Easy corrections requested by reviewer #2.

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index 5559af7edc44594e909e52e27ed79f2e3e9076e2..4ff46b6d368fe5a7b4efe4b8f0f994baf496a3c6 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -1,4 +1,3 @@
-

 \documentclass[conference]{IEEEtran}
 
 \usepackage[T1]{fontenc}
 \documentclass[conference]{IEEEtran}
 
 \usepackage[T1]{fontenc}


@@ -77,10 +76,10 @@
 Synchronous  iterative  algorithms  are  often less  scalable  than  asynchronous
 iterative  ones.  Performing  large  scale experiments  with  different kind  of
 network parameters is not easy  because with supercomputers such parameters are
 Synchronous  iterative  algorithms  are  often less  scalable  than  asynchronous
 iterative  ones.  Performing  large  scale experiments  with  different kind  of
 network parameters is not easy  because with supercomputers such parameters are

-fixed. So one  solution consists in using simulations first  in order to analyze
-what parameters  could influence or not  the behaviors of an  algorithm. In this
-paper, we show  that it is interesting to use SimGrid  to simulate the behaviors
-of asynchronous  iterative algorithms. For that,  we compare the  behaviour of a
+fixed. So, one  solution consists in using simulations first  in order to analyze
+what parameters  could influence or not  the behavior of an  algorithm. In this
+paper, we show  that it is interesting to use SimGrid  to simulate the behavior
+of asynchronous  iterative algorithms. For that,  we compare the  behavior of a

 synchronous  GMRES  algorithm  with  an  asynchronous  multisplitting  one  with
 simulations  which let us easily choose  some parameters.   Both  codes  are real  MPI
 codes and simulations allow us to see when the asynchronous multisplitting algorithm can be more
 synchronous  GMRES  algorithm  with  an  asynchronous  multisplitting  one  with
 simulations  which let us easily choose  some parameters.   Both  codes  are real  MPI
 codes and simulations allow us to see when the asynchronous multisplitting algorithm can be more


@@ -93,8 +92,8 @@ efficient than the GMRES one to solve a 3D Poisson problem.
 
 \section{Introduction}
 
 
 \section{Introduction}
 

-Parallel computing and high performance computing (HPC) are becoming  more and more imperative for solving various
-problems raised by  researchers on various scientific disciplines but also by industrial in  the field. Indeed, the
+Parallel computing and high performance computing (HPC) are becoming  more and more imperative to solve various
+problems raised by  researchers on various scientific disciplines but also by industrialists in  the field. Indeed, the

 increasing complexity of these requested  applications combined with a continuous increase of their sizes lead to  write
 distributed and parallel algorithms requiring significant hardware  resources (grid computing, clusters, broadband
 network, etc.) but also a non-negligible CPU execution time. We consider in this paper a class of highly efficient
 increasing complexity of these requested  applications combined with a continuous increase of their sizes lead to  write
 distributed and parallel algorithms requiring significant hardware  resources (grid computing, clusters, broadband
 network, etc.) but also a non-negligible CPU execution time. We consider in this paper a class of highly efficient


@@ -107,39 +106,37 @@ Parallelization of such algorithms generally involves the division of the proble
 into  several  \emph{blocks}  that  will  be  solved  in  parallel  on  multiple
 processing units. The latter will communicate each intermediate results before a
 new  iteration starts  and until  the  approximate solution  is reached.   These
 into  several  \emph{blocks}  that  will  be  solved  in  parallel  on  multiple
 processing units. The latter will communicate each intermediate results before a
 new  iteration starts  and until  the  approximate solution  is reached.   These

-parallel computations can be performed either in \emph{synchronous} mode where a
-new iteration  begins only  when all nodes  communications are completed,  or in
+parallel computations can be performed either in a \emph{synchronous} mode, where a
+new iteration  begins only  when all nodes  communications are completed,  or in an

 \emph{asynchronous}  mode where  processors can  continue independently  with no
 synchronization points~\cite{bcvc06:ij}. In this case, local computations do not
 need to  wait for  required data. Processors  can then perform  their iterations
 \emph{asynchronous}  mode where  processors can  continue independently  with no
 synchronization points~\cite{bcvc06:ij}. In this case, local computations do not
 need to  wait for  required data. Processors  can then perform  their iterations

-with the  data present at that time.  Even if the number  of iterations required
-before  the convergence  is generally  greater  than for  the synchronous  case,
+with the  data present at that time.  Even if the number of required iterations 
+before  the convergence  is generally  greater  than in  the synchronous  case,

 asynchronous  iterative algorithms  can significantly  reduce  overall execution
 times by  suppressing idle  times due to  synchronizations especially in  a grid
 computing context (see~\cite{Bahi07} for more details).
 
 asynchronous  iterative algorithms  can significantly  reduce  overall execution
 times by  suppressing idle  times due to  synchronizations especially in  a grid
 computing context (see~\cite{Bahi07} for more details).
 

