X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/01a38307eb4e07ea93979965602956371d2a04aa..34ef1c761f30fa1a22267a93d9aeaabb90869ada:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index 5b8ada3..beb3140 100644
--- a/paper.tex
+++ b/paper.tex
@@ -21,10 +21,11 @@
 \usepackage{algpseudocode}
 %\usepackage{amsthm}
 \usepackage{graphicx}
-\usepackage[american]{babel}
 % 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}}
@@ -71,48 +72,50 @@
 
 
 
-\begin{document} \RCE{Titre a confirmer.} \title{Comparative performance
-analysis of simulated grid-enabled numerical iterative algorithms}
+\begin{document}
+\title{Grid-enabled simulation of large-scale linear iterative solvers}
 %\itshape{\journalnamelc}\footnotemark[2]}
 
-\author{    Charles Emile Ramamonjisoa and
-    David Laiymani and
-    Arnaud Giersch and
-    Lilia Ziane Khodja and
-    RaphaÃ«l Couturier
+\author{Charles Emile Ramamonjisoa\affil{1},
+    David Laiymani\affil{1},
+    Arnaud Giersch\affil{1},
+    Lilia Ziane Khodja\affil{2} and
+    RaphaÃ«l Couturier\affil{1}
 }
 
 \address{
-	\centering
-    Femto-ST Institute - DISC Department\\
-    UniversitÃ© de Franche-ComtÃ©\\
-    Belfort\\
-    Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
+  \affilnum{1}%
+  Femto-ST Institute, DISC Department,
+  University of Franche-ComtÃ©,
+  Belfort, France.
+  Email:~\email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}\break
+  \affilnum{2}
+  Department of Aerospace \& Mechanical Engineering,
+  Non Linear Computational Mechanics,
+  University of Liege, Liege, Belgium.
+  Email:~\email{l.zianekhodja@ulg.ac.be}
 }
 
-%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
-
-\begin{abstract}   The behavior of 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 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 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 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 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.
+\begin{abstract} %% 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 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 developers to better tune their applications for a given multi-core
+%% architecture.
+
+%% In this paper we focus our attention on the simulation of iterative algorithms to solve sparse linear systems on large clusters. We study the behavior of the widely used GMRES algorithm and two different variants of the Multisplitting algorithms: one using synchronous iterations and another one with 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 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. 
+
+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}
 
