DL: corrections sur section 4

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 397decc12af88aba0e11d0f4c954cde9f731e27d..93f215d5bab1aad50151e30a5ad25dab9d926025 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -114,7 +114,7 @@ their  applications  using  a simulation tool before.
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;

-%performance} 
+%performance}

 \keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}
 
 \maketitle
 \keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}
 
 \maketitle


@@ -131,28 +131,28 @@ 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,
 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. In counterpart, the simulation results need to be consistent with the real ones.
+code debugging, ability to obtain results quickly~\ldots. In counterpart, the simulation results need to be consistent 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
 
 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
+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
 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 the method. Several well-known studies 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
 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 time, 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).
+from its neighbors. We say that the iteration computation follows a
+\textit{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 \textit{asynchronous}
+inducing that there is no more idle time, 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
 
 Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is very  time
 consuming to find optimal configuration  and deployment requirements for a given


@@ -223,22 +223,22 @@ consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}

-In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$
+In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:

 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}
 \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. 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. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). Two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. 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. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). Two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows:

 \begin{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}
 \begin{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, has been studied by many authors for example~\cite{Bru95,bahi07}.

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


@@ -259,19 +259,19 @@ where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are compute
 %\end{algorithm}
 \end{figure}
 
 %\end{algorithm}
 \end{figure}
 

-In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the 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
+In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the asynchronous model which allows 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}
 \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. 
+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}
 \end{equation}
 \begin{equation}
 S=[x^1,x^2,\ldots,x^s],~s\ll n.
 \label{eq:05}
 \end{equation}

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


@@ -304,11 +304,11 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 %\end{algorithm}
 \end{figure}
 
 %\end{algorithm}
 \end{figure}
 

-\subsection{Simulation of two-stage methods using SimGrid framework}
+\subsection{Simulation of the two-stage methods using SimGrid toolkit}

 \label{sec:04.02}
 
 One of our objectives when simulating the  application in Simgrid is, as in real
 \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
+life, to  get accurate results  (solutions of the  problem) but also to ensure the

 test reproducibility  under the same  conditions.  According to  our experience,
 very  few modifications  are required  to adapt  a MPI  program for  the Simgrid
 simulator using SMPI (Simulator MPI). The  first modification is to include SMPI
 test reproducibility  under the same  conditions.  According to  our experience,
 very  few modifications  are required  to adapt  a MPI  program for  the Simgrid
 simulator using SMPI (Simulator MPI). The  first modification is to include SMPI


@@ -316,11 +316,13 @@ libraries  and related  header files  (smpi.h).  The  second modification  is to
 suppress all global variables by replacing  them with local variables or using a
 Simgrid      selector       called      "runtime       automatic      switching"
 (smpi/privatize\_global\_variables). Indeed, global  variables can generate side
 suppress all global variables by replacing  them with local variables 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
-Simgrid  to simulate the  grid environment.  \RC{On vire cette  phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}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.
+effects on runtime between the threads running in the same process and generated by
+Simgrid  to simulate the  grid environment.
+
+%\RC{On vire cette  phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}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}


@@ -343,19 +345,19 @@ In addition, the following arguments are given to the programs at runtime:
 \begin{itemize}
        \item maximum number of inner and outer iterations;
        \item inner and outer precisions;
 \begin{itemize}
        \item maximum number of inner and outer iterations;
        \item inner and outer precisions;

-       \item maximum number of the gmres's restarts in the Arnorldi process;
-       \item maximum number of iterations qnd the tolerance threshold in classical GMRES;
+       \item maximum number of the GMRES restarts in the Arnorldi process;
+       \item maximum number of iterations and the tolerance threshold in classical GMRES;

        \item tolerance threshold for outer and inner-iterations;
        \item tolerance threshold for outer and inner-iterations;

-       \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on x, y, z axis;
-       \item matrix diagonal value = 6.0 for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments; \RC{CE tu vérifies, je dis ca de tête}
+       \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on $x, y, z$ axis;
+       \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments; \RC{CE tu vérifies, je dis ca de tête}

        \item matrix off-diagonal value;
        \item execution mode: synchronous or asynchronous;
        \item matrix off-diagonal value;
        \item execution mode: synchronous or asynchronous;

-       \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les  arguments pour CGLS ci dessous}
+       \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les  arguments pour CGLS ci dessous} \RC{Vu que tu n'as pas fait varier ce paramètre, on peut ne pas en parler}

        \item Size of matrix S;
        \item Size of matrix S;

-       \item Maximum number of iterations and tolerance threshold for CGLS. 
+       \item Maximum number of iterations and tolerance threshold for CGLS.

