modifs tables

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index c6b34a2ba1397cfd4cad34e0268388b79cc6dfb3..22fa0472b9b38ef08bc932da3c7c42705fc9fb68 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -81,8 +81,8 @@ 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
 synchronous  GMRES  algorithm  with  an  asynchronous  multisplitting  one  with
 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  in  which we  choose  some parameters.   Both  codes  are real  MPI
-codes. Simulations allow us to see when the multisplitting algorithm can be more
+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

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


@@ -102,7 +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


@@ -143,11 +143,11 @@ execution time.
 
 To our knowledge,  there is no existing work on the  large-scale simulation of a
 real asynchronous  iterative application.  {\bf The contribution  of the present
 
 To our knowledge,  there is no existing work on the  large-scale simulation of a
 real asynchronous  iterative application.  {\bf The contribution  of the present

-  paper can be  summarised in two main points}.  First we  give a first approach
+  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
 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)
+performance   with  the   synchronous  GMRES   (Generalized   Minimal  Residual) method

 \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
 modifications of the initial MPI code,  the SimGrid toolkit allows us to perform
 \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
 modifications of the initial MPI code,  the SimGrid toolkit allows us to perform


@@ -227,13 +227,13 @@ 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


@@ -250,8 +250,8 @@ 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


@@ -287,6 +287,8 @@ These two techniques can help to run simulations at a very large scale.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}

+
+\subsection{The multisplitting method}

 %Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
 Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping  
 \begin{equation*}
 %Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
 Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping  
 \begin{equation*}


@@ -381,7 +383,7 @@ 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}
 


@@ -392,9 +394,9 @@ 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 $i$
+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
 sets the token to \textit{True} if the local convergence is achieved or to

-\textit{False} otherwise, and sends it to master processor $i+1$. Finally, the
+\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,
 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,


@@ -419,17 +421,17 @@ 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. The general iteration scheme of our multisplitting method in a 3D domain using a seven point stencil 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{equation}

-\begin{array}{ll}
-u^{k+1}(x,y,z)= & u^k(x,y,z) - \frac{1}{6}\times\\
-               & (u^k(x-1,y,z) + u^k(x+1,y,z) + \\
-               & u^k(x,y-1,z) + u^k(x,y+1,z) + \\
-               & u^k(x,y,z-1) + u^k(x,y,z+1)),
+\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),
+%u(x,y,z)= & \frac{1}{6}\times [u(x-1,y,z) + u(x+1,y,z) + \\
+ %         & u(x,y-1,z) + u(x,y+1,z) + \\
+  %        & u(x,y,z-1) + u(x,y,z+1) - \\ & h^2f(x,y,z)],

 \end{array}
 \label{eq:03}
 \end{equation} 
 \end{array}
 \label{eq:03}
 \end{equation} 

-where 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. 
+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 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. 
 


@@ -441,23 +443,24 @@ The parallel solving of the 3D Poisson problem with our multisplitting method re
 \end{figure}
 
 
 \end{figure}
 
 

+\subsection{Simulation of the multisplitting method using SimGrid and SMPI}
+

 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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. In synchronous 
-mode, the execution of the program raised no particular issue but in asynchronous mode, the review of the sequence of MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions
-and with the addition of the primitive MPI\_Test was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
+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.
 %\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. First, the scope of all declared 
+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 
 shared memory used by threads simulating each computing unit in the SimGrid architecture. 
 global variables have been moved to local to subroutine. 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?}
+%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?}

 In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real 
 In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real 

-environment. We have successfully executed the code in synchronous mode using parallel GMRES algorithm compared with our multisplitting algorithm in asynchronous mode after few modifications. 
+environment. We have successfully executed the code for the synchronous GMRES algorithm compared with our asynchronous multisplitting algorithm after few modifications. 

 
 
 
 
 
 


@@ -472,15 +475,14 @@ study that the results depend on the following parameters:
 \item Hosts processors power (GFlops) can also influence on 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
 \item Hosts processors power (GFlops) can also influence on 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 in asynchronous communication less than in
-  synchronous mode. The ratio between the execution time of synchronous
-  compared to the asynchronous mode ($t_\text{sync} / t_\text{async}$) is defined as the \emph{relative gain}. So,
-  our objective running the algorithm in SimGrid is to obtain a relative gain
-  greater than 1.
-\end{itemize}
+  algorithm but also to get the main objective in getting an execution time with the asynchronous multisplitting  less than with synchronous GMRES. 
+  \end{itemize}

 
 

+The ratio between the simulated execution time of synchronous GMRES algorithm
+compared to the asynchronous multisplitting algorithm ($t_\text{GMRES} / t_\text{Multisplitting}$) is defined as the \emph{relative gain}. So,
+our objective running the algorithm in SimGrid is to obtain a relative gain greater than 1.

