X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/18767a457a4de2e57af831774829e95aa27adae1..01a38307eb4e07ea93979965602956371d2a04aa:/paper.tex

diff --git a/paper.tex b/paper.tex
index 397decc..5b8ada3 100644
--- a/paper.tex
+++ b/paper.tex
@@ -45,6 +45,8 @@
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RCE}[2][inline]{%
   \todo[color=yellow!10,#1]{\sffamily\textbf{RCE:} #2}\xspace}
+\newcommand{\DL}[2][inline]{%
+    \todo[color=pink!10,#1]{\sffamily\textbf{DL:} #2}\xspace}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
@@ -96,25 +98,26 @@ 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.
+applications. The main contribution of this paper is to show that the use of a
+simulation tool (here we have decided to use the SimGrid toolkit) can really
+help developpers to better tune their applications for a given multi-core
+architecture.
 
-In this paper, we focus our attention on two parallel iterative algorithms based
+In particular 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  variants of
+These algorithms  are used to  solve linear  systems. Two different  variants of
 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.
-
-
-
+one  with asynchronous iterations. For each algorithm we have simulated
+different architecture parameters to evaluate their influence on the overall
+execution time.  The obtain simulated results confirm the real results
+previously obtained on different real multi-core architectures and also confirm
+the efficiency of the asynchronous multisplitting algorithm compared to the
+synchronous GMRES method.
 
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
-%performance} 
+%performance}
 \keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}
 
 \maketitle
@@ -131,28 +134,28 @@ 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,
-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
-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
-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
-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
@@ -171,10 +174,22 @@ 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
+solver~\cite{saad86} in synchronous mode. 
+
+\LZK{Pas trop convainquant comme argument pour valider l'approche de simulation. \\On peut dire par exemple: on a pu simuler différents algos itératifs à large échelle (le plus connu GMRES et deux variantes de multisplitting) et la simulation nous a permis (sans avoir le vrai matériel) de déterminer quelle serait la meilleure solution pour une telle configuration de l'archi ou vice versa.\\A revoir...}
+
+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  compared to the synchronous  GMRES. In
+on non simulated architectures.  
+
+\LZK{Il n y a pas dans la partie expé cette comparaison et confirmation des résultats entre la simulation et l'exécution réelle des algos sur les vrais clusters.\\ Sinon on pourrait ajouter dans la partie expé une référence vers le journal supercomput de krylov multi pour confirmer que cette méthode est meilleure que GMRES sur les clusters large échelle.}
+
+We also confirm  the efficiency  of the
+asynchronous  multisplitting algorithm  compared to the synchronous  GMRES. 
+
+\LZK{P.S.: Pour tout le papier, le principal objectif n'est pas de faire des comparaisons entre des méthodes itératives!!\\Sinon, les deux algorithmes Krylov multisplitting synchrone et multisplitting asynchrone sont plus efficaces que GMRES sur des clusters à large échelle.\\Et préciser, si c'est vraiment le cas, que le multisplitting asynchrone est plus efficace et adapté aux clusters distants par rapport aux deux autres algos (je n'ai pas encore lu la partie expé)}
+
+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
@@ -188,8 +203,10 @@ 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.
 
+\LZK{Proposition d'un titre pour le papier: Grid-enabled simulation of large-scale linear iterative solvers.}
 
-\section{The asynchronous iteration model}
+
+\section{The asynchronous iteration model and the motivations of our work}
 \label{sec:asynchro}
 
 Asynchronous iterative methods have been  studied for many years theoritecally and
@@ -213,6 +230,21 @@ point. In the  asynchronous model, the convergence detection is  more tricky as
 it   must  not   synchronize  all   the  processors.   Interested  readers   can
 consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
 
