Concluding sentence.

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index d8c9a8c7a5823060c1917dfd8488f55aa5cdb837..83753da163938e5424590613cb5f4189d1c6d3aa 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -1,4 +1,3 @@
-

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


@@ -83,7 +82,7 @@ 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
 synchronous  GMRES  algorithm  with  an  asynchronous  multisplitting  one  with
 simulations  which let us easily choose  some parameters.   Both  codes  are real  MPI
 of asynchronous  iterative algorithms. For that,  we compare the  behaviour 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 ans simulations allow us to see when the asynchronous multisplitting algorithm can be more
+codes and simulations allow us to see when the asynchronous multisplitting algorithm can be more

 efficient than the GMRES one to solve a 3D Poisson problem.
 
 
 efficient than the GMRES one to solve a 3D Poisson problem.
 
 


@@ -103,7 +102,7 @@ suggests, these algorithms solve a given problem by successive iterations ($X_{n
 $X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods
 demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.
 
 $X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods
 demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.
 

-Parallelization of such algorithms generally involve the division of the problem
+Parallelization of such algorithms generally involves the division of the problem

 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


@@ -163,7 +162,8 @@ 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
 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

 
 
 
 
 
 


@@ -228,13 +228,13 @@ In the context of asynchronous algorithms, the number of iterations to reach the
 convergence depends on  the delay of 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
 convergence depends on  the delay of 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

-asynchronous iteratie algorithms comes from the fact it is necessary to run the algorithm
+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
 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
 with real data. In fact, from an execution to another 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

-nodes,  inter  and  intra clusrters  bandwith  and  latency,  ....) and  of  the
-algorithm  (number   of  splitting  with  the   multisplitting  algorithm),  the
-multisplitting code  will obtain the solution  more or less  quickly. Or course,
+nodes,  inter  and  intra clusrters  bandwith  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,

 the GMRES method also depends of the same parameters. As it is difficult to have
 access to  many clusters,  grids or supercomputers  with many  different network
 parameters,  it  is  interesting  to  be  able  to  simulate  the  behaviors  of
 the GMRES method also depends of the same parameters. As it is difficult to have
 access to  many clusters,  grids or supercomputers  with many  different network
 parameters,  it  is  interesting  to  be  able  to  simulate  the  behaviors  of


@@ -251,8 +251,8 @@ 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
 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
+date 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
 simulation for large-scale distributed systems.
 
 SimGrid provides several programming interfaces: MSG to simulate Concurrent


@@ -261,24 +261,26 @@ 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
 this paper.  The SMPI interface implements about \np[\%]{80} of the MPI 2.0
 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
 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
+Within SimGrid, the execution of a distributed application is simulated by a
+single process.  The application code is really executed, but some operations

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

-their bandwidth and latency, and the routing strategy.  The 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 is
+computed according to these properties.

 
 To compute the durations of the operations in the simulated world, and to take
 into account resource sharing (e.g. bandwidth sharing between competing
 communications), SimGrid uses a fluid model.  This allows to run relatively fast
 simulations, while still keeping accurate
 
 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
 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
 skipped, but the results have no importance for the simulation results, there is
 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


@@ -286,6 +288,17 @@ also the possibility to share dynamically allocated data structures between
 several simulated processes, and thus to reduce the whole memory consumption.
 These two techniques can help to run simulations at a very large scale.
 
 several simulated processes, and thus to reduce the whole memory consumption.
 These two techniques can help to run simulations at 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}
 


@@ -384,7 +397,7 @@ exchanged by message passing using MPI non-blocking communication routines.
 \begin{figure}[!t]
 \centering
   \includegraphics[width=60mm,keepaspectratio]{clustering}
 \begin{figure}[!t]
 \centering
   \includegraphics[width=60mm,keepaspectratio]{clustering}

-\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
+\caption{Example of three distant clusters of processors.}

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


@@ -422,7 +435,7 @@ 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 the 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),


@@ -450,7 +463,7 @@ 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.}
 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.}