 \end{itemize}
 
 \end{itemize}
 

-It should also be noticed that both solvers have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.
+It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.

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


@@ -363,7 +365,10 @@ It should also be noticed that both solvers have been executed with the Simgrid
 \section{Experimental Results}
 \label{sec:expe}
 
 \section{Experimental Results}
 \label{sec:expe}
 

-In this section, experiments for both Multisplitting algorithms are reported. First the problem used in our experiments is described.
+In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
+
+\subsection{3D Poisson}
+

 
 We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
 \begin{equation}
 
 We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
 \begin{equation}


@@ -374,19 +379,16 @@ such that
 \begin{equation*}
 \phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
 \end{equation*}
 \begin{equation*}
 \phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
 \end{equation*}

-where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that      
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that

 \begin{equation}
 \begin{array}{ll}
 \begin{equation}
 \begin{array}{ll}

-\phi^\star(x,y,z)= & \frac{1}{6}(\phi(x-h,y,z)+\phi(x+h,y,z) \\
-                  & +\phi(x,y-h,z)+\phi(x,y+h,z) \\
-                  & +\phi(x,y,z-h)+\phi(x,y,z+h)\\
-                  & -h^2f(x,y,z))
+\phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))

 \end{array}
 \label{eq:08}
 \end{equation}
 \end{array}
 \label{eq:08}
 \end{equation}

-until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid. 
+until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.

 
 

-In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic sub-problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries. 
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic block of the problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries.

 
 \subsection{Study setup and Simulation Methodology}
 
 
 \subsection{Study setup and Simulation Methodology}
 


@@ -459,30 +461,28 @@ transit between the clusters and nodes during the code execution.
  speed.  The network  between distant  clusters might  be a  bottleneck for  the
  global performance of the application.
 
  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}
+\subsection{Comparison of GMRES and Krylov 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
-\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.
+In the scope  of this paper, our  first objective is to analyze  when the Krylov
+Multisplitting  method   has  better  performances  than   the  classical  GMRES
+method. With an  iterative method, better performances mean a  smaller number of
+iterations and 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.

 
 The following paragraphs present the test conditions, the output results
 and our comments.\\
 
 
 
 The following paragraphs present the test conditions, the output results
 and our comments.\\
 
 

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

 architecture and scaling up the input matrix size}
 architecture and scaling up the input matrix size}

-\\
-
+\ \\

 % environment
 % environment

-\begin{footnotesize}
+
+\begin{figure} [ht!]
+\begin{center}

 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline


@@ -490,26 +490,34 @@ architecture and scaling up the input matrix size}
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
  - &  N$_{x}$ x N$_{y}$ x N$_{z}$  =170 x 170 x 170    \\ \hline
  \end{tabular}
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
  - &  N$_{x}$ x N$_{y}$ x N$_{z}$  =170 x 170 x 170    \\ \hline
  \end{tabular}

-Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
+\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\end{center}
+\end{figure}

 
 

-\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}
 
 

-In this section, we compare the algorithms performance running on various grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in figure 3 show for all grid configuration 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.
+In this  section, we analyze the  performences of algorithms running  on various
+grid configuration  (2x16, 4x8, 4x16  and 8x8). First,  the results in  Figure~\ref{fig:01}
+show for all grid configuration 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.
+
+\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
+\RC{Les légendes ne sont pas explicites...}
+

 
 

-%\begin{wrapfigure}{l}{100mm}

 \begin{figure} [ht!]
 \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}
-%\label{overflow}}
+  \begin{center}
+    \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+  \end{center}
+  \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
+  \label{fig:01}

 \end{figure}
 \end{figure}

-%\end{wrapfigure}
+

 
 The execution time difference between the two algorithms is important when
 comparing between different grid architectures, even with the same number of
 
 The execution time difference between the two algorithms is important when
 comparing between different grid architectures, even with the same number of


@@ -674,7 +682,7 @@ 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,
 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. 
+after adding more powerful CPU.

 
 \subsection{Comparing GMRES in native synchronous mode and
 Multisplitting algorithms in asynchronous mode}
 
 \subsection{Comparing GMRES in native synchronous mode and
 Multisplitting algorithms in asynchronous mode}