+The number of iterations required to reach the convergence is generally greater
+for the asynchronous scheme (this number depends depends on  the delay of the
+messages). Note that, it is not the case in the synchronous mode where the
+number of iterations is the same than in the sequential mode. In this way, the
+set of the parameters  of the  platform (number  of nodes,  power of nodes,
+inter and  intra clusters  bandwidth  and  latency \ldots) and  of  the
+application can drastically change the number of iterations required to get the
+convergence. It follows that asynchronous iterative algorithms are difficult to
+optimize since the financial and deployment costs on large scale multi-core
+architecture are often very important. So, prior to delpoyment and tests it
+seems very promising to be able to simulate the behavior of asynchronous
+iterative algorithms. The problematic is then to show that the results produce
+by simulation are in accordance with reality i.e. of the same order of
+magnitude. To our knowledge, there is no study on this problematic.
+
 \section{SimGrid}
  \label{sec:simgrid}
 
@@ -223,22 +255,22 @@ consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}
-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}
-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}
-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}
-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{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]
@@ -259,19 +291,19 @@ where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are compute
 %\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}
-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}
-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}
@@ -304,11 +336,11 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 %\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
-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
@@ -316,11 +348,13 @@ 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
-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}
@@ -341,21 +375,21 @@ nodes/processors for each cluster).
 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;
-	\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 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 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}
-	\item Size of matrix S;
-	\item Maximum number of iterations and tolerance threshold for CGLS. 
+	\item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
+	\item inner precision $\TOLG$ and outer precision $\TOLM$,
+	\item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
+	\item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
+	\item matrix off-diagonal value is fixed to $-1.0$,
+	\item number of vectors in matrix $S$ (i.e. value of $s$),
+	\item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
+        \item maximum number of iterations and precision for the classical GMRES method,
+        \item maximum number of restarts for the Arnorldi process in GMRES method,
+      	\item execution mode: synchronous or asynchronous.
 \end{itemize}
+\LZK{CE pourrais tu vérifier et confirmer les valeurs des éléments diag et off-diag de la matrice?}
+\RCE{oui, les valeurs de diag et off-diag donnees sont ok}
 
-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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
@@ -363,32 +397,32 @@ It should also be noticed that both solvers have been executed with the Simgrid
 \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{The 3D Poisson problem}
+
 
-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
+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}
 \frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
 \label{eq:07}
 \end{equation}
-such that
+such that:
 \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}
-\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}
-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}
 
 First, to conduct our study, we propose the following methodology
 which can be reused for any grid-enabled applications.\\
@@ -397,10 +431,12 @@ which can be reused for any grid-enabled applications.\\
 the application to be tested. Numerical parallel iterative algorithms
 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 the 3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting method. In addition, the Simgrid simulator has been chosen to simulate the behaviors of the
-distributed applications. Simgrid is running on the Mesocentre datacenter in the University of  Franche-Comte and also in a virtual machine on a simple laptop. \\
+\textbf{Step 2}: Collect the software materials needed for the experimentation.
+In our case, we have two variants algorithms for the resolution of the
+3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting
+method. In addition, the Simgrid simulator has been chosen to simulate the
+behaviors of the distributed applications. Simgrid is running in a virtual
+machine on a simple 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
@@ -427,13 +463,13 @@ input data.  \\
 a grid environment}
 
 When running a distributed application in a computational grid, many factors may
-have a strong impact on the performances.  First of all, the architecture of the
+have a strong impact on the performance.  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 performances  of the application
+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:
 \begin{enumerate}
@@ -441,290 +477,307 @@ the program output results:
     capacity" of the network is defined as  the maximum of data that can transit
     from one point to another in a unit of time.
 \item 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.
+  start time to send  a simple data from a source to a destination.
 \end{enumerate}
-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  and
-transit between the clusters and nodes during the code execution.
+Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster
+and  between distant  clusters.  This parameter is application dependent.
 
