Corrections section 5.4.6

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index c4c39393aeefed46eabaa715b5271a533557d8ad..35ab88f5db1f14578b1986ad4f5a986abea301d9 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -24,6 +24,8 @@
 % Extension pour les liens intra-documents (tagged PDF)
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}
 % Extension pour les liens intra-documents (tagged PDF)
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}

+\usepackage{multirow}
+

 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}


@@ -111,10 +113,10 @@
 %% execution time. 
 %% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
 
 %% execution time. 
 %% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
 

-

 The behavior of multi-core 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. 
 
 The behavior of multi-core 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. 
 

-In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the Multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the GMRES algorithm.
+In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm.
+

 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;


@@ -149,10 +151,10 @@ task cannot begin a new iteration while it has not received data dependencies
 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
 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}
+neighbors. Both communications 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
 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
+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
 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


@@ -165,7 +167,7 @@ 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  a small
 parameter variation of the execution platform and of the application data can
 challenging and  labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
 problematic is  even more difficult  for the  asynchronous scheme where  a small
 parameter variation of the execution platform and of the application data can

-lead to very different numbers of iterations to reach the converge and so to
+lead to very different numbers of iterations to reach the convergence 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.
 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.


@@ -173,16 +175,16 @@ platform scenarios.
 The  {\bf 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
 The  {\bf 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
+better tune their  applications for a given multi-core architecture.  To show the

 validity of this approach we first compare the simulated execution of the Krylov
 validity of this approach we first compare the simulated execution of the Krylov

-multisplitting  algorithm   with  the   GMRES  (Generalized   Minimal  Residual)
+multisplitting  algorithm   with  the   GMRES  (Generalized   Minimal  RESidual)

 solver~\cite{saad86} in  synchronous mode.  The simulation  results allow  us to
 solver~\cite{saad86} in  synchronous mode.  The simulation  results allow  us to

-determine  which method  to choose  given a  specified multi-core  architecture.
+determine  which method  to choose  for a given multi-core  architecture.

 Moreover the  obtained results  on different simulated  multi-core architectures
 confirm the  real results  previously obtained  on non  simulated architectures.
 More precisely the simulated results are in accordance (i.e. with the same order
 of magnitude)  with the works  presented in~\cite{couturier15}, which  show that
 Moreover the  obtained results  on different simulated  multi-core architectures
 confirm the  real results  previously obtained  on non  simulated architectures.
 More precisely the simulated results are in accordance (i.e. with the same order
 of magnitude)  with the works  presented in~\cite{couturier15}, which  show that

-the synchronous  multisplitting method  is more efficient  than GMRES  for large
+the synchronous  Krylov multisplitting method  is more efficient  than GMRES  for large

 scale  clusters.   Simulated   results  also  confirm  the   efficiency  of  the
 asynchronous  multisplitting   algorithm  compared  to  the   synchronous  GMRES
 especially in case of geographically distant clusters.
 scale  clusters.   Simulated   results  also  confirm  the   efficiency  of  the
 asynchronous  multisplitting   algorithm  compared  to  the   synchronous  GMRES
 especially in case of geographically distant clusters.


@@ -195,20 +197,20 @@ 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
 
 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:simgrid} the SimGrid simulation toolkit is presented.

 Section~\ref{sec:04} details the different solvers that we use.  Finally our
 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
+experimental results are presented in Section~\ref{sec:expe} followed by some

 concluding remarks and perspectives.
 
 
 \section{The asynchronous iteration model and the motivations of our work}
 \label{sec:asynchro}
 
 concluding remarks and perspectives.
 
 
 \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
+Asynchronous iterative methods have been  studied for many years theoretically and

 practically. Many methods have been considered and convergence results have been
 proved. These  methods can  be used  to solve, in  parallel, fixed  point problems
 (i.e. problems  for which  the solution is  $x^\star =f(x^\star)$.  In practice,
 practically. Many methods have been considered and convergence results have been
 proved. These  methods can  be used  to solve, in  parallel, fixed  point problems
 (i.e. problems  for which  the solution is  $x^\star =f(x^\star)$.  In practice,

-asynchronous iterations  methods can be used  to solve, for example,  linear and
+asynchronous iteration  methods can be used  to solve, for example,  linear and

 non-linear systems of equations or optimization problems, interested readers are
 invited to read~\cite{BT89,bahi07}.
 
 non-linear systems of equations or optimization problems, interested readers are
 invited to read~\cite{BT89,bahi07}.
 


