X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/56dc4d55704617d8f459826573f8bd2beeb5b5b3..refs/heads/master:/paper.tex

diff --git a/paper.tex b/paper.tex
index 7fd9704..bc1b1b4 100644
--- a/paper.tex
+++ b/paper.tex
@@ -1,4 +1,4 @@
-\documentclass[times]{cpeauth}
+ \documentclass[times]{cpeauth}
 
 \usepackage{moreverb}
 
@@ -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}
+\usepackage{multirow}
+
 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
@@ -75,10 +77,10 @@
 %\itshape{\journalnamelc}\footnotemark[2]}
 
 \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}
+  Lilia Ziane Khodja\affil{2},
+  David Laiymani\affil{1},
+  RaphaÃ«l Couturier\affil{1} and
+  Arnaud Giersch\affil{1}
 }
 
 \address{
@@ -86,7 +88,7 @@
   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
+  Email:~\email{{charles.ramamonjisoa,david.laiymani,raphael.couturier,arnaud.giersch}@univ-fcomte.fr}\break
   \affilnum{2}
   Department of Aerospace \& Mechanical Engineering,
   Non Linear Computational Mechanics,
@@ -105,13 +107,13 @@
 %% 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.  
+%% 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. 
+%% 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 always proves quite challenging to predict, especially with a new architecture for which no experiment has yet 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.
 
@@ -131,11 +133,11 @@ complex parallel applications operating on a large amount of data.
 Unfortunately,  users (industrials or scientists),  who need such computational
 resources, may not have an easy access to such efficient architectures. The cost
 of using the platform and/or the cost of  testing and deploying an application
-are often very important. So, in this context it is difficult to optimize a
+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\dots{} In counterpart, the simulation results need to be consistent with the real ones.
+simulation environment.  The advantages are numerous: life cycle development,
+code debugging, ability to obtain results quickly\dots{} In return, 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
@@ -146,26 +148,26 @@ data dependencies to/from its neighbors and to iterate this process until the
 convergence of the method. Several well-known studies demonstrate the
 convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
 task cannot begin a new iteration while it has not received data dependencies
-from its neighbors. We say that the iteration computation follows a
+from its neighbors. The iteration computation is said to follow 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 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}. Even if the number of
 iterations required to converge is generally  greater  than for the synchronous
 case, it appears that the asynchronous  iterative scheme  can significantly
 reduce  overall execution times by  suppressing idle  times due to
 synchronizations~(see~\cite{bahi07} for more details).
 
-Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is very  time
-consuming to find optimal configuration  and deployment requirements for a given
+Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is extremely  time
+consuming to find optimal configurations  and deployment requirements for a given
 application  on   a  given   multi-core  architecture.  Finding   good  resource
 allocations policies under  varying CPU power, network speeds and  loads is very
 challenging and  labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
 problematic is  even more difficult  for the  asynchronous scheme where  a small
 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
+lead to very different numbers of iterations to reach the convergence and consequently 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.
@@ -178,8 +180,8 @@ 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.
+Moreover, the  obtained results  on different simulated  multi-core architectures
+confirm the  real results  previously obtained  on real physical 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
@@ -187,8 +189,8 @@ 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
+Thus, with a simple computing architecture (a laptop) SimGrid allows us
+to run a test campaign  of  real parallel iterative  applications on
 different simulated multi-core architectures.  To our knowledge, there is no
 related work on the large-scale multi-core simulation of a real synchronous and
 asynchronous iterative application.
@@ -204,20 +206,20 @@ 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 theoretically and
+Asynchronous iterative methods have been  studied for many years both 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,
+(i.e. problems  for which  the solution is  $x^\star =f(x^\star)$).  In practice,
 asynchronous iteration  methods can be used  to solve, for example,  linear and
-non-linear systems of equations or optimization problems, interested readers are
+non-linear systems of equations or optimization problems. Interested readers are
 invited to read~\cite{BT89,bahi07}.
 