  In  a grid  environment, it  is common  to distinguish,  on the  one hand,  the
- "intra-network" which refers  to the links between nodes within  a cluster and,
+ "intra-network" which refers  to the links between nodes within  a cluster and
  on  the other  hand, the  "inter-network" which  is the  backbone link  between
- clusters.  In   practice,  these  two   networks  have  different   speeds.  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  throuth 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}
-
-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.
+ clusters.  In   practice,  these  two   networks  have  different   speeds.
+ 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  throuth internet with
+ a lower speed.  The network  between distant  clusters might  be a  bottleneck
+ for  the global performance of the application.
+
+\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
+
+In the scope  of this paper, our  first objective is to analyze  when the Krylov
+Multisplitting  method   has  better  performance  than   the  classical  GMRES
+method. With a synchronous  iterative method, better performance 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.\\
 
 
-\textit{3.a Executing the algorithms on various computational grid
-architecture and scaling up the input matrix size}
-\\
-
+\subsubsection{Execution of the algorithms on various computational grid
+architectures and scaling up the input matrix size}
+\ \\
 % environment
-\begin{footnotesize}
+
+\begin{table} [ht!]
+\begin{center}
 \begin{tabular}{r c }
  \hline
- Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
+ Grid Architecture & 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
  - &  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{Test conditions: Various grid configurations with the input matix size 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)}}
+\label{tab:01}
+\end{center}
+\end{table}
 
-\end{footnotesize}
 
 
 
 %\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  performance of algorithms running  on various
+grid configurations  (2x16, 4x8, 4x16  and 8x8). First,  the results in  Figure~\ref{fig:01}
+show for all grid configurations 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!]
-\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{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170}
+  \label{fig:01}
 \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
-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 the number of the processors in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs 40\% better (resp. 48\%) less when running from 2x16=32 to 8x8=64 processors.
 
-\textit{\\3.b Running on two different speed cluster inter-networks\\}
+The execution  times between  the two algorithms  is significant  with different
+grid architectures, even  with the same number of processors  (for example, 2x16
+and  4x8). We  can  observ  the low  sensitivity  of  the Krylov multisplitting  method
+(compared with the classical GMRES) when scaling up the number of the processors
+in the  grid: in  average, the GMRES  (resp. Multisplitting)  algorithm performs
+$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors.
 
-% environment
-\begin{footnotesize}
+\subsubsection{Running on two different inter-clusters network speeds \\}
+
+\begin{table} [ht!]
+\begin{center}
 \begin{tabular}{r c }
  \hline
- Grid & 2x16, 4x8\\ %\hline
+ Grid Architecture & 2x16, 4x8\\ %\hline
  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
  \end{tabular}
-Table 2 : Clusters x Nodes - Networks N1 x N2 \\
-
- \end{footnotesize}
+\caption{Test conditions: Grid 2x16 and 4x8 - Networks N1 vs N2}
+\label{tab:02}
+\end{center}
+\end{table}
 
+These experiments  compare the  behavior of  the algorithms  running first  on a
+speed inter-cluster  network (N1) and  also on  a less performant  network (N2).
+Figure~\ref{fig:02} 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  by a factor of  $2$. The results depict  also that when
+the  network speed  drops down (variation of 12.5\%), the  difference between  the two Multisplitting algorithms execution times can reach more than 25\%.
+%\RC{c'est pas clair : la différence entre quoi et quoi?}
+%\DL{pas clair}
+%\RCE{Modifie}
 
 
 %\begin{wrapfigure}{l}{100mm}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Cluster x Nodes N1 x N2}
-%\label{overflow}}
+\caption{Grid 2x16 and 4x8 - Networks N1 vs N2}
+\label{fig:02}
 \end{figure}
 %\end{wrapfigure}
 
-The experiments compare the behavior of the algorithms running first on
-a speed inter- cluster network (N1) and also on 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 (12.5\%), the difference between the execution
-times can reach more than 25\%.
 