@@ -134,7 +137,7 @@ 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\dots{} 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
@@ -148,10 +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
-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
-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
@@ -163,34 +166,30 @@ application  on   a  given   multi-core  architecture.  Finding   good  resource
 allocations policies under  varying CPU power, network speeds and  loads is very
 challenging and  labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
 problematic is  even more difficult  for the  asynchronous scheme where  a small
-parameter variation of the execution platform can lead to very different numbers
-of iterations to reach the converge and so to very different execution times. In
-this challenging context we think that the  use of a simulation tool can greatly
-leverage the possibility of testing various platform scenarios.
-
-The main contribution of this paper is to show that the use of a simulation tool
-(i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real  parallel
-applications (i.e. large linear system solvers) can help developers to better
-tune their application for a given multi-core architecture. To show the validity
-of this approach we first compare the simulated execution of the multisplitting
-algorithm  with  the  GMRES   (Generalized   Minimal  Residual)
-solver~\cite{saad86} in synchronous mode. 
-
-\LZK{Pas trop convainquant comme argument pour valider l'approche de simulation. \\On peut dire par exemple: on a pu simuler diffÃ©rents algos itÃ©ratifs Ã  large Ã©chelle (le plus connu GMRES et deux variantes de multisplitting) et la simulation nous a permis (sans avoir le vrai matÃ©riel) de dÃ©terminer 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.  
-
-\LZK{Il n y a pas dans la partie expÃ© cette comparaison et confirmation des rÃ©sultats entre la simulation et l'exÃ©cution rÃ©elle des algos sur les vrais clusters.\\ Sinon on pourrait ajouter dans la partie expÃ© une rÃ©fÃ©rence vers le journal supercomput de krylov multi pour confirmer que cette mÃ©thode est meilleure que GMRES sur les clusters large Ã©chelle.}
-
-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 mÃ©thodes itÃ©ratives!!\\Sinon, les deux algorithmes Krylov multisplitting synchrone et multisplitting asynchrone sont plus efficaces que GMRES sur des clusters Ã  large Ã©chelle.\\Et prÃ©ciser, si c'est vraiment le cas, que le multisplitting asynchrone est plus efficace et adaptÃ© aux clusters distants par rapport aux deux autres algos (je n'ai pas encore lu la partie expÃ©)}
-
-In
-this way and with a simple computing architecture (a laptop) SimGrid allows us
+parameter variation of the execution platform and of the application data can
+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.
+
+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  applications for a given multi-core architecture.  To show the
+validity of this approach we first compare the simulated execution of the Krylov
+multisplitting  algorithm   with  the   GMRES  (Generalized   Minimal  RESidual)
+solver~\cite{saad86} in  synchronous mode.  The simulation  results allow  us to
+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
+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.
+
+In this way and with a simple computing architecture (a laptop) SimGrid allows us
 to run a test campaign  of  a  real parallel iterative  applications on
 different simulated multi-core architectures.  To our knowledge, there is no
 related work on the large-scale multi-core simulation of a real synchronous and
@@ -198,22 +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
-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
-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.
 
-\LZK{Proposition d'un titre pour le papier: Grid-enabled simulation of large-scale linear iterative solvers.}
-
 
 \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,
-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}.
 
@@ -223,7 +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
-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
@@ -231,24 +228,76 @@ 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
+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,
-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
-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
-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.
 
 \section{SimGrid}
- \label{sec:simgrid}
+\label{sec:simgrid}
+SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile} is a discrete event simulation framework to study the behavior of large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds and High Performance Computation systems. It is widely used to simulate and evaluate heuristics, prototype applications or even assess legacy MPI applications. It is still actively developed by the scientific community and distributed as an open source software.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
+% SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
+% is a simulation framework to study the behavior of large-scale distributed
+% systems.  As its name suggests, it emanates from the grid computing community,
+% but is nowadays used to study grids, clouds, HPC or peer-to-peer systems.  The
+% early versions of SimGrid date back from 1999, but it is still actively
+% developed and distributed as an open source software.  Today, it is one of the
+% major generic tools in the field of simulation for large-scale distributed
+% systems.
+
+SimGrid provides several programming interfaces: MSG to simulate Concurrent
+Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
+run real applications written in MPI~\cite{MPI}.  Apart from the native C
+interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
+languages.  SMPI is the interface that has been used for the work described in
+this paper.  The SMPI interface implements about \np[\%]{80} of the MPI 2.0
+standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports
+applications written in C or Fortran, with little or no modifications (cf Section IV - paragraph B).
+
+Within SimGrid, the execution of a distributed application is simulated by a
+single process.  The application code is really executed, but some operations,
+like communications, are intercepted, and their running time is computed
+according to the characteristics of the simulated execution platform.  The
+description of this target platform is given as an input for the execution, by
+means of an XML file.  It describes the properties of the platform, such as
+the computing nodes with their computing power, the interconnection links with
+their bandwidth and latency, and the routing strategy.  The scheduling of the
+simulated processes, as well as the simulated running time of the application
+are computed according to these properties.
+
+To compute the durations of the operations in the simulated world, and to take
+into account resource sharing (e.g. bandwidth sharing between competing
+communications), SimGrid uses a fluid model.  This allows users to run relatively fast
+simulations, while still keeping accurate
+results~\cite{bedaride+degomme+genaud+al.2013.toward,
+  velho+schnorr+casanova+al.2013.validity}.  Moreover, depending on the
+simulated application, SimGrid/SMPI allows to skip long lasting computations and
+to only take their duration into account.  When the real computations cannot be
+skipped, but the results are unimportant for the simulation results, it is
+also possible to share dynamically allocated data structures between
+several simulated processes, and thus to reduce the whole memory consumption.
+These two techniques can help to run simulations on a very large scale.
+
+The validity of simulations with SimGrid has been asserted by several studies.
+See, for example, \cite{velho+schnorr+casanova+al.2013.validity} and articles
+referenced therein for the validity of the network models.  Comparisons between
+real execution of MPI applications on the one hand, and their simulation with
+SMPI on the other hand, are presented in~\cite{guermouche+renard.2010.first,
+  clauss+stillwell+genaud+al.2011.single,
+  bedaride+degomme+genaud+al.2013.toward}.  All these works conclude that
+SimGrid is able to simulate pretty accurately the real behavior of the
+applications.
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Two-stage multisplitting methods}
@@ -270,9 +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}
-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}.
+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]
@@ -310,7 +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}).
 