 Before  using  an  asynchronous  iterative   method,  the  convergence  must  be
-studied. Otherwise, the  application is not ensure to reach  the convergence. An
+studied. Otherwise, there is no guarantee that the  application will reach  the convergence. An
 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
+practice, only  the communications management and  the 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
 used in both  modes. Concerning the convergence  detection, synchronous variants
 can use  a global convergence procedure  which acts as a  global synchronization
@@ -228,7 +230,7 @@ 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 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
+number of iterations is the same as in the sequential mode. Thus, the
 set of the parameters  of the  platform (number  of nodes,  power of nodes,
 inter and  intra clusters  bandwidth  and  latency,~\ldots) and  of  the
 application can drastically change the number of iterations required to get the
@@ -237,72 +239,157 @@ optimize since the financial and deployment costs on large scale multi-core
 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 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.
 
 \section{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.
+In the scope of this paper, we have chosen the SimGrid
+toolkit~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
+to simulate the behavior of parallel iterative linear solvers on different
+computational grid configurations. In opposite to most of the simulators which
+are stayed very application-oriented, the SimGrid framework is designed to study
+the behavior of many large-scale distributed computing platforms as Grids,
+Peer-to-Peer systems, Clouds or High Performance Computation systems. It is
+still actively developed by the scientific community and distributed as an open
+source software.
+
+SimGrid provides four user interfaces which can be convenient for different
+distributed applications.  In this paper we are interested on the SMPI
+(Simulated MPI) user interface which implements about \np[\%]{80} of the MPI 2.0
+standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and allows minor
+modifications of the initial code (see Section~\ref{sec:04.02}). SMPI enables
+the direct simulation of the execution, as in the real life, of an unmodified
+MPI distributed application, and gets accurate results with the detailed
+resources consumption.
+
+SimGrid simulator uses an XML input file describing the computational grid
+resources: the number of clusters in the grid, the number of processors/cores in
+each cluster, the detailed description of the intra and inter networks and the
+list of the hosts in each cluster (see the details in
+Section~\ref{sec:expe}). SimGrid employs a fluid model to simulate the use of
+these resources along the program execution.  This model produces accurate
+results while still running relatively
+fast~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}.
+During the simulation, the computations are really executed, but the communications
+are intercepted and their execution time evaluated according to the parameters
+of the simulated platform. It is also possible for SimGrid/SMPI to only keep the
+duration of large computations by skipping them.  Moreover, when applicable, the
+application can be run by sharing some in-memory structures between the
+simulated processes and thus allowing the use of very large-scale data.
+
+The choice of SimGrid/SMPI as a simulator tool in this study has been emphasized
+by the results obtained by several studies to validate, in the real
+environments, the behavior of different network models simulated in
+SimGrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline
+the comparison between the real MPI application executions and the SimGrid/SMPI
+ones~\cite{guermouche+renard.2010.first,clauss+stillwell+genaud+al.2011.single,bedaride+degomme+genaud+al.2013.toward}. These
+works show the accuracy of SimGrid simulations compared to the executions on
+real physical architectures.
+
+%% In the scope of this paper, the SimGrid toolkit~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile},
+%% an open source framework actively developed by its scientific community, has been chosen to simulate the behavior of iterative linear solvers in different computational grid configurations. SimGrid pretends to be non-specialized in opposite to some other simulators which stayed to be very specific oriented-application. One of the well-known SimGrid advantage is its SMPI (Simulated MPI) user interface. SMPI purpose is to execute by simulation in a similar way as in real life, an MPI distributed application and to get accurate results with the detailed resources
+%% consumption.Several studies have demonstrated the accuracy of the simulation
+%% compared with execution on real physical architectures. In addition of SMPI,
+%% Simgrid provides other API which can be convienent for different distrbuted
+%% applications: computational grid applications, High Performance Computing (HPC),
+%% P2P but also clouds applications. In this paper we use the SMPI API. It
+%% implements about \np[\%]{80} of the MPI 2.0 standard and allows minor
+%% modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward}
+%% (see Section~\ref{sec:04.02}).
+
+
+%%  Provided as an input to the simulator, at least $3$ XML files describe the
+%%  computational grid resources: number of clusters in the grid, number of
+%%  processors/cores in each cluster, detailed description of the intra and inter
+%%  networks and the list of the hosts in each cluster (see the details in Section~\ref{sec:expe}). Simgrid uses a fluid model to simulate the program execution.
+%%  This gives several simulation modes which produce accurate
+%%  results~\cite{bedaride+degomme+genaud+al.2013.toward,
+%%  velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode
+%%  really executes the computation but "intercepts" the communications (running
+%%  time is then evaluated according to the parameters of the simulated platform).
+%%  It is also possible for SimGrid/SMPI to only keep duration of large
+%%  computations by skipping them. Moreover the application can be run "in vitro"
+%%  by sharing some in-memory structures between the simulated processes and
+%%  thus allowing the use of very large data scale.
+
+
+%% The choice of Simgrid/SMPI as a simulator tool in this study has been emphasized
+%% by the results obtained by several studies to validate, in real environments,
+%% the behavior of different network models simulated in
+%% Simgrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline
+%% the comparison between real MPI executions  and SimGrid/SMPI
+%% ones\cite{guermouche+renard.2010.first, clauss+stillwell+genaud+al.2011.single,
+%% bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of
+%% SimGrid simulations.
+
+
+
+
+
+
+% 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}
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}
-In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
+In this paper we focus on two-stage multisplitting methods in both their versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
 \begin{equation}
 Ax=b,
 \label{eq:01}
@@ -312,14 +399,14 @@ where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side
 x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
 \label{eq:02}
 \end{equation}
-where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel so that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:
 \begin{equation}
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{equation}
-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
+where the 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. Line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using the 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]
@@ -338,7 +425,7 @@ where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are compute
 %\end{algorithm}
 \end{figure}
 