-Parallel applications  based on a (synchronous or  asynchronous) iteration model
+Parallel applications  based on a synchronous or  asynchronous iteration model

 may have different configuration and deployment requirements.  Quantifying their
 resource  allocation  policies and  application  scheduling  algorithms in  grid
 may have different configuration and deployment requirements.  Quantifying their
 resource  allocation  policies and  application  scheduling  algorithms in  grid

-computing environments under varying load,  CPU power and network speeds is very
+computing environments under varying load,  CPU power and network speeds are very

 costly,       very        labor       intensive       and        very       time
 consuming~\cite{Calheiros:2011:CTM:1951445.1951450}.   The case  of asynchronous
 costly,       very        labor       intensive       and        very       time
 consuming~\cite{Calheiros:2011:CTM:1951445.1951450}.   The case  of asynchronous

-iterative algorithms  is even more problematic  since they are  very sensible to
+iterative algorithms  is even more problematic  since they are  very sensitive to

 the  execution environment  context.  For instance,  variations  in the  network
 bandwidth (intra and  inter-clusters), in the number and the  power of nodes, in
 the number  of clusters\dots{} can lead  to very different  number of iterations
 and so  to very  different execution times.   Then, it  appears that the  use of
 simulation tools to explore various  platform scenarios and to run large numbers
 the  execution environment  context.  For instance,  variations  in the  network
 bandwidth (intra and  inter-clusters), in the number and the  power of nodes, in
 the number  of clusters\dots{} can lead  to very different  number of iterations
 and so  to very  different execution times.   Then, it  appears that the  use of
 simulation tools to explore various  platform scenarios and to run large numbers

-of  experiments quickly  can  be very  promising.  In  this  way, the  use of  a
-simulation  environment  to execute  parallel  iterative  algorithms found  some
-interests in reducing the highly cost  of access to computing resources: (1) for
+of  experiments quickly  can  be very  promising.  
+
+Thus, using a simulation environment to execute parallel iterative algorithms can prove to be very interesting to reduce the highly cost  of access to computing resources: (1) for

 the  applications  development life  cycle  and in  code  debugging  (2) and  in
 production  to get  results  in a  reasonable  execution time  with a  simulated
 the  applications  development life  cycle  and in  code  debugging  (2) and  in
 production  to get  results  in a  reasonable  execution time  with a  simulated

-infrastructure not  accessible with physical  resources.  Indeed, the  launch of
-distributed  iterative asynchronous  algorithms to  solve a  given problem  on a
-large-scale  simulated  environment challenges  to  find optimal  configurations
-giving  the best  results  with  a lowest  residual  error and  in  the best  of
-execution time.
+infrastructure not  accessible with physical  resources. Indeed, to  find optimal  configurations
+giving  the best  results  with  a lowest  residual  error and  in  the best 
+execution time is very challenging  for large scale distributed  iterative asynchronous  algorithms

 
 
 To our knowledge,  there is no existing work on the  large-scale simulation of a
 
 
 To our knowledge,  there is no existing work on the  large-scale simulation of a


@@ -147,31 +144,32 @@ real asynchronous  iterative application.  {\bf The contribution  of the present
   paper can be  summarized in two main points}.  First we  give a first approach
 of the simulation  of asynchronous iterative algorithms using  a simulation tool
 (i.e.    the   SimGrid   toolkit~\cite{SimGrid}).    Second,  we   confirm   the
   paper can be  summarized in two main points}.  First we  give a first approach
 of the simulation  of asynchronous iterative algorithms using  a simulation tool
 (i.e.    the   SimGrid   toolkit~\cite{SimGrid}).    Second,  we   confirm   the

-effectiveness  of the  asynchronous  multisplitting algorithm  by comparing  its
-performance   with  the   synchronous  GMRES   (Generalized   Minimal  Residual) method
+efficiency  of the  asynchronous  multisplitting algorithm  by comparing  its
+performances   with  the   synchronous  GMRES   (Generalized   Minimal  Residual) method

 \cite{ref1}.  Both  these codes can  be used to  solve large linear  systems. In
 \cite{ref1}.  Both  these codes can  be used to  solve large linear  systems. In

-this  paper, we  focus  on  a 3D  Poisson  problem.  We  show,  that with  minor
+this  paper, we  focus  on  a 3D  Poisson  problem.  We  show  that, with  minor

 modifications of the initial MPI code,  the SimGrid toolkit allows us to perform
 a  test campaign  of  a  real asynchronous  iterative  application on  different
 computing architectures.
 % The  simulated results  we
 %obtained are  in line with real  results exposed in  ??\AG[]{ref?}. 
 modifications of the initial MPI code,  the SimGrid toolkit allows us to perform
 a  test campaign  of  a  real asynchronous  iterative  application on  different
 computing architectures.
 % The  simulated results  we
 %obtained are  in line with real  results exposed in  ??\AG[]{ref?}. 