@@ -493,7 +506,7 @@ simulates the case of distant clusters linked with long distance network as in g
 
 
 Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
 
 
 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
+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 $\text{150}^\text{3} =
 \text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one. 
 %\AG{Expliquer comment lire les tableaux.}
 $\text{62}^\text{3} = \text{\np{238328}}$ 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. 
 %\AG{Expliquer comment lire les tableaux.}


@@ -509,7 +522,7 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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 +530,14 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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)
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      \\
-    \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 +555,14 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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)
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\ %      & 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} \\


@@ -561,13 +574,15 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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?}
+

 %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
 %clusters. In the same way as above, a judicious choice of key parameters has
 %permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the
 %relative gains greater than 1 with a matrix size from 62 to 100 elements.
 
 %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
 %clusters. In the same way as above, a judicious choice of key parameters has
 %permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the
 %relative gains greater than 1 with a matrix size from 62 to 100 elements.
 

-\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}
+%\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}

 %\begin{table}[!t]
 %  \centering
 %  \caption{3 clusters, each with 33 nodes}
 %\begin{table}[!t]
 %  \centering
 %  \caption{3 clusters, each with 33 nodes}


@@ -634,8 +649,8 @@ Note that the program was run with the following parameters:
   \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};

-  \item Intra-cluster network bandwidth: \np[Gbit/s]{1.25} and latency: \np[$\mu$s]{0.05};
-  \item Inter-cluster network bandwidth: \np[Mbit/s]{5} or \np[Mbit/s]{50} and latency: \np[$\mu$s]{20};
+  \item Intra-cluster network bandwidth: \np[Gbit/s]{1.25} and latency: \np[$\mu$s]{50};
+  \item Inter-cluster network bandwidth: \np[Mbit/s]{5} or \np[Mbit/s]{50} and latency: \np[ms]{20};

   \end{itemize}
 \end{itemize}
 
   \end{itemize}
 \end{itemize}
 


@@ -645,11 +660,11 @@ 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}
 
 \item Communication mode: asynchronous.
 \end{itemize}
 


@@ -660,14 +675,14 @@ the results have given a relative gain more than 2.5, showing the effectiveness
 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
+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 we vary the residual error precision from \np{E-5} to \np{E-9}. By
 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
+increasing the matrix size up to $100^3$ 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
 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
+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,


@@ -677,8 +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


@@ -691,21 +706,18 @@ 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 
-reach the following three objectives: 
+In this work, we show that SimGrid is an efficient simulation tool that allows us to 
+reach the following two objectives: 

 
 \begin{enumerate}
 
 \begin{enumerate}

-\item To have a flexible configurable execution platform resolving the 
-hard exercise to access to very limited but so solicited physical 
-resources;
-\item to ensure the algorithm convergence with a reasonable time and
-iteration number ;
-\item and finally and more importantly, to find the correct combination 
-of the cluster and network specifications permitting to save time in 
-executing the algorithm in asynchronous mode.
+\item  To have  a flexible  configurable execution  platform that  allows  us to
+  simulate algorithms for  which execution of all parts of
+  the  code is  necessary. Using  simulations before  real executions  is  a nice
+  solution to detect potential scalability problems.
+
+\item To test the combination of the cluster and network specifications permitting to execute an asynchronous algorithm faster than a synchronous one.

 \end{enumerate}
 \end{enumerate}

-Our results have shown that in certain conditions, asynchronous mode is 
-speeder up to \np[\%]{40} comparing to the synchronous GMRES method
+Our results have shown that with two distant clusters, the asynchronous multisplitting method 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.
 
 which is not negligible for solving complex practical problems with more 
 and more increasing size.
 


@@ -715,8 +727,8 @@ demonstrated an original solution to optimize the use of a simulation
 tool to run efficiently an iterative parallel algorithm in asynchronous 
 mode in a grid architecture. 
 
 tool to run efficiently an iterative parallel algorithm in asynchronous 
 mode in a grid architecture. 
 

-For our futur 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.
+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 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}