-\begin{figure}[t]
+\begin{figure}[htbp]
 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]
@@ -339,26 +388,20 @@ 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}
 
-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,
-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
-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
-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
-Simgrid  to simulate the  grid environment.
+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}
-\  \\ \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:
@@ -366,10 +409,10 @@ inter-cluster communications. In the following, these parameters are described:
 \begin{itemize}
 	\item hostfile: hosts description file.
 	\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
-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:
@@ -377,8 +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 $\TOLM$,
-	\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,
@@ -386,20 +429,19 @@ In addition, the following arguments are given to the programs at runtime:
         \item maximum number of restarts for the Arnorldi process in GMRES method,
       	\item execution mode: synchronous or asynchronous.
 \end{itemize}
-\LZK{CE pourrais tu vÃ©rifier et confirmer les valeurs des Ã©lÃ©ments 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 \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}
 
-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}
+\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:
@@ -433,9 +475,9 @@ 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
-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
@@ -444,13 +486,16 @@ 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
+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$50. The first number
 represents the number  of clusters in the grid and  the second number represents
-the number  of hosts (processors/cores)  in each  cluster. The network  has been
+the number  of hosts (processors/cores)  in each  cluster. The network has been
 designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
 latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links
-(resp.  inter-clusters backbone links). \\
+(resp.  inter-clusters backbone links).  \\
+
+\LZK{Il me semble que le bw et lat des deux rÃ©seaux varient dans les expÃ©s d'une simu Ã  l'autre. On vire la derniÃ¨re phrase?}
+\RC{il me semble qu'on peut laisser ca}
 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 within these configurations by varying the key parameters, especially
@@ -459,8 +504,7 @@ input data.  \\
 
 \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
@@ -473,10 +517,10 @@ 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}
-\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.
-\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
@@ -487,109 +531,122 @@ 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
- 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.
 
-\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
+\subsection{Comparison of GMRES and Krylov two-stage 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
+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.
-For a systematic study,  the experiments  should figure  out  that, for  various
-grid  parameters values, the simulator will confirm  the targeted outcomes,
-particularly for poor and slow  networks, focusing on the  impact on the
-communication  performance on the chosen class of algorithm.
+In what follows, we will present the test conditions, the output results and our comments.
 