 A priori, obtaining a relative gain greater than 1 would be difficult in a local
 A priori, obtaining a relative gain greater than 1 would be difficult in a local

-area network configuration where the synchronous mode will take advantage on the
+area network configuration where the synchronous GMRES method will take advantage on the

 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
 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


@@ -488,14 +490,13 @@ 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 network as in grid computing context.
 
 

-% As a first step, 
-The algorithm was run 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 algorithm convergence with a 3D
-matrix size ranging 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, 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 $N=N_x = N_y = N_z = \text{62}$ 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), is obtained in asynchronous in average 2.5 times faster than in the synchronous mode.
-\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}
+\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.}
+%\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
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
 \newenvironment{mytable}[1]{% #1: number of columns for data
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
 \newenvironment{mytable}[1]{% #1: number of columns for data


@@ -506,7 +507,8 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
 
 \begin{table}[!t]
   \centering
 
 \begin{table}[!t]
   \centering

-  \caption{2 clusters, each with 50 nodes}
+  \caption{Relative gain  of the multisplitting algorithm compared  to GMRES for
+    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}


@@ -514,14 +516,14 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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} \\


@@ -539,14 +541,14 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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} \\


@@ -558,13 +560,15 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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}


@@ -627,13 +631,12 @@ 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 : two clusters (cluster1 and cluster2) with the following characteristics :
+\item PLATFORM: XML file description of the platform architecture whith the following characteristics: %two clusters (cluster1 and cluster2) with the following characteristics :

   \begin{itemize}
   \begin{itemize}

-  \item Processor unit power: \np[GFlops]{1.5};
-  \item Intracluster network bandwidth: \np[Gbit/s]{1.25} and latency:
-    \np[$\mu$s]{0.05};
-  \item Intercluster network bandwidth: \np[Mbit/s]{5} and latency:
-    \np[$\mu$s]{5};
+  \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]{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}
 


@@ -642,12 +645,12 @@ Note that the program was run with the following parameters:
 
 \begin{itemize}
 \item Description of the cluster architecture matching the format <Number of
 
 \begin{itemize}
 \item Description of the cluster architecture matching the format <Number of

-  cluster> <Number of hosts in cluster1> <Number of hosts in cluster2>;
-\item Maximum number of iterations;
-\item Precisions on the residual error;
+  clusters> <Number of hosts in cluster1> <Number of hosts in cluster2>;
+\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: \np{1.0} (See~(\ref{eq:03}));
-\item Matrix off-diagonal value: \np{-1}/\np{6} (See~(\ref{eq:03}));
+\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}
 


@@ -655,17 +658,17 @@ Note that the program was run with the following parameters:
 
 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
 
 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

-asynchronous performance compared to the synchronous mode.
+asynchronous multisplitting  compared to GMRES with two distant clusters.

 
 With these settings, Table~\ref{tab.cluster.2x50} shows
 
 With these settings, Table~\ref{tab.cluster.2x50} shows

-that after a deterioration of inter cluster network with a bandwidth of \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
+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
 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
+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,


@@ -675,8 +678,8 @@ 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


@@ -688,39 +691,35 @@ elements.
 %\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
 %\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
 \section{Conclusion}
 %\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
 %\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
 \section{Conclusion}

-The experimental results on executing a parallel iterative algorithm in 
-asynchronous mode on an environment simulating a large scale of virtual 
-computers organized with interconnected clusters have been presented. 
-Our work has demonstrated that using such a simulation tool allow us to 
-reach the following three objectives: 
+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 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} than executing the algorithm in synchronous mode
+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.
 

- Several studies have already addressed the performance execution time of 
+Several studies have already addressed the performance execution time of 

 this class of algorithm. The work presented in this paper has 
 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. 
 
 this class of algorithm. The work presented in this paper has 
 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. 
 

-\LZK{Perspectives???}
+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}
 
 This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 
 \section*{Acknowledgment}
 
 This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).

-\todo[inline]{The authors would like to thank\dots{}}
+%\todo[inline]{The authors would like to thank\dots{}}

 
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