-In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the asynchronous model which allows communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged:
+In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the asynchronous scheme which allows communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged:
 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}
@@ -355,9 +442,9 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde
 \min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
 \label{eq:06}
 \end{equation}
-The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
+The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using the CGLS method~\cite{Hestenes52} sosuch 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]
@@ -386,37 +473,31 @@ 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}
 
-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
-simulator using SMPI (Simulator MPI). The  first modification is to include SMPI
-libraries  and related  header files  (smpi.h).  The  second modification  is to
+test reproducibility  under similar  conditions.  According to  our experience,
+very  few modifications  are required  to adapt  a MPI  program for  the SimGrid
+simulator using SMPI (Simulated MPI). The  first modification is to include SMPI
+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.
-
-%\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{Simulation parameters for 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
+in which  several clusters are  geographically distant,  so that 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,
-\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:
@@ -424,8 +505,8 @@ 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,
@@ -434,19 +515,18 @@ In addition, the following arguments are given to the programs at runtime:
       	\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}
 
-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:
 \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
@@ -465,7 +545,13 @@ where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$
 \end{equation}
 until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.
 
-In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic block of the problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries.
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$
+sub-problems such that $L$ is the number of clusters and $p$ is the number of
+processors in each cluster. We apply the three-dimensional partitioning instead
+of the row-by-row one in order to reduce the size of the data shared at the
+sub-problems boundaries. In this case, each processor is in charge of
+parallelepipedic block of the problem and has at most six neighbors in the same
+cluster or in distant clusters with which it shares data at boundaries.
 
 \subsection{Study setup and simulation methodology}
 
@@ -477,10 +563,10 @@ the application to be tested. Numerical parallel iterative algorithms
 have been chosen for the study in this paper. \\
 
 \textbf{Step 2}: Collect the software materials needed for the experimentation.
-In our case, we have two variants algorithms for the resolution of the
-3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting
-method. In addition, the Simgrid simulator has been chosen to simulate the
-behaviors of the distributed applications. Simgrid is running in a virtual
+In our case, we have two variants for the resolution of the
+3D-Poisson problem: (1) using the classical GMRES; (2) using 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
@@ -488,24 +574,20 @@ 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. \\
 
-\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$50. The first number
 represents the number  of clusters in the grid and  the second number represents
-the number  of hosts (processors/cores)  in each  cluster. The network  has been
-designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
-latency of 8.10$^{-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
+\textbf{Step 5}: Conduct  extensive and comprehensive testings
 within these configurations by varying the key parameters, especially
 the CPU power capacity, the network parameters and also the size of the
 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
@@ -518,272 +600,187 @@ 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
-    capacity" of the network is defined as  the maximum of data that can transit
+\item  the network  bandwidth  ($bw$ in Gbits/s) also  known  as "the  data-carrying
+    capacity" of the network is defined as  the maximum amount of data that can transit
     from one point to another in a unit of time.
-\item the  network latency  (lat :  microsecond) defined as  the delay  from the
-  start time to send  a simple data from a source to a destination.
+\item the  network latency  ($lat$ in microseconds) defined as  the delay  from the
+  starting 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
+Among  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster
 and  between distant  clusters.  This parameter is application dependent.
 