-\textit{\\3.c Network latency impacts on performance\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Network latency impacts on performance}
+\ \\
+\begin{table} [ht!]
+\centering
 \begin{tabular}{r c }
  \hline
- Grid & 2x16\\ %\hline
+ Grid Architecture & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline\\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
  \end{tabular}
-Table 3 : Network latency impact \\
-
-\end{footnotesize}
+\caption{Test conditions: Network latency impacts}
+\label{tab:03}
+\end{table}
 
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impact on execution time}
-%\label{overflow}}
+\caption{Network latency impacts on execution time}
+\label{fig:03}
 \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
-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$^{-6}$.
+According to the results  of  Figure~\ref{fig:03}, a  degradation  of the  network
+latency from $8.10^{-6}$  to $6.10^{-5}$ implies an absolute  time increase of more
+than $75\%$  (resp. $82\%$) of the  execution for the classical  GMRES (resp. Krylov
+multisplitting)   algorithm.   In   addition,   it  appears   that  the   Krylov
+multisplitting method tolerates  more the network latency variation  with a less
+rate  increase  of  the  execution   time.   Consequently,  in  the  worst  case
+($lat=6.10^{-5 }$), the  execution time for GMRES is almost  the double than the
+time of the Krylov 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}
+\subsubsection{Network bandwidth impacts on performance}
+\ \\
+\begin{table} [ht!]
+\centering
 \begin{tabular}{r c }
  \hline
- Grid & 2x16\\ %\hline
+ Grid Architecture & 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 \\
  \end{tabular}
-Table 4 : Network bandwidth impact \\
-
-\end{footnotesize}
+\caption{Test conditions: Network bandwidth impacts}
+\label{tab:04}
+\end{table}
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
-\caption{Network bandwith impact on execution time}
-%\label{overflow}
+\caption{Network bandwith impacts on execution time}
+\label{fig:04}
 \end{figure}
 
+The results  of increasing  the network  bandwidth show  the improvement  of the
+performance  for   both  algorithms   by  reducing   the  execution   time  (see
+Figure~\ref{fig:04}). However,  in this  case, the Krylov  multisplitting method
+presents a better  performance in the considered bandwidth interval  with a gain
+of $40\%$ which is only around $24\%$ for the classical GMRES.
 
-
-The results of increasing the network bandwidth show the improvement
-of the performance for both of the two algorithms by reducing the execution time (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\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Input matrix size impacts on performance}
+\ \\
+\begin{table} [ht!]
+\centering
 \begin{tabular}{r c }
  \hline
- Grid & 4x8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\
+ Grid Architecture & 4x8\\ %\hline
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
+ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
  \end{tabular}
-Table 5 : Input matrix size impact\\
-
-\end{footnotesize}
+\caption{Test conditions: Input matrix size impacts}
+\label{tab:05}
+\end{table}
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
-\caption{Pb size impact on execution time}
-%\label{overflow}}
+\caption{Problem size impacts on execution time}
+\label{fig:05}
 \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
-iinput 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\\}
+In these experiments, 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 Figure~\ref{fig:05},  the execution
+time for  both algorithms increases when  the input matrix size  also increases.
+But the interesting results are:
+\begin{enumerate}
+  \item the drastic increase ($10$ times) \RC{Je ne vois pas cela sur la figure}
+\RCE{Corrige} of the  number of  iterations needed  to reach the  convergence for  the classical
+GMRES algorithm when  the matrix size go beyond $N_{x}=150$;
+\item the  classical GMRES execution time  is almost the double  for $N_{x}=140$
+  compared with the Krylov multisplitting method.
+\end{enumerate}
 