@@ -218,7 +220,7 @@ algorithm that supports both the synchronous or the asynchronous iteration model
 requires very few modifications  to be able to be executed  in both variants. In
 practice, only  the communications and  convergence detection are  different. In
 the synchronous  mode, iterations are  synchronized whereas in  the asynchronous
 requires very few modifications  to be able to be executed  in both variants. In
 practice, only  the communications and  convergence detection are  different. In
 the synchronous  mode, iterations are  synchronized whereas in  the asynchronous

-one, they are not.  It should be noticed that non blocking communications can be
+one, they are not.  It should be noticed that non-blocking communications can be

 used in both  modes. Concerning the convergence  detection, synchronous variants
 can use  a global convergence procedure  which acts as a  global synchronization
 point. In the  asynchronous model, the convergence detection is  more tricky as
 used in both  modes. Concerning the convergence  detection, synchronous variants
 can use  a global convergence procedure  which acts as a  global synchronization
 point. In the  asynchronous model, the convergence detection is  more tricky as


@@ -226,17 +228,17 @@ 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
 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
+for the asynchronous scheme (this number 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,
 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
+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
 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
+architectures are often very important. So, prior to deployment and tests it

 seems very promising to be able to simulate the behavior of asynchronous
 seems very promising to be able to simulate the behavior of asynchronous

-iterative algorithms. The problematic is then to show that the results produce
+iterative algorithms. The problematic is then to show that the results produced

 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.
 
 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.
 


@@ -317,9 +319,9 @@ where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{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, has been 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~\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{figure}[htpb]

 %\begin{algorithm}[t]
 %\caption{Block Jacobi two-stage multisplitting method}
 \begin{algorithmic}[1]
 %\begin{algorithm}[t]
 %\caption{Block Jacobi two-stage multisplitting method}
 \begin{algorithmic}[1]


@@ -357,7 +359,7 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde
 \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}).
 
 \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}).
 

-\begin{figure}[t]
+\begin{figure}[htbp]

 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]
 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]


@@ -386,37 +388,31 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \subsection{Simulation of the two-stage methods using SimGrid toolkit}
 \label{sec:04.02}
 
 \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
+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 to ensure the
 test reproducibility  under the same  conditions.  According to  our experience,
 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
+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
 simulator using SMPI (Simulator MPI). The  first modification is to include SMPI

-libraries  and related  header files  (smpi.h).  The  second modification  is to
+libraries  and related  header files  (\verb+smpi.h+).  The  second modification  is to

 suppress all global variables by replacing  them with local variables or using a
 suppress all global variables by replacing  them with local variables or using a

-Simgrid      selector       called      "runtime       automatic      switching"
+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 and generated by
 (smpi/privatize\_global\_variables). Indeed, global  variables can generate side
 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.
+SimGrid  to simulate the  grid environment.

 
 

-
-\paragraph{Simgrid Simulator parameters}
-\  \\ \noindent  Before running  a Simgrid  benchmark, many  parameters for  the
+\paragraph{Parameters of the simulation in SimGrid}
+\  \\ \noindent  Before running  a SimGrid  benchmark, many  parameters for  the

 computation platform must be defined. For our experiments, we consider platforms
 in which  several clusters are  geographically distant,  so there are  intra and
 inter-cluster communications. In the following, these parameters are described:
 
 \begin{itemize}
 computation platform must be defined. For our experiments, we consider platforms
 in which  several clusters are  geographically distant,  so there are  intra and
 inter-cluster communications. In the following, these parameters are described:
 
 \begin{itemize}

-       \item hostfile: hosts description file.
+       \item hostfile: hosts description file,

        \item platform: file describing the platform architecture: clusters (CPU power,
        \item platform: file describing the platform architecture: clusters (CPU power,

-\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
-latency lat, \dots{}).
+\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
+latency $lat$, \dots{}),

        \item archi   : grid computational description (number of clusters, number of
        \item archi   : grid computational description (number of clusters, number of

-nodes/processors for each cluster).
+nodes/processors in each cluster).

 \end{itemize}
 \noindent
 In addition, the following arguments are given to the programs at runtime:
 \end{itemize}
 \noindent
 In addition, the following arguments are given to the programs at runtime:


@@ -424,8 +420,8 @@ In addition, the following arguments are given to the programs at runtime:
 \begin{itemize}
        \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
        \item inner precision $\TOLG$ and outer precision $\TOLM$,
 \begin{itemize}
        \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 sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+       \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,

        \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 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,


@@ -434,19 +430,18 @@ 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}
 

-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.
+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.