-SimGrid  had  allowed us  to  launch the  application  from  a modest  computing
+SimGrid  has  allowed us  to  launch the  application  from  a modest  computing

 infrastructure  by simulating  different distributed  architectures  composed by
 clusters  nodes interconnected by  variable speed  networks.  Parameters  of the
 network  platforms  are   the  bandwidth  and  the  latency   of  inter  cluster
 network. Parameters on the cluster's architecture are the number of machines and
 the  computation power  of a  machine.  Simulations show  that the  asynchronous
 multisplitting algorithm  can solve the  3D Poisson problem  approximately twice
 infrastructure  by simulating  different distributed  architectures  composed by
 clusters  nodes interconnected by  variable speed  networks.  Parameters  of the
 network  platforms  are   the  bandwidth  and  the  latency   of  inter  cluster
 network. Parameters on the cluster's architecture are the number of machines and
 the  computation power  of a  machine.  Simulations show  that the  asynchronous
 multisplitting algorithm  can solve the  3D Poisson problem  approximately twice

-faster than GMRES with two distant clusters.
+faster than GMRES with two distant clusters. In this way, we present an original solution to optimize the use of a simulation 
+tool to run efficiently an  asynchronous iterative parallel algorithm in a grid architecture

 
 
 
 This article is structured as follows: after this introduction, the next section
 
 
 
 This article is structured as follows: after this introduction, the next section

-will  give a  brief  description  of iterative  asynchronous  model.  Then,  the
+will  give a  brief  description  of the iterative  asynchronous  model.  Then,  the

 simulation framework  SimGrid is presented  with the settings to  create various
 distributed architectures.  Then, the  multisplitting method is presented, it is
 simulation framework  SimGrid is presented  with the settings to  create various
 distributed architectures.  Then, the  multisplitting method is presented, it is

-based  on GMRES to  solve each  block obtained  of the  splitting. This  code is
+based  on GMRES to  solve each  block obtained  from the  splitting. This  code is

 written with MPI  primitives and its adaptation to  SimGrid with SMPI (Simulated
 MPI) is  detailed in the next  section. At last, the  simulation results carried
 out will be presented before some concluding remarks and future works.
 written with MPI  primitives and its adaptation to  SimGrid with SMPI (Simulated
 MPI) is  detailed in the next  section. At last, the  simulation results carried
 out will be presented before some concluding remarks and future works.


@@ -179,29 +177,29 @@ out will be presented before some concluding remarks and future works.
  
 \section{Motivations and scientific context}
 
  
 \section{Motivations and scientific context}
 

-As exposed in  the introduction, parallel iterative methods  are now widely used
-in  many scientific  domains.   They can  be  classified in  three main  classes
-depending on  how iterations  and communications are  managed (for  more details
-readers  can refer  to~\cite{bcvc06:ij}). In  the synchronous  iterations model,
-data are exchanged  at the end of each iteration. All  the processors must begin
-the same iteration  at the same time and important idle  times on processors are
+As described in the introduction, parallel iterative methods are now widely used
+in many scientific domains.  They can be classified in three main classes
+depending on how iterations and communications are managed (for more details
+readers can refer to~\cite{bcvc06:ij}). In the synchronous iterations model,
+data are exchanged at the end of each iteration. All the processors must begin
+the same iteration at the same time and important idle times on processors are

 generated.  It is possible to use asynchronous communications, in this case, the
 generated.  It is possible to use asynchronous communications, in this case, the

-model can be  compared to the previous one except that  data required on another
-processor are  sent asynchronously i.e.  without  stopping current computations.
-This technique  allows to partially  overlap communications by  computations but
-unfortunately, the overlapping is only  partial and important idle times remain.
-It is clear that, in a grid computing context, where the number of computational
-nodes is large,  heterogeneous and widely distributed, the  idle times generated
-by synchronizations are very penalizing. One  way to overcome this problem is to
-use the asynchronous iterations model.   Here, local computations do not need to
-wait for  required data. Processors can  then perform their  iterations with the
-data present  at that time.  Figure~\ref{fig:aiac} illustrates  this model where
-the gray blocks represent the  computation phases.  With this algorithmic model,
-the number  of iterations required  before the convergence is  generally greater
-than  for the  two former  classes.  But,  and as  detailed in~\cite{bcvc06:ij},
-asynchronous  iterative algorithms  can significantly  reduce  overall execution
-times by  suppressing idle  times due to  synchronizations especially in  a grid
-computing context.
+model can be compared to the previous one except that data required on another
+processor are sent asynchronously i.e.  without stopping current computations.
+This technique allows communications to be partially overlapped by computations
+but unfortunately, the overlapping is only partial and important idle times
+remain.  It is clear that, in a grid computing context, where the number of
+computational nodes is large, heterogeneous and widely distributed, the idle
+times generated by synchronizations are very penalizing. One way to overcome
+this problem is to use the asynchronous iterations model.  Here, local
+computations do not need to wait for required data. Processors can then perform
+their iterations with the data present at that time.  Figure~\ref{fig:aiac}
+illustrates this model where the gray blocks represent the computation phases.
+With this algorithmic model, the number of iterations required before the
+convergence is generally greater than for the two former classes.  But, and as
+detailed in~\cite{bcvc06:ij}, asynchronous iterative algorithms can
+significantly reduce overall execution times by suppressing idle times due to
+synchronizations especially in a grid computing context.

