Merge

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index e043c784c66f453ccdd8386102c4bf0c49964511..640f3ae56c6fb43afd652aaa84139a68b3b29d0b 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -135,6 +135,35 @@ 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.
 
 simulated environment challenges to  find optimal configurations giving the best
 results with a lowest residual error and in the best of execution time.
 

+<<<<<<< HEAD
+To our knowledge, there is no existing work on the large-scale simulation of a
+real AIAC application. The aim of this paper is twofold. First we give a first
+approach of the simulation of AIAC algorithms using a simulation tool (i.e. the
+SimGrid toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of
+asynchronous mode algorithms by comparing their performance with the synchronous
+mode. More precisely, we had implemented a program for solving large
+linear system of equations by numerical method GMRES (Generalized
+Minimal Residual) \cite{ref1}. We show, that with minor modifications of the
+initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a
+real AIAC 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
+infrastructure by simulating different distributed architectures composed by
+clusters nodes interconnected by variable speed networks. In the simulated environment, after setting appropriate 
+network and cluster parameters like the network bandwidth, latency or the processors power, 
+the experimental results have demonstrated a asynchronous execution time saving up to \np[\%]{40} in
+compared to the synchronous mode.
+\AG{Il faudrait revoir la phrase précédente (couper en deux?).  Là, on peut
+  avoir l'impression que le gain de \np[\%]{40} est entre une exécution réelle
+  et une exécution simulée!}
+\CER{La phrase a été modifiée}
+
+This article is structured as follows: after this introduction, the next  section will give a brief description of
+iterative asynchronous model.  Then, the simulation framework SimGrid is presented with the settings to create various
+distributed architectures. The algorithm of  the multisplitting method based on GMRES \LZK{??? GMRES n'utilise pas la méthode de multisplitting! Sinon ne doit on pas expliquer le choix d'une méthode de multisplitting?} \CER{La phrase a été corrigée} written with MPI primitives and
+its adaptation to SimGrid with SMPI (Simulated MPI) is detailed in the next section. At last, the experiments results
+carried out will be presented before some concluding remarks and future works.
+=======

 To our knowledge,  there is no existing work on the  large-scale simulation of a
 real  AIAC application.   {\bf  The contribution  of  the present  paper can  be
   summarised  in two  main  points}.  First  we  give a  first  approach of  the
 To our knowledge,  there is no existing work on the  large-scale simulation of a
 real  AIAC application.   {\bf  The contribution  of  the present  paper can  be
   summarised  in two  main  points}.  First  we  give a  first  approach of  the


@@ -167,6 +196,7 @@ based  on GMRES to  solve each  block obtained  of 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.

+>>>>>>> 6785b9ef58de0db67c33ca901c7813f3dfdc76e0

  
 \section{Motivations and scientific context}
 
  
 \section{Motivations and scientific context}
 


@@ -402,25 +432,63 @@ where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the
 tolerance threshold of the error computed between two successive local solution
 $X_\ell^k$ and $X_\ell^{k+1}$.
 
 tolerance threshold of the error computed between two successive local solution
 $X_\ell^k$ and $X_\ell^{k+1}$.
 

+
+
+In this paper, we solve the 3D Poisson problem whose the mathematical model is 
+\begin{equation}
+\left\{
+\begin{array}{l}
+\nabla^2 u = f \text{~in~} \Omega \\
+u =0 \text{~on~} \Gamma =\partial\Omega
+\end{array}
+\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 
+\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)),
+\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. 
+
+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. 
+
+\begin{figure}[!t]
+\centering
+  \includegraphics[width=80mm,keepaspectratio]{partition}
+\caption{Example of the 3D data partitioning between two clusters of processors.}
+\label{fig:4.2}
+\end{figure}
+
+
+
+

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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.

