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.
 
 on the chosen class of algorithm.
 

-The following paragraphs present the test conditions, the output results 
+The following paragraphs present the test conditions, the output results

 and our comments.\\
 
 
 and our comments.\\
 
 

-\textit{3.a Executing the algorithms on various computational grid 
+\textit{3.a Executing the algorithms on various computational grid

 architecture scaling up the input matrix size}
 \\
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
 architecture scaling up the input matrix size}
 \\
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
  Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
  Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline


@@ -397,37 +430,37 @@ Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
 \end{footnotesize}
 
 
 \end{footnotesize}
 
 

- 
+

 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 
 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 

-The results in figure 3 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. 
+The results in figure 3 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}{100mm}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
 
 %\begin{wrapfigure}{l}{100mm}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}

-\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170} 
+\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}

 %\label{overflow}}
 \end{figure}
 %\end{wrapfigure}
 
 %\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. 
+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 }
 
 \textit{\\3.b Running on various computational grid architecture\\}
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16, 4x8\\ %\hline
  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
  Grid & 2x16, 4x8\\ %\hline
  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\


@@ -448,20 +481,20 @@ Table 2 : Clusters x Nodes - Networks N1 x N2 \\
 \end{figure}
 %\end{wrapfigure}
 
 \end{figure}
 %\end{wrapfigure}
 

-The experiments compare the behavior of the algorithms running first on 
-a speed inter- cluster network (N1) and a less performant network (N2). 
-Figure 4 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\%. 
+The experiments compare the behavior of the algorithms running first on
+a speed inter- cluster network (N1) and a less performant network (N2).
+Figure 4 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 }
 
 \textit{\\3.c Network latency impacts on performance\\}
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline\\
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline\\


@@ -480,22 +513,22 @@ Table 3 : Network latency impact \\
 \end{figure}
 
 
 \end{figure}
 
 

-According the results in figure 5, 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 
+According the results in figure 5, 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 of the execution time. Consequently, in the worst case (lat=6.10$^{-5
 a less rate increase of the execution time. Consequently, in the worst case (lat=6.10$^{-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}$. 
+}$), 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 }
 
 \textit{\\3.d Network bandwidth impacts on performance\\}
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\


@@ -514,10 +547,10 @@ Table 4 : Network bandwidth impact \\
 
 
 
 
 
 

-The results of increasing the network bandwidth depict the improvement 
-of the performance by reducing the execution time for both of the two 
-algorithms (Figure 6). However, and again in this case, the multisplitting method 
-presents a better performance in the considered bandwidth interval with 
+The results of increasing the network bandwidth depict the improvement
+of the performance by reducing the execution time for both of the two
+algorithms (Figure 6). 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\\}
 a gain of 40\% which is only around 24\% for classical GMRES.
 
 \textit{\\3.e Input matrix size impacts on performance\\}


@@ -525,7 +558,7 @@ a gain of 40\% which is only around 24\% for classical GMRES.
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 4x8\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\
  Grid & 4x8\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\


@@ -542,18 +575,18 @@ Table 5 : Input matrix size impact\\
 %\label{overflow}}
 \end{figure}
 
 %\label{overflow}}
 \end{figure}
 

-In this experimentation, the input matrix size has been set from 
-N$_{x}$ = N$_{y}$ = N$_{z}$ = 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 7, 
-the execution time for the two algorithms convergence increases with the 
-input matrix size. But the interesting results 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 N$_{x}$=150; (ii) the classical GMRES execution time also almost 
-the double from N$_{x}$=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 
+In this experimentation, the input matrix size has been set from
+N$_{x}$ = N$_{y}$ = N$_{z}$ = 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 7,
+the execution time for the two algorithms convergence increases with the
+input matrix size. But the interesting results 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 N$_{x}$=150; (ii) the classical GMRES execution time also almost
+the double from N$_{x}$=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\\}
 same test has been done with the grid 2x16 getting the same conclusion.
 
 \textit{\\3.f CPU Power impact on performance\\}


@@ -561,7 +594,7 @@ same test has been done with the grid 2x16 getting the same conclusion.
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  Grid & 2x16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline


@@ -578,39 +611,39 @@ Table 6 : CPU Power impact \\
 %\label{overflow}}
 \end{figure}
 
 %\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 
+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.
 
 figure is in logarithmic scale.
 

-\subsection{Comparing GMRES in native synchronous mode and 
+\subsection{Comparing GMRES in native synchronous mode and

 Multisplitting algorithms in asynchronous mode}
 
 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 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
 the performance of the two proposed methods in \textbf{synchronous mode

-}. In this section, with the same previous methodology, the goal is to 
+}. In this section, with the same previous methodology, the goal is to

 demonstrate the efficiency of the multisplitting method in \textbf{
 demonstrate the efficiency of the multisplitting method in \textbf{

-asynchronous mode} compare with the classical GMRES staying in the 
+asynchronous mode} compare with the classical GMRES staying in the

 synchronous mode.
 
 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 
+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.
 
 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. 
+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 : \\
 
 
 The test conditions are summarized in the table below : \\


@@ -618,7 +651,7 @@ The test conditions are summarized in the table below : \\
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x50 totaling 100 processors\\ %\hline
  Processors & 1 GFlops to 1.5 GFlops\\
    Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
  Grid & 2x50 totaling 100 processors\\ %\hline
  Processors & 1 GFlops to 1.5 GFlops\\
    Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline


@@ -628,15 +661,15 @@ The test conditions are summarized in the table below : \\
  \end{tabular}
 \end{footnotesize}
 
  \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 
+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
 internet.
 
 % use the same column width for the following three tables


@@ -649,7 +682,7 @@ internet.
 
 \begin{table}[!t]
   \centering
 
 \begin{table}[!t]
   \centering

-  \caption{Relative gain of the multisplitting algorithm compared with 
+  \caption{Relative gain of the multisplitting algorithm compared with

 the classical GMRES}
   \label{"Table 7"}
 
 the classical GMRES}
   \label{"Table 7"}