fin correct

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index f1247fe8886326bd3fc53b1b60b9435440680285..bc1b1b4c9870e3f8a12a9a0f203ca84ef1160653 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -1,4 +1,4 @@
-\documentclass[times]{cpeauth}
+ \documentclass[times]{cpeauth}

 
 \usepackage{moreverb}
 
 
 \usepackage{moreverb}
 


@@ -88,7 +88,7 @@
   Femto-ST Institute, DISC Department,
   University of Franche-Comté,
   Belfort, France.
   Femto-ST Institute, DISC Department,
   University of Franche-Comté,
   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,
   \affilnum{2}
   Department of Aerospace \& Mechanical Engineering,
   Non Linear Computational Mechanics,


@@ -113,7 +113,7 @@
 %% execution time.
 %% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
 
 %% execution time.
 %% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
 

-The behavior of multi-core applications is always a challenge to predict, especially with a new architecture for which no experiment has been performed. With some applications, it is difficult, if not impossible, to build accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications.
+The behavior of multi-core applications 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.
 
 
 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.
 


@@ -133,11 +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
 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
 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
 
 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


@@ -148,26 +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
 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
 \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).
 
 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
 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.
 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.


@@ -180,8 +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.
 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
 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


@@ -189,8 +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.
 
 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.
 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.


@@ -206,20 +206,20 @@ concluding remarks and perspectives.
 \section{The asynchronous iteration model and the motivations of our work}
 \label{sec:asynchro}
 
 \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
 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
 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
 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
 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
 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


@@ -230,7 +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
 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
 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


@@ -239,54 +239,90 @@ 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
 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}
 
 
 \section{SimGrid}
 \label{sec:simgrid}
 

-In the scope of this paper, the Simgrid
-toolkitSimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile},
-an open source framework actively developped by its community, has been choosen
-to simulate the behavior of the solvers algorithms in different grid
-computational 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). 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.
+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.

 
 
 
 
 
 


@@ -353,7 +389,7 @@ SimGrid simulations.
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}

-In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
+In this paper we focus on two-stage multisplitting methods in 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}
 \begin{equation}
 Ax=b,
 \label{eq:01}


@@ -363,12 +399,12 @@ 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}
 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}
 \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~\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}[htpb]
 %\begin{algorithm}[t]
 
 \begin{figure}[htpb]
 %\begin{algorithm}[t]


@@ -389,7 +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}
 
 %\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}
 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}


@@ -406,7 +442,7 @@ 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}
 \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}[htbp]
 %\begin{algorithm}[t]
 
 \begin{figure}[htbp]
 %\begin{algorithm}[t]


@@ -439,9 +475,9 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 
 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
 
 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,
+test reproducibility  under similar  conditions.  According to  our experience,

 very  few modifications  are required  to adapt  a MPI  program for  the SimGrid
 very  few modifications  are required  to adapt  a MPI  program for  the SimGrid

-simulator using SMPI (Simulator MPI). The  first modification is to include SMPI
+simulator using SMPI (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"
 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"


@@ -449,10 +485,10 @@ SimGrid selector       called      "runtime       automatic      switching"
 effects on runtime between the threads running in the same process and generated by
 SimGrid  to simulate the  grid environment.
 
 effects on runtime between the threads running in the same process and generated by
 SimGrid  to simulate the  grid environment.
 

-\paragraph{Parameters of the simulation in SimGrid}
+\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
 \  \\ \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}
 inter-cluster communications. In the following, these parameters are described:
 
 \begin{itemize}


@@ -527,8 +563,8 @@ 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.
 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
+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. \\
 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. \\


@@ -544,7 +580,7 @@ have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times
 represents the number  of clusters in the grid and  the second number represents
 the number  of hosts (processors/cores)  in each  cluster. \\
 
 represents the number  of clusters in the grid and  the second number represents
 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.  \\
 within these configurations by varying the key parameters, especially
 the CPU power capacity, the network parameters and also the size of the
 input data.  \\


@@ -564,39 +600,38 @@ Another important factor  impacting the overall performance  of the application
 is the network configuration. Two main network parameters can modify drastically
 the program output results:
 \begin{enumerate}
 is the network configuration. Two main network parameters can modify drastically
 the program output results:
 \begin{enumerate}

-\item  the network  bandwidth  ($bw$ in 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  ($lat$ in microseconds) defined as  the delay  from the
     from one point to another in a unit of time.
 \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.
+  starting time to send  a simple data from a source to a destination.

 \end{enumerate}
 \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.
 
 and  between distant  clusters.  This parameter is application dependent.
 

- In  a grid  environment, it  is common  to distinguish,  on one hand,  the
+ 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
  \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. In opposite, the
- inter-network connects clusters sometime via  heterogeneous networks components
- through internet with a lower speed.  The network  between distant  clusters
+ 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
  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 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
+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
 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 $bw$=10Gbs and the latency
+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.
 
 $lat=8\mu$s. In what follows, we will present the test conditions, the output
 results and our comments.
 


@@ -605,8 +640,8 @@ results and our comments.
 \begin{tabular}{ll}
 \hline
 Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
 \begin{tabular}{ll}
 \hline
 Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\

-\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
-                                        & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
+\multirow{2}{*}{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}
 \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}


@@ -624,16 +659,16 @@ 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
 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 splitting (in our case, it is equal to the number of clusters) influences on the
-convergence speed. The higher the number  of splitting is, the slower the
+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).
 
 convergence of the algorithm is (see the output results obtained from
 configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
 8$\times$8).
 