 
 \begin{figure}[!t]
   \centering
 
 \begin{figure}[!t]
   \centering


@@ -225,20 +223,20 @@ computing context.
 %% \AG{Several works\dots{} what?\\
 %  Le paragraphe suivant se trouve déjà dans l'intro ?}
 In the context of asynchronous algorithms, the number of iterations to reach the
 %% \AG{Several works\dots{} what?\\
 %  Le paragraphe suivant se trouve déjà dans l'intro ?}
 In the context of asynchronous algorithms, the number of iterations to reach the

-convergence depends on  the delay of messages. With  synchronous iterations, the
+convergence depends on  the delay of the messages. With  synchronous iterations, the

 number of  iterations is exactly  the same than  in the sequential mode  (if the
 parallelization process does  not change the algorithm). So  the difficulty with
 number of  iterations is exactly  the same than  in the sequential mode  (if the
 parallelization process does  not change the algorithm). So  the difficulty with

-asynchronous iterative algorithms comes from the fact it is necessary to run the algorithm
-with real data. In fact, from an execution to another the order of messages will
+asynchronous iterative algorithms comes from the fact that it is necessary to run the algorithm
+with real data. Indeed, from one execution to the other the order of messages will

 change and the  number of iterations to reach the  convergence will also change.
 According  to all  the parameters  of the  platform (number  of nodes,  power of
 change and the  number of iterations to reach the  convergence will also change.
 According  to all  the parameters  of the  platform (number  of nodes,  power of

-nodes,  inter  and  intra clusrters  bandwith  and  latency, etc.) and  of  the
+nodes,  inter  and  intra clusters  bandwidth  and  latency, etc.) and  of  the

 algorithm  (number   of  splittings  with  the   multisplitting  algorithm),  the
 multisplitting code  will obtain the solution  more or less  quickly. Of course,
 algorithm  (number   of  splittings  with  the   multisplitting  algorithm),  the
 multisplitting code  will obtain the solution  more or less  quickly. Of course,

-the GMRES method also depends of the same parameters. As it is difficult to have
+the GMRES method also depends on the same parameters. As it is difficult to have

 access to  many clusters,  grids or supercomputers  with many  different network
 access to  many clusters,  grids or supercomputers  with many  different network

-parameters,  it  is  interesting  to  be  able  to  simulate  the  behaviors  of
-asynchronous iterative algoritms before being able to runs real experiments.
+parameters,  it  is  interesting  to  be  able  to  simulate  the  behavior  of
+asynchronous iterative algorithms before being able to run real experiments.

 
 
 
 
 
 


@@ -247,44 +245,58 @@ asynchronous iterative algoritms before being able to runs real experiments.
 
 \section{SimGrid}
 
 
 \section{SimGrid}
 

-SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid} is a simulation
-framework to study the behavior of large-scale distributed systems.  As its name
-says, 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 from 1999, but it's still actively developed and distributed as an open
-source software.  Today, it's one of the major generic tools in the field of
-simulation for large-scale distributed systems.
+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
 
 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 exposed in
+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
 this paper.  The SMPI interface implements about \np[\%]{80} of the MPI 2.0

-standard~\cite{bedaride:hal-00919507}, and supports applications written in C or
-Fortran, with little or no modifications.
+standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports
+applications written in C or Fortran, with little or no modifications.

 
 

-Within SimGrid, the execution of a distributed application is simulated on a
-single machine.  The application code is really executed, but some operations
-like the communications are intercepted, and their running time is computed
+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
 according to the characteristics of the simulated execution platform.  The
 description of this target platform is given as an input for the execution, by

-the mean of an XML file.  It describes the properties of the platform, such as
+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
 the computing nodes with their computing power, the interconnection links with

-their bandwidth and latency, and the routing strategy.  The simulated running
-time of the application is computed according to these properties.
+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
 
 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 to run relatively fast
+communications), SimGrid uses a fluid model.  This allows users to run relatively fast

 simulations, while still keeping accurate
 simulations, while still keeping accurate

-results~\cite{bedaride:hal-00919507,tomacs13}.  Moreover, depending on the
+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
 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 have no importance for the simulation results, there is
-also the possibility to share dynamically allocated data structures between
+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.
 several simulated processes, and thus to reduce the whole memory consumption.