-The following paragraphs present the test conditions, the output results
-and our comments.\\
+%%RAPH : on vire ca, c'est pas clair et pas important
+%For a systematic study,  the experiments  should figure  out  that, for  various
+%grid  parameters values, the simulator will confirm Multisplitting method  better performance compared to classical GMRES, particularly on poor and slow networks.
+%\LZK{Pas du tout claire la derniÃ¨re phrase (For a systematic...)!!}
+%\RCE { Reformule autrement}
 
 
-\subsubsection{Execution of the algorithms on various computational grid
-architectures and scaling up the input matrix size}
+
+%\subsubsection{Execution of the algorithms on various computational grid architectures and scaling up the input matrix size}
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
 \ \\
 % environment
 
+\RC{Je ne comprends plus rien CE : pourquoi dans 5.4.1 il y a 2 network et aussi dans 5.4.2. Quelle est la diffÃ©rence? Dans la figure 3 de la section 5.4.1 pourquoi il n'y a pas N1 et N2?}
+
 \begin{table} [ht!]
 \begin{center}
-\begin{tabular}{r c }
+\begin{tabular}{ll }
  \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
+ Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
+ \multirow{2}{*}{Network} & Inter (N2): $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
+                          & Intra (N1): $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ \multirow{2}{*}{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{je ne comprends pas la lÃ©gende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\caption{Test conditions: various grid configurations with the matrix sizes 150$^3$ or 170$^3$}
+%\LZK{Ce sont les caractÃ©ristiques du rÃ©seau intra ou inter clusters? Ce n'est pas prÃ©cisÃ©...}
+%\RCE{oui c est precise}
 \label{tab:01}
 \end{center}
 \end{table}
 
 
-
-
-%\RCE{J'ai voulu mettre les tableaux des donnÃ©es mais je pense que c'est inutile et Ã§a va surcharger}
-
-
-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 vÃ©rifie...}
-\RC{Les lÃ©gendes ne sont pas explicites...}
-
-
-\begin{figure} [ht!]
+In  this  section,  we  analyze   the  simulations  conducted  on  various  grid
+configurations presented  in Table~\ref{tab:01}. It  should be noticed  that two
+networks are considered: N1 is  the network between clusters (inter-cluster) and
+N2 is the network inside  a cluster (intra-cluster).  Figure~\ref{fig:01} shows,
+for all  grid configurations  and a  given matrix size,  a non-variation  in the
+number of iterations for the classical GMRES algorithm, which is not the case of
+the Krylov two-stage algorithm.
+%% 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 vÃ©rifie...}
+%\RC{Les lÃ©gendes ne sont pas explicites...}
+%\RCE{Corrige}
+
+\begin{figure} [htbp]
   \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}
+  \caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+%\AG{Utiliser le point comme sÃ©parateur dÃ©cimal et non la virgule.  Idem dans les autres figures.}
+%\LZK{Pour quelle taille du problÃ¨me sont calculÃ©s les nombres d'itÃ©rations? Que reprÃ©sente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?}
+  %\RCE {Corrige}
+    \RC{IdÃ©alement dans la lÃ©gende il faudrait insiquer Pb size=$150^3$ ou $170^3$  car pour l'instant Nx=150 ca n'indique rien concernant Ny et Nz}
   \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
+grid architectures, even  with the same number of processors  (for example, 2 $\times$ 16
+and  4 $\times  8$). We  can  observe  a better  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.
+$40\%$ better (resp. $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors/cores (grid 8 $\times$ 8). Note that even with a grid 8 $\times$ 8 having the maximum number of clusters, the execution time of the multisplitting method is in average 32\% less compared to GMRES. 
+\RC{pas trÃ¨s clair, c'est pas prÃ©cis de dire qu'un algo perform mieux qu'un autre, selon quel critÃ¨re?}
+\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exÃ©cution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?}
+\RCE{Remarquez que meme avec une grille 8x8, le multi est toujours plus performant}
 