-% environment
-\begin{footnotesize}
+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.  It  should be noticed that the same test has  been done with the
+grid 2x16 leading to the same conclusion.
+
+\subsubsection{CPU Power impacts on performance}
+
+\begin{table} [ht!]
+\centering
 \begin{tabular}{r c }
  \hline
- Grid & 2x16\\ %\hline
+ Grid architecture & 2x16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}
-Table 6 : CPU Power impact \\
-
-\end{footnotesize}
-
+\caption{Test conditions: CPU Power impacts}
+\label{tab:06}
+\end{table}
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
-\caption{CPU Power impact on execution time}
-%\label{overflow}}
-s\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. 
-
-\subsection{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. The study compares
-the performance of the two proposed algorithms both in \textit{synchronous mode
-}. In this section, following the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textit{
-asynchronous mode} compared with the classical GMRES staying in
-\textit{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 \textit{"relative gain"} (exec\_time$_{GMRES}$ / exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
-
-
-The test conditions are summarized in the table below : \\
+\caption{CPU Power impacts on execution time}
+\label{fig:06}
+\end{figure}
 
-% environment
-\begin{footnotesize}
+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 Figure~\ref{fig:06}  confirm the
+performance gain,  around $95\%$ for  both of the  two methods, after  adding more
+powerful CPU.
+
+\DL{il faut une conclusion sur ces tests : ils confirment les résultats déjà
+obtenus en grandeur réelle. Donc c'est une aide précieuse pour les dev. Pas
+besoin de déployer sur une archi réelle}
+
+
+\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm 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.  In  this
+section, following  the same previous  methodology, our  goal is to  compare the
+efficiency of the multisplitting method  in \textit{ asynchronous mode} compared with the
+classical GMRES in \textit{synchronous mode}.
+
+The  interest of  using  an asynchronous  algorithm  is that  there  is no  more
+synchronization. With  geographically distant  clusters, this may  be essential.
+In  this case,  each  processor can  compute its  iteration  freely without  any
+synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
+theoretically reduce  the overall execution  time and can improve  the algorithm
+performance.
+
+\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
+In this section, Simgrid simulator tool has been successfully used to show
+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    \textit{"relative    gain"}   (exec\_time$_{GMRES}$    /
+exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
+
+
+The test conditions are summarized in the table~\ref{tab:07}: \\
+
+\begin{table} [ht!]
+\centering
 \begin{tabular}{r c }
  \hline
- Grid & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2x50 totaling 100 processors\\ %\hline
  Processors Power & 1 GFlops to 1.5 GFlops\\
    Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
    Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
  Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}
-\end{footnotesize}
+\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
+\label{tab:07}
+\end{table}
 
-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 7 below has recorded the best grid configurations
-allowing the multisplitting method execution time more performant 2.5 times than
-the classical GMRES execution and convergence time. The experimentation has demonstrated the relative multisplitting algorithm tolerance when using a low speed network that we encounter usually with distant clusters thru the internet.
+Again,  comprehensive and  extensive tests  have been  conducted with  different
+parameters as  the CPU power, the  network parameters (bandwidth and  latency)
+and with different problem size. The  relative gains greater than $1$  between the
+two algorithms have  been captured after  each step  of the test.   In
+Figure~\ref{fig:07}  are  reported the  best  grid  configurations allowing
+the  multisplitting method to  be more than  $2.5$ times faster  than the
+classical  GMRES.  These  experiments also  show the  relative tolerance  of the
+multisplitting algorithm when using a low speed network as usually observed with
+geographically distant clusters through the internet.
 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
@@ -735,14 +788,12 @@ the classical GMRES execution and convergence time. The experimentation has demo
     \end{tabular}}
 
 
-\begin{table}[!t]
-  \centering
+\begin{figure}[!t]
+\centering
+%\begin{table}
 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
 %  \label{"Table 7"}
-Table 7. Relative gain of the multisplitting algorithm compared with
-the classical GMRES \\
-
-  \begin{mytable}{11}
+ \begin{mytable}{11}
     \hline
     bandwidth (Mbit/s)
     & 5     & 5     & 5         & 5         & 5  & 50        & 50        & 50        & 50        & 50 \\
@@ -763,7 +814,11 @@ the classical GMRES \\
     & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\
     \hline
   \end{mytable}
-\end{table}
+%\end{table}
+ \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+ \label{fig:07}
+\end{figure}
+
 
 \section{Conclusion}
 CONCLUSION