-The execution times between both algorithms is significant with different grid
+The execution times between both algorithms are significant with different grid

 architectures. The synchronous Krylov two-stage algorithm presents better
 architectures. The synchronous Krylov two-stage algorithm presents better

-performances than the GMRES algorithm, even for a high number of clusters (about
-$32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can
+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
 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


@@ -669,7 +704,7 @@ efficient for distributed systems with high latency networks.
 \end{figure}
 
 \subsubsection{Network latency impacts on performances\\}
 \end{figure}
 
 \subsubsection{Network latency impacts on performances\\}

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


@@ -682,9 +717,8 @@ Figure~\ref{fig:03} shows the impact of the network latency on the performances
 
 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
 
 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 $150^3$. The results of increasing the
-network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
-both algorithms by reducing the execution times. However, the Krylov two-stage
+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.
 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.


@@ -698,7 +732,7 @@ GMRES algorithm.
 
 \subsubsection{Matrix size impacts on performances\\}
 
 
 \subsubsection{Matrix size impacts on performances\\}
 

-In these experiments, the matrix size of the 3D Poisson problem is varied from
+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
 $50^3$ 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


@@ -709,7 +743,7 @@ 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.
 observe that for some problem sizes, the convergence (and thus the execution
 time) of the Krylov two-stage algorithm varies quite a lot.
 %This is due to the 3D partitioning of the 3D matrix of the Poisson problem.

-These findings may help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up.
+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
 
 \begin{figure}[ht]
 \centering


@@ -739,14 +773,14 @@ processors.
 
 To conclude these series of experiments, with  SimGrid we have been able to make
 many simulations  with many parameters  variations. Doing all  these experiments
 
 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 or very costly. Moreover
+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{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
 
 the behavior of both GMRES and  Krylov two-stage algorithms is in accordance
 with larger real executions on large scale supercomputers~\cite{couturier15}.
 
 
 \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
 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


@@ -755,8 +789,8 @@ 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  iterations  freely without  any
 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  iterations  freely without  any

-synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
-theoretically reduce  the overall execution  time and can improve  the algorithm
+synchronization  with   the  other   processors.  Thus,  an   asynchronous algorithm  may
+theoretically reduce  the overall execution  time and can also improve  the algorithm

 performance.
 
 In this section,  the SimGrid simulator is  used to compare the  behavior of the
 performance.
 
 In this section,  the SimGrid simulator is  used to compare the  behavior of the


@@ -777,12 +811,12 @@ summarized in Table~\ref{tab:02}.
  \hline
  Grid architecture                       & 2$\times$50 totaling 100 processors\\
  Processors Power                        & 1 GFlops to 1.5 GFlops \\
  \hline
  Grid architecture                       & 2$\times$50 totaling 100 processors\\
  Processors Power                        & 1 GFlops to 1.5 GFlops \\

- \multirow{2}{*}{Network inter-clusters} & $bw$=1.25 Gbits, $lat=50\mu$s \\
-                                         & $bw$=5 Mbits, $lat=20ms$\\
+ \multirow{2}{*}{Network inter-clusters} & $bw$: 5 Mbits to 50 Mbits\\
+                                         & $lat$: 20 ms\\

  Matrix size                             & from $62^3$ to $150^3$\\
  Matrix size                             & from $62^3$ to $150^3$\\

- Residual error precision                & $10^{-5}$ to $10^{-9}$\\ \hline \\
+ Residual error precision                & $10^{-5}$ to $10^{-11}$\\ \hline \\

  \end{tabular}
  \end{tabular}

-\caption{Test conditions: GMRES in synchronous mode vs. Krylov two-stage in asynchronous mode}
+\caption{Test conditions: GMRES in synchronous mode vs. two-stage multisplitting in asynchronous mode}

 \label{tab:02}
 \end{table}
 
 \label{tab:02}
 \end{table}
 


@@ -823,7 +857,7 @@ summarized in Table~\ref{tab:02}.
     \hline
   \end{mytable}
 %\end{table}
     \hline
   \end{mytable}
 %\end{table}

- \caption{Relative gains of the two-stage multisplitting algorithm compared with the classical GMRES}
+ \caption{Relative gains of the asynchronous two-stage multisplitting algorithm compared to the classical synchronous GMRES algorithm}

  \label{tab:03}
 \end{table}
 
  \label{tab:03}
 \end{table}
 


@@ -831,8 +865,8 @@ summarized in Table~\ref{tab:02}.
 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
 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.  In  average,  the  two-stage
-multisplitting algorithm to  be more than $2.5$ times faster  than the classical
+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.
 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.


@@ -851,19 +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
 
 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
 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
 
 
 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
 of cores,  it is not possible  to do that with  SimGrid. In fact, this  tool will
 execution of the solvers of this paper with thousand or even dozens of thousands
 of cores,  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.
+make the real computation. That is why, we plan to focus our research on that problematic.

 
 
 
 
 
 


@@ -883,3 +917,7 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 %%% fill-column: 80
 %%% ispell-local-dictionary: "american"
 %%% End:
 %%% fill-column: 80
 %%% ispell-local-dictionary: "american"
 %%% End:

+
+%  LocalWords:  Ramamonjisoa Ziane Khodja Laiymani Raphaël Arnaud Giersch Femto
+%  LocalWords:  Franche Comté Belfort GMRES multisplitting SimGrid Krylov SMPI
+%  LocalWords:  MPI