-\subsubsection{Running on two different inter-clusters network speeds \\}
+\subsubsection{Simulations for two different inter-clusters network speeds \\}
 
 \begin{table} [ht!]
 \begin{center}
-\begin{tabular}{r c }
+\begin{tabular}{ll}
  \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
+ Grid architecture        & 2$\times$16, 4$\times$8\\ %\hline
+ \multirow{2}{*}{Inter Network} & N1: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
+                          & N2: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ 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 - Networks N1 vs N2}
+\caption{Test conditions: grid configurations 2$\times$16 and 4$\times$8 with 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
+In this section, the experiments  compare the  behavior of  the algorithms  running on a
+speeder inter-cluster  network (N2) and  also on  a less performant  network (N1) respectively defined in the test conditions Table~\ref{tab:02}.
+%\RC{Il faut dÃ©finir cela avant...}
+Figure~\ref{fig:02} shows that end users will reduce the execution time
+for  both  algorithms when using  a  grid  architecture  like  4 $\times$ 16 or  8 $\times$ 8: the reduction factor is around $2$. 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 diffÃ©rence entre quoi et quoi?}
-%\DL{pas clair}
-%\RCE{Modifie}
+
 
 
 %\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\begin{figure} [htbp]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Grid 2x16 and 4x8 - Networks N1 vs N2}
+\caption{Various grid configurations with networks N1 vs N2
+\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
+\RCE{Corrige}
 \label{fig:02}
 \end{figure}
 %\end{wrapfigure}
@@ -601,33 +658,30 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \centering
 \begin{tabular}{r c }
  \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
+ Grid Architecture & 2 $\times$ 16\\ %\hline
+ \multirow{2}{*}{Inter Network N1} & $bw$=1Gbs, \\ %\hline
+                          & $lat$= From 8$\times$10$^{-6}$ to  $6.10^{-5}$ second \\
+ 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}
+\caption{Test conditions: network latency impacts}
 \label{tab:03}
 \end{table}
 
-
-
-\begin{figure} [ht!]
+\begin{figure} [htbp]
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impacts on execution time}
+\caption{Network latency impacts on execution time
+\AG{\np{E-6}}}
 \label{fig:03}
 \end{figure}
 
+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 which means that the GMRES seems tolerate more the network latency variation with a less  rate increase  of  the  execution time. However, the execution time factor between the two algorithms varies from 2.2 to 1.5 times with a network latency decreasing from $8.10^{-6}$ to  $6.10^{-5}$.
 
-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 $8.10^{-6}$.
+\RC{Les  2  prÃ©cÃ©dentes phrases  me  semblent en contradiction....}  
+\RCE{Reformule}
 
 \subsubsection{Network bandwidth impacts on performance}
 \ \\
@@ -635,19 +689,23 @@ order of magnitude with a latency of $8.10^{-6}$.
 \centering
 \begin{tabular}{r c }
  \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 \\
+ Grid Architecture & 2 $\times$ 16\\ %\hline
+\multirow{2}{*}{Inter Network N1} & $bw$=From 1Gbs to 10 Gbs \\ %\hline
+                          & $lat$= 5.10$^{-5}$ second \\
+ 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}
+\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
+\RCE{C est le bw}
 \label{tab:04}
 \end{table}
 
 
-\begin{figure} [ht!]
+\begin{figure} [htbp]
 \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.}
+%\RCE{Corrige}
 \label{fig:04}
 \end{figure}
 
@@ -663,16 +721,16 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
 \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
+ Grid Architecture & 4 $\times$ 8\\ %\hline
+ Inter Network & $bw$=1Gbs - $lat$=5.10$^{-5}$ \\
+ Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 40$^{3}$ to 200$^{3}$\\ \hline
  \end{tabular}
 \caption{Test conditions: Input matrix size impacts}
 \label{tab:05}
 \end{table}
 
 
-\begin{figure} [ht!]
+\begin{figure} [htbp]
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
 \caption{Problem size impacts on execution time}
@@ -685,9 +743,11 @@ In these experiments, the input matrix size  has been set from $N_{x} = N_{y}
 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 important increase ($10$ times)  of the number of iterations needed to
+    reach the convergence for the classical GMRES algorithm particularly, when the matrix size
+    go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itÃ©rations est 10 fois plus long mais la suite de la phrase ne veut rien dire}
+    \RCE{Le nombre d'iterations augmente de 10 fois, cela surtout a partir de N=150}
+    
 \item the  classical GMRES execution time  is almost the double  for $N_{x}=140$
   compared with the Krylov multisplitting method.
 \end{enumerate}
@@ -695,17 +755,18 @@ GMRES algorithm when  the matrix size go beyond $N_{x}=150$;
 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.
+grid 2 $\times$ 16 leading to the same conclusion.
 