- In  a grid  environment, it  is common  to distinguish,  on the  one hand,  the
- "intra-network" which refers  to the links between nodes within  a cluster and
- on  the other  hand, the  "inter-network" which  is the  backbone link  between
- clusters.  In   practice,  these  two   networks  have  different   speeds.
- The intra-network  generally works  like a  high speed  local network  with a
- high bandwith and very low latency. In opposite, the inter-network connects
- clusters sometime via  heterogeneous networks components  throuth internet with
- a lower speed.  The network  between distant  clusters might  be a  bottleneck
- for  the global performance of the application.
-
-\subsection{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.
-
-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
+ In  a grid  environment, it  is common  to distinguish,  on the one hand,  the
+ \textit{intra-network} which refers  to the links between nodes within  a
+ cluster and on  the other  hand, the  \textit{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 bandwidth and very low latency. On the contrary, the
+ inter-network connects clusters sometimes via  heterogeneous networks components the through internet with a lower speed.  The network  between distant  clusters
+ might  be a  bottleneck for  the global performance of the application.
+
+
+\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 performances than the
+classical GMRES method. With a synchronous iterative method, better performances
+mean a smaller number of iterations and execution time before reaching the
+convergence.
+
+Table~\ref{tab:01} summarizes the parameters used in the different simulations:
+the grid architectures (i.e. the number of clusters and the number of nodes per
+cluster), 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 $bw$=10Gbit/s 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{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 dÃ©fini..}\RC{Nx est dÃ©fini, 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} & $N1$: $bw$=10Gbit/s, $lat=8\mu$s \\
+                                        & $N2$: $bw$=1Gbit/s, $lat=50\mu$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}
 
-
-
-
-
-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!]
-  \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 sÃ©parateur dÃ©cimal 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 trÃ¨s clair, c'est pas prÃ©cis de dire qu'un algo perform mieux qu'un autre, selon quel critÃ¨re?}
-
-\subsubsection{Running on two different inter-clusters network speeds \\}
-
-\begin{table} [ht!]
+\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 of 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 splittings (in our case, it is equal to the number of clusters) influences on the
+convergence speed. The higher the number  of splittings 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 are 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 (it is about
+$32\%$ more efficient on a grid of 8$\times$8 than the 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$16).
+
+\begin{figure}[ht]
 \begin{center}
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8.10$^{-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{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 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  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 two 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}
-\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 two networks parameters: $N1$ vs. $N2$}
+%\LZK{CE, remplacer les ``,'' des dÃ©cimales par un ``.''}
+%\RCE{ok}
 \label{fig:02}
 \end{figure}
-%\end{wrapfigure}
-
-
-\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}
-
 
+\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 $bw$=1Gbit/s 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.
 
-\begin{figure} [ht!]
+\begin{figure}[ht]
 \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}
 
+\subsubsection{Network bandwidth impacts on performances\\}
 
-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  prÃ©cÃ©dentes 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}
-
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
+$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
+solve a 3D Poisson problem of size $150^3$. Increasing the
+network bandwidth from $1$Gbit/s to $10$Gbit/s results in improving the performances of both algorithms by reducing the execution times. However, the Krylov two-stage
+algorithm presents a better performance gain in the considered bandwidth
+interval with a gain of $40\%$ compared to only about $24\%$ for the classical
+GMRES algorithm.
 
-\begin{figure} [ht!]
+\begin{figure}[ht]
 \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}
 
-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{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!]
+\subsubsection{Matrix size impacts on performances\\}
+
+In these experiments, the matrix size of the 3D Poisson problem varies 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}). As shown in Figure~\ref{fig:05}, the execution
+times for both algorithms increase with increased matrix sizes.  For all problem
+sizes, the 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 greatly 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.
+
+\begin{figure}[ht]
 \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}
 
-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'itÃ©rations 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 performances\\}
 
-\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}
+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 on a grid of $2\times16$ 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}
-\caption{CPU Power impacts on execution time}
+\caption{CPU Power impacts on performances}
 \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 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}.
+with a real platform is most of the time impossible or very costly. Moreover
+the behavior of 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{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
 
-The previous paragraphs  put in evidence the interests to  simulate the behavior
+The previous paragraphs  put in evidence the interest to  simulate the behavior
 of  the application  before  any  deployment in  a  real  environment.  In  this
 section, following  the same previous  methodology, our  goal is to  compare the
 efficiency of the multisplitting method  in \textit{ asynchronous mode} compared with the
@@ -791,45 +788,38 @@ classical GMRES in \textit{synchronous mode}.
 