-These two techniques can help to run simulations at a very large scale.
+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{Simulation of the multisplitting method}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}


@@ -367,7 +379,7 @@ used iterative method by many researchers.
 \label{algo:01}
 \end{figure}
 
 \label{algo:01}
 \end{figure}
 

-Algorithm on Figure~\ref{algo:01} shows the main key points of the
+The algorithm in Figure~\ref{algo:01} shows the main key points of the

 multisplitting method to solve a large sparse linear system. This algorithm is
 based on an outer-inner iteration method where the parallel synchronous GMRES
 method is used to solve the inner iteration. It is executed in parallel by each
 multisplitting method to solve a large sparse linear system. This algorithm is
 based on an outer-inner iteration method where the parallel synchronous GMRES
 method is used to solve the inner iteration. It is executed in parallel by each


@@ -383,8 +395,8 @@ exchanged by message passing using MPI non-blocking communication routines.
 
 \begin{figure}[!t]
 \centering
 
 \begin{figure}[!t]
 \centering

-  \includegraphics[width=60mm,keepaspectratio]{clustering2}
-\caption{Example of two distant clusters of processors.}
+  \includegraphics[width=60mm,keepaspectratio]{clustering}
+\caption{Example of three distant clusters of processors.}

 \label{fig:4.1}
 \end{figure}
 
 \label{fig:4.1}
 \end{figure}
 


@@ -395,12 +407,12 @@ processor is designated (for example the processor with rank 1) and masters of
 all clusters are interconnected by a virtual unidirectional ring network (see
 Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around
 the virtual ring from a master processor to another until the global convergence
 all clusters are interconnected by a virtual unidirectional ring network (see
 Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around
 the virtual ring from a master processor to another until the global convergence

-is achieved. So starting from the cluster with rank 1, each master processor $\ell$
+is achieved. So, starting from the cluster with rank 1, each master processor $\ell$

 sets the token to \textit{True} if the local convergence is achieved or to
 \textit{False} otherwise, and sends it to master processor $\ell+1$. Finally, the
 global convergence is detected when the master of cluster 1 receives from the
 master of cluster $L$ a token set to \textit{True}. In this case, the master of
 sets the token to \textit{True} if the local convergence is achieved or to
 \textit{False} otherwise, and sends it to master processor $\ell+1$. Finally, the
 global convergence is detected when the master of cluster 1 receives from the
 master of cluster $L$ a token set to \textit{True}. In this case, the master of

-cluster 1 broadcasts a stop message to masters of other clusters. In this work,
+cluster 1 broadcasts a stop message to the masters of other clusters. In this work,

 the local convergence on each cluster $\ell$ is detected when the following
 condition is satisfied
 \begin{equation*}
 the local convergence on each cluster $\ell$ is detected when the following
 condition is satisfied
 \begin{equation*}


@@ -412,7 +424,7 @@ $X_\ell^k$ and $X_\ell^{k+1}$.
 
 
 
 
 
 

-In this paper, we solve the 3D Poisson problem whose the mathematical model is 
+In this paper, we solve the 3D Poisson problem whose mathematical model is 

 \begin{equation}
 \left\{
 \begin{array}{l}
 \begin{equation}
 \left\{
 \begin{array}{l}


@@ -422,7 +434,7 @@ u =0 \text{~on~} \Gamma =\partial\Omega
 \right.
 \label{eq:02}
 \end{equation}
 \right.
 \label{eq:02}
 \end{equation}

-where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
+where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite differences scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose general expression could be written as

 \begin{equation}
 \begin{array}{l}
 u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
 \begin{equation}
 \begin{array}{l}
 u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),


@@ -434,7 +446,7 @@ u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x
 \end{equation} 
 where $h$ is the distance between two adjacent elements in the spatial discretization scheme and the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite. 
 
 \end{equation} 
 where $h$ is the distance between two adjacent elements in the spatial discretization scheme and the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite. 
 

-The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries. 
+The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning one in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries. 

 
 \begin{figure}[!t]
 \centering
 
 \begin{figure}[!t]
 \centering


@@ -450,13 +462,13 @@ The parallel solving of the 3D Poisson problem with our multisplitting method re
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code 

-debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method , the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
+debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method, the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions

 and with the addition of the primitive \texttt{MPI\_Test} was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
 %\CER{On voulait en fait montrer la simplicité de l'adaptation de l'algo a SimGrid. Les problèmes rencontrés décrits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais préféré avoir un peu plus de détails sur l'adaptation de la version async} 
 %\CER{Le problème majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a été le plantage en SMPI de Waitall après un Isend et Irecv. J'avais proposé un workaround en utilisant un MPI\_wait séparé pour chaque échange a la place d'un waitall unique pour TOUTES les échanges, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais après, tu as modifié le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de détection de la convergence et du coup, l'échange global avec waitall a aussi fonctionné.}
 Note here that the use of SMPI functions optimizer for memory footprint and CPU usage is not recommended knowing that one wants to get real results by simulation.
 and with the addition of the primitive \texttt{MPI\_Test} was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
 %\CER{On voulait en fait montrer la simplicité de l'adaptation de l'algo a SimGrid. Les problèmes rencontrés décrits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais préféré avoir un peu plus de détails sur l'adaptation de la version async} 
 %\CER{Le problème majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a été le plantage en SMPI de Waitall après un Isend et Irecv. J'avais proposé un workaround en utilisant un MPI\_wait séparé pour chaque échange a la place d'un waitall unique pour TOUTES les échanges, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais après, tu as modifié le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de détection de la convergence et du coup, l'échange global avec waitall a aussi fonctionné.}
 Note here that the use of SMPI functions optimizer for memory footprint and CPU usage is not recommended knowing that one wants to get real results by simulation.