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

-\section{Experimental Results}
+\section{Experimental results}

 \label{sec:expe}
 
 \label{sec:expe}
 

-In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson 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}
 
 \subsection{The 3D Poisson problem}

-
-
+\label{3dpoisson}

 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
 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


@@ -478,9 +473,9 @@ 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
 
 \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
+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
 machine on a simple laptop. \\
 
 \textbf{Step 3}: Fix the criteria which will be used for the future


@@ -488,14 +483,11 @@ results comparison and analysis. In the scope of this study, we retain
 on the  one hand the algorithm execution mode (synchronous and asynchronous)
 and on the other hand the execution time and the number of iterations to reach the convergence. \\
 
 on the  one hand the algorithm execution mode (synchronous and asynchronous)
 and on the other hand the execution time and the number of iterations to reach the convergence. \\
 

-\textbf{Step 4  }: Set up the  different grid testbed environments  that will be
-simulated in the  simulator tool to run the program.  The following architecture
-has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. The first number
+\textbf{Step 4}: Set up the  different grid testbed environments  that will be
+simulated in the  simulator tool to run the program.  The following architectures
+have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number

 represents the number  of clusters in the grid and  the second number represents
 represents the number  of clusters in the grid and  the second number represents

-the number  of hosts (processors/cores)  in each  cluster. The network  has been
-designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
-latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links
-(resp.  inter-clusters backbone links). \\
+the number  of hosts (processors/cores)  in each  cluster. \\

 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 within these configurations by varying the key parameters, especially
 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 within these configurations by varying the key parameters, especially


@@ -504,8 +496,7 @@ input data.  \\
 
 \textbf{Step 6} : Collect and analyze the output results.
 
 
 \textbf{Step 6} : Collect and analyze the output results.
 

-\subsection{Factors impacting distributed applications performance in
-a grid environment}
+\subsection{Factors impacting distributed applications performance in a grid environment}

 
 When running a distributed application in a computational grid, many factors may
 have a strong impact on the performance.  First of all, the architecture of the
 
 When running a distributed application in a computational grid, many factors may
 have a strong impact on the performance.  First of all, the architecture of the


@@ -518,10 +509,10 @@ 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}
 is the network configuration. Two main network parameters can modify drastically
 the program output results:
 \begin{enumerate}

-\item  the network  bandwidth  (bw=bits/s) also  known  as "the  data-carrying
+\item  the network  bandwidth  ($bw$ in bits/s) also  known  as "the  data-carrying

     capacity" of the network is defined as  the maximum of data that can transit
     from one point to another in a unit of time.
     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
+\item the  network latency  ($lat$ in microseconds) defined as  the delay  from the

   start time to send  a simple data from a source to a destination.
 \end{enumerate}
 Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster
   start time to send  a simple data from a source to a destination.
 \end{enumerate}
 Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster


@@ -532,253 +523,120 @@ and  between distant  clusters.  This parameter is application dependent.
  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
  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
+ high bandwidth and very low latency. In opposite, the inter-network connects
+ clusters sometime via  heterogeneous networks components  through internet with

  a lower speed.  The network  between distant  clusters might  be a  bottleneck
  for  the global performance of the application.
 
  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 means 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.
+\subsection{Comparison between GMRES and two-stage multisplitting algorithms in synchronous mode}
+In the scope of this paper, our first objective is to analyze when the synchronous Krylov two-stage method has better performance than the classical GMRES method. With a synchronous iterative method, better performance means a smaller number of iterations and execution time before reaching the convergence.

 
 

-The following paragraphs present the test conditions, the output results
-and our comments.\\
-
-
-\subsubsection{Execution of the algorithms on various computational grid
-architectures and scaling up the input matrix size}
-\ \\
-% environment
+Table~\ref{tab:01} summarizes the parameters used in the different simulations: the grid architectures, the network of inter-clusters backbone links and the matrix sizes of the 3D Poisson problem. However, for all simulations we fix the network parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency $lat=8\mu$s. In what follows, we will present the test conditions, the output results and our comments. 

 
 \begin{table} [ht!]
 \begin{center}
 
 \begin{table} [ht!]
 \begin{center}

-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
- Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
- - &  N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$  =170 $\times$ 170 $\times$ 170    \\ \hline
- \end{tabular}
-\caption{Test conditions: various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}
-\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}.  Idem dans le texte, les figures, etc.}}
+\begin{tabular}{ll}
+\hline
+Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ 
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
+                                        & $N2$: $bw$=1Gbs, $lat=50\mu$s \\ 
+\multirow{2}{*}{Matrix size}            & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
+                                        & $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
+\end{tabular}
+\caption{Parameters for the different simulations}