 \subsubsection{CPU Power impacts on performance}
 
-\begin{table} [ht!]
+\begin{table} [htbp]
 \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 x 150 x 150\\ \hline
+ Grid architecture & 2 $\times$ 16\\ %\hline
+ Inter Network & N2 : $bw$=1Gbs - $lat$=5.10$^{-5}$ \\ %\hline
+ Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ 
+ CPU Power & From 3 to 19 GFlops \\ \hline
  \end{tabular}
 \caption{Test conditions: CPU Power impacts}
 \label{tab:06}
@@ -723,10 +784,16 @@ 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 rÃ©sultats dÃ©jÃ 
+%obtenus en grandeur rÃ©elle. Donc c'est une aide prÃ©cieuse pour les dev. Pas
+%besoin de dÃ©ployer sur une archi rÃ©elle}
 
-\DL{il faut une conclusion sur ces tests : ils confirment les rÃ©sultats dÃ©jÃ 
-obtenus en grandeur rÃ©elle. Donc c'est une aide prÃ©cieuse pour les dev. Pas
-besoin de dÃ©ployer sur une archi rÃ©elle}
+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}.
 
 
 \subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
@@ -744,25 +811,27 @@ 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.
+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  ration  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
- 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$^{-2}$\\
- Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
+ Input matrix size & $N_{x}$ = From 62 to 150\\ %\hline
  Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}
 \caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
@@ -773,7 +842,7 @@ 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
+Table~\ref{tab:08}  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
@@ -788,7 +857,7 @@ geographically distant clusters through the internet.
     \end{tabular}}
 
 
-\begin{figure}[!t]
+\begin{table}[!t]
 \centering
 %\begin{table}
 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
@@ -804,7 +873,7 @@ 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
-    size (N)
+    size ($N^3$)
     & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\
     \hline
     Precision
@@ -816,22 +885,48 @@ geographically distant clusters through the internet.
   \end{mytable}
 %\end{table}
  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
- \label{fig:07}
-\end{figure}
+ \label{tab:08}
+\end{table}
 
 
 \section{Conclusion}
-CONCLUSION
 
+In this paper we have presented the simulation of the execution of three
+different parallel solvers on some multi-core architectures. We have show that
+the SimGrid toolkit is an interesting simulation tool that has allowed us to
+determine  which method  to choose  given a  specified multi-core  architecture.
+Moreover the simulated results are in accordance (i.e. with the same order of
+magnitude)  with the works  presented in~\cite{couturier15}. Simulated   results
+also  confirm  the   efficiency  of  the asynchronous  multisplitting
+algorithm  compared  to  the   synchronous  GMRES especially in case of
+geographically distant clusters.
 
-\section*{Acknowledgment}
+These results are important since it is very  time consuming to find optimal
+configuration  and deployment requirements for a given application  on   a given
+multi-core  architecture. Finding   good  resource allocations policies under
+varying CPU power, network speeds and  loads is very challenging and  labor
+intensive. 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 very different execution times.
 
-This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 
+In future works, we  plan to investigate how to simulate  the behavior of really
+large scale  applications. For  example, if  we are  interested to  simulate the
+execution of the solvers of this paper with thousand or even dozens of thousands
+or core,  it is not possible  to do that with  SimGrid. In fact, this  tool will
+make the real computation. So we plan to focus our research on that problematic.
+
+
+
+%\section*{Acknowledgment}
+\ack
+This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 
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+
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