-As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. The scope of all declared 
-global variables have been moved to local to subroutine. Indeed, global variables generate side effects arising from the concurrent access of 
+As mentioned, upon this adaptation, the algorithm is executed as in real life in the simulated environment after the following minor changes. The scope of all declared 
+global variables have been moved to local subroutines. Indeed, global variables generate side effects arising from the concurrent access of 

 shared memory used by threads simulating each computing unit in the SimGrid architecture. 
 %Second, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
 %\AG{compilation or run-time error?}
 shared memory used by threads simulating each computing unit in the SimGrid architecture. 
 %Second, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
 %\AG{compilation or run-time error?}


@@ -473,7 +485,7 @@ study that the results depend on the following parameters:
 \begin{itemize}
 \item At the network level, we found that the most critical values are the
   bandwidth and the network latency.
 \begin{itemize}
 \item At the network level, we found that the most critical values are the
   bandwidth and the network latency.

-\item Hosts processors power (GFlops) can also influence on the results.
+\item Host processor power (GFlops) can also influence the results.

 \item Finally, when submitting job batches for execution, the arguments values
   passed to the program like the maximum number of iterations or the precision are critical. They allow us to ensure not only the convergence of the
   algorithm but also to get the main objective in getting an execution time with the asynchronous multisplitting  less than with synchronous GMRES. 
 \item Finally, when submitting job batches for execution, the arguments values
   passed to the program like the maximum number of iterations or the precision are critical. They allow us to ensure not only the convergence of the
   algorithm but also to get the main objective in getting an execution time with the asynchronous multisplitting  less than with synchronous GMRES. 


@@ -488,14 +500,14 @@ rapid exchange of information on such high-speed links. Thus, the methodology
 adopted was to launch the application on a clustered network. In this
 configuration, degrading the inter-cluster network performance will penalize the
 synchronous mode allowing to get a relative gain greater than 1.  This action
 adopted was to launch the application on a clustered network. In this
 configuration, degrading the inter-cluster network performance will penalize the
 synchronous mode allowing to get a relative gain greater than 1.  This action

-simulates the case of distant clusters linked with long distance network as in grid computing context.
+simulates the case of distant clusters linked with long distance networks as in grid computing context.

 
 
 
 
 
 

-Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem  ranges from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
+Both codes were simulated on a two clusters based network with 50 hosts each, totalling 100 hosts. Various combinations of the above
+factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem  ranges from $N=N_x = N_y = N_z = \text{62}$ to 150 elements (that is from

 $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
 $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =

-\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one. 
+\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is on average 2.5 times faster than with the synchronous GMRES one. 

 %\AG{Expliquer comment lire les tableaux.}
 %\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
 % use the same column width for the following three tables
 %\AG{Expliquer comment lire les tableaux.}
 %\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
 % use the same column width for the following three tables


@@ -509,7 +521,7 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
 \begin{table}[!t]
   \centering
   \caption{Relative gain  of the multisplitting algorithm compared  to GMRES for
 \begin{table}[!t]
   \centering
   \caption{Relative gain  of the multisplitting algorithm compared  to GMRES for

-    different configurations with 2 clusters, each one composed of 50 nodes.}
+    different configurations with 2 clusters, each one composed of 50 nodes. Latency = $20$ms}

   \label{tab.cluster.2x50}
 
   \begin{mytable}{5}
   \label{tab.cluster.2x50}
 
   \begin{mytable}{5}


@@ -517,14 +529,14 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
     bandwidth (Mbit/s)
     & 5         & 5         & 5         & 5         & 5         \\
     \hline
     bandwidth (Mbit/s)
     & 5         & 5         & 5         & 5         & 5         \\
     \hline

-    latency (ms)
-    & 20      &  20      & 20      & 20      & 20      \\
-    \hline
+  %  latency (ms)
+   % & 20      &  20      & 20      & 20      & 20      \\
+    %\hline

     power (GFlops)
     & 1         & 1         & 1         & 1.5       & 1.5       \\
     \hline
     power (GFlops)
     & 1         & 1         & 1         & 1.5       & 1.5       \\
     \hline

-    size $(n^3)$
-    & 62        & 62        & 62        & 100       & 100       \\
+    size $(N)$
+    & $62^3$        & $62^3$        & $62^3$        & $100^3$       & $100^3$       \\

     \hline
     Precision
     & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} \\
     \hline
     Precision
     & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} \\


@@ -542,14 +554,14 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
     bandwidth (Mbit/s)
     & 50        & 50        & 50        & 50        & 50 \\ %       & 10        & 10 \\
     \hline
     bandwidth (Mbit/s)
     & 50        & 50        & 50        & 50        & 50 \\ %       & 10        & 10 \\
     \hline