 \label{tab:01}
 \end{center}
 \end{table}
 
 \label{tab:01}
 \end{center}
 \end{table}
 

+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}

 
 

+In  this  section,  we  analyze   the  simulations  conducted  on  various  grid
+configurations and for different sizes of the 3D Poisson problem. The parameters
+of    the    network    between    clusters    is    fixed    to    $N2$    (see
+Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and a
+given matrix size 170$^3$ elements, a  non-variation in the number of iterations
+for the classical GMRES algorithm, which is not the case of the Krylov two-stage
+algorithm. In fact, with multisplitting  algorithms, the number of splitting (in
+our case, it is the number of clusters) influences on the convergence speed. The
+higher the number  of splitting is, the slower the  convergence of the algorithm
+is (see the output results obtained from configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs. 8$\times$8).

 
 

+The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16). 

 
 

-
-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{figure} [ht!]
-  \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 $N_{x}=170$\RC{idem}
-\AG{Utiliser le point comme séparateur décimal et non la virgule.  Idem dans les autres figures.}}
-  \label{fig:01}
-\end{figure}
-
-
-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. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
-
-\subsubsection{Running on two different inter-clusters network speeds \\}
-
-\begin{table} [ht!]
+\begin{figure}[ht]

 \begin{center}
 \begin{center}

-\begin{tabular}{r c }
- \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 & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
- \end{tabular}
-\caption{Test conditions: grid 2x16 and 4x8 with  networks N1 vs N2}
-\label{tab:02}
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}

 \end{center}
 \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). \RC{Il faut définir cela avant...}
-Figure~\ref{fig:02} shows that end users will reduce the execution time
-for  both  algorithms when using  a  grid  architecture  like  4x16 or  8x8: the reduction is about $2$. 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\%.
-
-
+\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\label{fig:01}
+\end{figure}

 
 

-%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\subsubsection{Simulations for two different inter-clusters network speeds\\}
+In  Figure~\ref{fig:02} we  present the  execution times  of both  algorithms to
+solve a  3D Poisson problem of  size $150^3$ on two  different simulated network
+$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
+this figure  that the Krylov two-stage  algorithm is sensitive to  the number of
+clusters (i.e. it is better to have a small number of clusters). However, we can
+notice an  interesting behavior of  the Krylov  two-stage algorithm. It  is less
+sensitive to bad network bandwidth and latency for the inter-clusters links than
+the  GMRES algorithms.  This  means  that the  multisplitting  methods are  more
+efficient for distributed systems with high latency networks.
+
+\begin{figure}[ht]

 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}

-\caption{Grid 2x16 and 4x8 with networks N1 vs N2
-\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
+\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\LZK{CE, remplacer les ``,'' des décimales par un ``.''}

 \label{fig:02}
 \end{figure}
 \label{fig:02}
 \end{figure}

-%\end{wrapfigure}

 
 

+\subsubsection{Network latency impacts on performances\\}
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm. 

 
 

-\subsubsection{Network latency impacts on performance}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16\\ %\hline
- Network & N1 : bw=1Gbs \\ %\hline
- Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
- \end{tabular}
-\caption{Test conditions: network latency impacts}
-\label{tab:03}
-\end{table}
-
-
-
-\begin{figure} [ht!]
+\begin{figure}[ht]

 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}

-\caption{Network latency impacts on execution time
-\AG{\np{E-6}}}
+\caption{Network latency impacts on performances}

 \label{fig:03}
 \end{figure}
 
 \label{fig:03}
 \end{figure}
 

+\subsubsection{Network bandwidth impacts on performances\\}
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of 2$\times$16 processors interconnected by a network of latency $lat=50\mu$s to solve a 3D Poisson problem of size $150^3$. The results of increasing the network bandwidth from 1Gbs to 10Gbs show the performances improvement for both algorithms by reducing the execution times. However, the Krylov two-stage algorithm presents a better performance in the considered bandwidth interval with a gain of $40\%$ compared to only about $24\%$ for the classical GMRES algorithm.