 The  interest of  using  an asynchronous  algorithm  is that  there  is no  more
 synchronization. With  geographically distant  clusters, this may  be essential.
-In  this case,  each  processor can  compute its  iteration  freely without  any
-synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
-theoretically reduce  the overall execution  time and can improve  the algorithm
+In  this case,  each  processor can  compute its  iterations  freely without  any
+synchronization  with   the  other   processors.  Thus,  an   asynchronous algorithm  may
+theoretically reduce  the overall execution  time and can also 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
+two-stage  algorithm  in  asynchronous  mode with  GMRES  in  synchronous  mode.
+Several benchmarks  have been  performed with various  combinations of  the grid
+resources  (CPU,  Network,  matrix  size,   \ldots).  The  test  conditions  are
+summarized in Table~\ref{tab:02}.
 
 
-The test conditions are summarized in the table~\ref{tab:07}: \\
 
-\begin{table} [ht!]
+%\LZK{Quelle table repporte les gains relatifs?? SÃ»rement pas Table II !!}
+%\RCE{Table III avec la nouvelle numerotation}
+
+
+\begin{table}[htbp]
 \centering
-\begin{tabular}{r c }
+\begin{tabular}{ll}
  \hline
- Grid Architecture & 2x50 totaling 100 processors\\ %\hline
- Processors Power & 1 GFlops to 1.5 GFlops\\
-   Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
-   Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
- Input matrix size & $N_{x}$ = From 62 to 150\\ %\hline
- Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
+ Grid architecture                       & 2$\times$50 totaling 100 processors\\
+ Processors Power                        & 1 GFlops to 1.5 GFlops \\
+ \multirow{2}{*}{Network inter-clusters} & $bw$: 5 Mbits to 50 Mbits\\
+                                         & $lat$: 20 ms\\
+ Matrix size                             & from $62^3$ to $150^3$\\
+ Residual error precision                & $10^{-5}$ to $10^{-11}$\\ \hline \\
  \end{tabular}
-\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
-\label{tab:07}
+\caption{Test conditions: GMRES in synchronous mode vs. two-stage multisplitting in asynchronous mode}
+\label{tab:02}
 \end{table}
 
-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
-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
-geographically distant clusters through the internet.
 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
@@ -856,7 +846,7 @@ 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
@@ -867,15 +857,24 @@ geographically distant clusters through the internet.
     \hline
   \end{mytable}
 %\end{table}
- \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
- \label{tab:08}
+ \caption{Relative gains of the asynchronous two-stage multisplitting algorithm compared to the classical synchronous GMRES algorithm}
+ \label{tab:03}
 \end{table}
 
 
-\section{Conclusion}
+Table~\ref{tab:03} reports  the relative gains  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.  On  average,  the  two-stage
+multisplitting algorithm is more than $2.5$ times faster  than the classical
+GMRES.  These experiments also show the relative tolerance of the multisplitting
+algorithm when using a low speed network as usually observed with geographically
+distant clusters through the internet.
+
 
+\section{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
+different parallel solvers on some multi-core architectures. We have shown 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
@@ -886,19 +885,19 @@ geographically distant clusters.
 
 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
+multi-core  architecture. Finding   good  resource allocation 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.
+convergeence and consequently to very different execution times.
 
 
 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
+large scale  applications. For  example, if  we are  interested in  simulating 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.
+of cores,  it is not possible  to do that with  SimGrid. In fact, this  tool will
+make the real computation. That is why, we plan to focus our research on that problematic.
 
 
 
@@ -918,3 +917,7 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 %%% fill-column: 80
 %%% ispell-local-dictionary: "american"
 %%% End:
+
+%  LocalWords:  Ramamonjisoa Ziane Khodja Laiymani RaphaÃ«l Arnaud Giersch Femto
+%  LocalWords:  Franche ComtÃ© Belfort GMRES multisplitting SimGrid Krylov SMPI
+%  LocalWords:  MPI