-    latency (ms)
-    & 20      & 20      & 20      & 20      & 20 \\ %      & 0.03      & 0.01 \\
-    \hline
+    %latency (ms)
+    %& 20      & 20      & 20      & 20      & 20 \\ %      & 0.03      & 0.01 \\
+    %\hline

     Power (GFlops)
     & 1.5       & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\
     \hline
     Power (GFlops)
     & 1.5       & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\
     \hline

-    size $(n^3)$
-    & 110       & 120       & 130       & 140       & 150  \\ %     & 171       & 171 \\
+    size $(N)$
+    & $110^3$       & $120^3$       & $130^3$       & $140^3$       & $150^3$  \\ %     & 171       & 171 \\

     \hline
     Precision
     & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5}  & \np{E-5} \\
     \hline
     Precision
     & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5}  & \np{E-5} \\


@@ -561,7 +573,7 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
   \end{mytable}
 \end{table}
   
   \end{mytable}
 \end{table}
   

-\RC{Du coup la latence est toujours la même, pourquoi la mettre dans la table?}
+%\RC{Du coup la latence est toujours la même, pourquoi la mettre dans la table?}

 
 %Then we have changed the network configuration using three clusters containing
 %respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
 
 %Then we have changed the network configuration using three clusters containing
 %respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the


@@ -632,7 +644,9 @@ Note that the program was run with the following parameters:
 
 \begin{itemize}
 \item HOSTFILE: Text file containing the list of the processors units name. Here 100 hosts;
 
 \begin{itemize}
 \item HOSTFILE: Text file containing the list of the processors units name. Here 100 hosts;

-\item PLATFORM: XML file description of the platform architecture whith the following characteristics: %two clusters (cluster1 and cluster2) with the following characteristics :
+\item PLATFORM: XML file description of the platform architecture with the
+  following characteristics:
+  % two clusters (cluster1 and cluster2) with the following characteristics:

   \begin{itemize}
   \item 2 clusters of 50 hosts each;
   \item Processor unit power: \np[GFlops]{1} or \np[GFlops]{1.5};
   \begin{itemize}
   \item 2 clusters of 50 hosts each;
   \item Processor unit power: \np[GFlops]{1} or \np[GFlops]{1.5};


@@ -647,29 +661,28 @@ Note that the program was run with the following parameters:
 \begin{itemize}
 \item Description of the cluster architecture matching the format <Number of
   clusters> <Number of hosts in cluster1> <Number of hosts in cluster2>;
 \begin{itemize}
 \item Description of the cluster architecture matching the format <Number of
   clusters> <Number of hosts in cluster1> <Number of hosts in cluster2>;

-\item Maximum number of iterations;
-\item Precisions on the residual error;
+\item Maximum numbers of outer and inner iterations;
+\item Outer and inner precisions on the residual error;

 \item Matrix size $N_x$, $N_y$ and $N_z$;
 \item Matrix size $N_x$, $N_y$ and $N_z$;

-\item Matrix diagonal value: $6$ (See Equation~(\ref{eq:03}));
-\item Matrix off-diagonal value: $-1$;
+\item Matrix diagonal value: $6$ (see Equation~(\ref{eq:03}));
+\item Matrix off-diagonal values: $-1$;

 \item Communication mode: asynchronous.
 \end{itemize}
 
 \paragraph*{Interpretations and comments}
 
 After analyzing the outputs, generally, for the two clusters including one hundred hosts configuration (Tables~\ref{tab.cluster.2x50}), some combinations of parameters affecting
 \item Communication mode: asynchronous.
 \end{itemize}
 
 \paragraph*{Interpretations and comments}
 
 After analyzing the outputs, generally, for the two clusters including one hundred hosts configuration (Tables~\ref{tab.cluster.2x50}), some combinations of parameters affecting

-the results have given a relative gain more than 2.5, showing the effectiveness of the
+the results, have given a relative gain of more than 2.5, showing the effectiveness of the

 asynchronous multisplitting  compared to GMRES with two distant clusters.
 
 With these settings, Table~\ref{tab.cluster.2x50} shows
 asynchronous multisplitting  compared to GMRES with two distant clusters.
 
 With these settings, Table~\ref{tab.cluster.2x50} shows

-that after setting the bandwidth of the  inter cluster network to  \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
-of one GFlops, an efficiency of about \np[\%]{40} is
-obtained in asynchronous mode for a matrix size of 62 elements. It is noticed that the result remains
-stable even we vary the residual error precision from \np{E-5} to \np{E-9}. By
-increasing the matrix size up to 100 elements, it was necessary to increase the
-CPU power of \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency.  Maintaining such processor power but increasing network throughput inter cluster up to
-\np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5 is obtained with
-high external precision of \np{E-11} for a matrix size from 110 to 150 side
+that after setting the bandwidth of the  inter cluster network to  \np[Mbit/s]{5}, the latency to $20$ millisecond and the processor power
+to one GFlops, an efficiency of about \np[\%]{40} is
+obtained in asynchronous mode for a matrix size of $62^3$ elements. It is noticed that the result remains
+stable even if the residual error precision varies from \np{E-5} to \np{E-9}. By
+increasing the matrix size up to $100^3$ elements, it was necessary to increase the
+CPU power by \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency.  Maintaining  a relative gain of $2.5$ and such processor power but increasing network throughput inter cluster up to \np[Mbit/s]{50},  is obtained with
+high external precision of \np{E-11} for a matrix size from $110^3$ to $150^3$ side

 elements.
 