 
 

-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.\RC{Les  2  précédentes phrases  me
-  semblent en contradiction....}  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 $8.10^{-6}$.
-
-\subsubsection{Network bandwidth impacts on performance}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16\\ %\hline
- Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
- \end{tabular}
-\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
-\label{tab:04}
-\end{table}
-
-
-\begin{figure} [ht!]
+\begin{figure}[ht]

 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}

-\caption{Network bandwith impacts on execution time
-\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}
+\caption{Network bandwith impacts on performances}

 \label{fig:04}
 \end{figure}
 
 \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.
+\subsubsection{Matrix size impacts on performances\\}
+In these experiments, the matrix size of the 3D Poisson problem is varied from $50^3$ to $190^3$ elements. The simulated computational grid is composed of 4 clusters of 8 processors each interconnected by the network $N2$ (see Table~\ref{tab:01}). Obviously, as shown in Figure~\ref{fig:05}, the execution times for both algorithms increase with increased matrix sizes.  For all problem sizes, GMRES algorithm is always slower than the Krylov two-stage algorithm. Moreover, for this benchmark, it seems that the greater the problem size is, the bigger the ratio between execution times of both algorithms is. We can also observe that for some problem sizes, the convergence (and thus the execution time) of the Krylov two-stage algorithm varies quite a lot. %This is due to the 3D partitioning of the 3D matrix of the Poisson problem. 
+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. 

 
 

-\subsubsection{Input matrix size impacts on performance}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \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}
-\caption{Test conditions: Input matrix size impacts}
-\label{tab:05}
-\end{table}
-
-
-\begin{figure} [ht!]
+\begin{figure}[ht]

 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}

-\caption{Problem size impacts on execution time}
+\caption{Problem size impacts on performances}

 \label{fig:05}
 \end{figure}
 
 \label{fig:05}
 \end{figure}
 

-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)  of the number of iterations needed to
-    reach the convergence for the classical GMRES algorithm when the matrix size
-    go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itérations est 10 fois plus long mais la suite de la phrase ne veut rien dire}
-\item the  classical GMRES execution time  is almost the double  for $N_{x}=140$
-  compared with the Krylov multisplitting method.
-\end{enumerate}
-
-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 architecture & 2x16\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline
- \end{tabular}
-\caption{Test conditions: CPU Power impacts}
-\label{tab:06}
-\end{table}
+\subsubsection{CPU power impacts on performances\\}
+Using the SimGrid simulator flexibility, we have tried to determine the impact of the CPU power of the processors in the different clusters on performances of both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The simulation is conducted in a grid of 2$\times$16 processors interconnected by the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size $150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance gain, about $95\%$ for both algorithms, after improving the CPU power of processors.

 
 

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

 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}

-\caption{CPU Power impacts on execution time}
+\caption{CPU Power impacts on performances}

 \label{fig:06}
 \end{figure}
 \label{fig:06}
 \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 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}
-

 To conclude these series of experiments, with  SimGrid we have been able to make
 many simulations  with many parameters  variations. Doing all  these experiments
 with a real platform is most of  the time not possible. Moreover the behavior of
 To conclude these series of experiments, with  SimGrid we have been able to make
 many simulations  with many parameters  variations. Doing all  these experiments
 with a real platform is most of  the time not possible. Moreover the behavior of

-both GMRES and  Krylov multisplitting methods is in accordance  with larger real
-executions on large scale supercomputer~\cite{couturier15}.
+both GMRES and  Krylov two-stage algorithms is in accordance  with larger real
+executions on large scale supercomputers~\cite{couturier15}.

 
 
 \subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
 
 
 \subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}


@@ -796,21 +654,23 @@ synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
 theoretically reduce  the overall execution  time and can improve  the algorithm
 performance.
 
 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.
+In this section,  the Simgrid simulator is  used to compare the  behavior of the
+multisplitting in  asynchronous mode  with GMRES  in synchronous  mode.  Several
+benchmarks have  been performed with  various combination of the  grid resources
+(CPU, Network, input  matrix size, \ldots ). The test  conditions are summarized
+in  Table~\ref{tab:07}. In  order to  compare  the execution  times, this  table
+reports the  relative gain between both  algorithms. It is defined  by the ratio
+between  the   execution  time  of   GMRES  and   the  execution  time   of  the
+multisplitting.  The  ratio  is  greater  than  one  because  the  asynchronous
+multisplitting version is faster than GMRES.

 
 
 
 

-The test conditions are summarized in the table~\ref{tab:07}: \\

 
 

-\begin{table} [ht!]
+\begin{table} [htbp]

 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline

- Grid Architecture & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2 $\times$ 50 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$^{-2}$\\
  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$^{-2}$\\


@@ -856,7 +716,7 @@ geographically distant clusters through the internet.
     power (GFlops)
     & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
     \hline
     power (GFlops)
     & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
     \hline

-    size (N)
+    size ($N^3$)

     & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\
     \hline
     Precision
     & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\
     \hline
     Precision