 %For the 3 clusters architecture including a total of 100 hosts,
 elements.
 
 %For the 3 clusters architecture including a total of 100 hosts,


@@ -679,8 +692,8 @@ elements.
 %(synchronous and asynchronous) is achieved with an inter cluster of
 %\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency greater than 1.2 with a matrix %size of 100 points, it was necessary to degrade the
 %inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
 %(synchronous and asynchronous) is achieved with an inter cluster of
 %\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency greater than 1.2 with a matrix %size of 100 points, it was necessary to degrade the
 %inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.

-\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
-  Quelle est la perte de perfs en faisant ça ?}
+%\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
+  %Quelle est la perte de perfs en faisant ça ?}

 
 %A last attempt was made for a configuration of three clusters but more powerful
 %with 200 nodes in total. The convergence with a relative gain around 1.1 was
 
 %A last attempt was made for a configuration of three clusters but more powerful
 %with 200 nodes in total. The convergence with a relative gain around 1.1 was


@@ -693,7 +706,7 @@ elements.
 %\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
 \section{Conclusion}
 The simulation of the execution of parallel asynchronous iterative algorithms on large scale  clusters has been presented. 
 %\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
 \section{Conclusion}
 The simulation of the execution of parallel asynchronous iterative algorithms on large scale  clusters has been presented. 

-In this work, we show that SIMGRID is an efficient simulation tool that allows us to 
+In this work, we show that SimGrid is an efficient simulation tool that has enabled us to 

 reach the following two objectives: 
 
 \begin{enumerate}
 reach the following two objectives: 
 
 \begin{enumerate}


@@ -704,9 +717,8 @@ reach the following two objectives:
 
 \item To test the combination of the cluster and network specifications permitting to execute an asynchronous algorithm faster than a synchronous one.
 \end{enumerate}
 
 \item To test the combination of the cluster and network specifications permitting to execute an asynchronous algorithm faster than a synchronous one.
 \end{enumerate}

-Our results have shown that with two distant clusters, the asynchronous multisplitting is faster to \np[\%]{40} compared to the synchronous GMRES method
-which is not negligible for solving complex practical problems with more 
-and more increasing size.
+Our results have shown that with two distant clusters, the asynchronous multisplitting method is faster by \np[\%]{40} compared to the synchronous GMRES method
+which is not negligible for solving complex practical problems with ever increasing size.

 
 Several studies have already addressed the performance execution time of 
 this class of algorithm. The work presented in this paper has 
 
 Several studies have already addressed the performance execution time of 
 this class of algorithm. The work presented in this paper has 


@@ -715,7 +727,7 @@ tool to run efficiently an iterative parallel algorithm in asynchronous
 mode in a grid architecture. 
 
 In future works, we plan to extend our experimentations to larger scale platforms by increasing the number of computing cores and the number of clusters. 
 mode in a grid architecture. 
 
 In future works, we plan to extend our experimentations to larger scale platforms by increasing the number of computing cores and the number of clusters. 

-We will also have to increase the size of the input problem which will require the use of a more powerful simulation platform. At last, we expect to compare our simulation results to real execution results on real architectures in order to experimentally validate our study.
+We will also have to increase the size of the input problem which will require the use of a more powerful simulation platform. At last, we expect to compare our simulation results to real execution results on real architectures in order to better experimentally validate our study. Finally, we also plan to study other problems with the multisplitting method and other asynchronous iterative methods.

 
 \section*{Acknowledgment}
 
 
 \section*{Acknowledgment}
 


@@ -745,6 +757,6 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 % LocalWords:  Ouest Vieille Talence cedex scalability experimentations HPC MPI
 % LocalWords:  Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua
 % LocalWords:  Fortran GFlops priori Mbit de du fcomte multisplitting scalable
 % LocalWords:  Ouest Vieille Talence cedex scalability experimentations HPC MPI
 % LocalWords:  Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua
 % LocalWords:  Fortran GFlops priori Mbit de du fcomte multisplitting scalable

-% LocalWords:  SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib
+% LocalWords:  SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib Gbit

 % LocalWords:  intra durations nonsingular Waitall discretization discretized
 % LocalWords:  intra durations nonsingular Waitall discretization discretized

-% LocalWords:  InnerSolver Isend Irecv
+% LocalWords:  InnerSolver Isend Irecv parallelization