-\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{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, all declared 
 global variables have been moved to local variables for each subroutine. In fact, 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, the alignment of certain types of variables such as ``long int'' had
 also to be reviewed.
 \AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.}
 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, all declared 
 global variables have been moved to local variables for each subroutine. In fact, 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, the alignment of certain types of variables such as ``long int'' had
 also to be reviewed.
 \AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.}

+\CER{Ce problème fait partie des modifications que j'ai dû faire dans l'adaptation du programme MPI vers SMPI. IL découle de la différence de la taille des mots en mémoire : en 32 bits, pour les variables declarees en long int, on garde dans les instructions de sortie (printf, sprintf, ...) le format \%lu sinon en 64 bits, on le substitue par \%llu.} 

  Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
 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. 
 
 
 
  Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
 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. 
 
 
 

-\section{Experimental results}
+\section{Simulation results}

 
 When the \textit{real} application runs in the simulation environment and produces the expected results, varying the input
 parameters and the program arguments allows us to compare outputs from the code execution. We have noticed from this
 
 When the \textit{real} application runs in the simulation environment and produces the expected results, varying the input
 parameters and the program arguments allows us to compare outputs from the code execution. We have noticed from this


@@ -428,73 +496,37 @@ 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 power (GFlops) can also influence on the results.
+\item Hosts processors power (GFlops) can also influence on the results.

 \item Finally, when submitting job batches for execution, the arguments values
 \item Finally, when submitting job batches for execution, the arguments values

-  passed to the program like the maximum number of iterations or the external
-  precision are critical. They allow to ensure not only the convergence of the
-  algorithm but also to get the main objective of the experimentation of the
-  simulation in having an execution time in asynchronous less than in
-  synchronous mode. The ratio between the execution time of asynchronous
-  compared to the synchronous mode is defined as the \emph{relative gain}. So,
+  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.
   \AG{$t_\text{async} / t_\text{sync} > 1$, l'objectif est donc que ça dure plus
     longtemps (que ça aille moins vite) en asynchrone qu'en synchrone ?
     Ce n'est pas plutôt l'inverse ?}
   our objective running the algorithm in SimGrid is to obtain a relative gain
   greater than 1.
   \AG{$t_\text{async} / t_\text{sync} > 1$, l'objectif est donc que ça dure plus
     longtemps (que ça aille moins vite) en asynchrone qu'en synchrone ?
     Ce n'est pas plutôt l'inverse ?}

+  \CER{J'ai modifie la phrase.}

 \end{itemize}
 
 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
 rapid exchange of information on such high-speed links. Thus, the methodology
 \end{itemize}
 
 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
 rapid exchange of information on such high-speed links. Thus, the methodology

-adopted was to launch the application on clustered network. In this last
+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
 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 like
-Internet.
+simulates the case of distant clusters linked with long distance network as in grid computing context.

 
 

-\AG{Cette partie sur le poisson 3D
-  % on sait donc que ce n'est pas une plie ou une sole (/me fatigué)
-  n'est pas à sa place.  Elle devrait être placée plus tôt.}
-In this paper, we solve the 3D Poisson problem whose the mathematical model is 
-\begin{equation}
-\left\{
-\begin{array}{l}
-\nabla^2 u = f \text{~in~} \Omega \\
-u =0 \text{~on~} \Gamma =\partial\Omega
-\end{array}
-\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 
-\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)),
-\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. 
-
-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. 
-
-\begin{figure}[!t]
-\centering
-  \includegraphics[width=80mm,keepaspectratio]{partition}
-\caption{Example of the 3D data partitioning between two clusters of processors.}
-\label{fig:4.2}
-\end{figure}

 
 

-
-As a first step, the algorithm was run on a network consisting of two clusters
-containing 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50} with a
-matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from
-$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
-\text{\np{5000211}}$ entries.
+% 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
+$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
+\text{\np{3375000}}$ entries), is obtained in asynchronous in average 2.5 times speeder than the synchronous mode. 

 \AG{Expliquer comment lire les tableaux.}
 \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


@@ -510,19 +542,19 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
 
   \begin{mytable}{6}
     \hline
 
   \begin{mytable}{6}
     \hline

-    bandwidth
+    bandwidth (Mbits/s)

     & 5         & 5         & 5         & 5         & 5         & 50 \\
     \hline
     & 5         & 5         & 5         & 5         & 5         & 50 \\
     \hline

-    latency
+    latency (ms)

     & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\
     \hline
     & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\
     \hline

-    power
+    power (GFlops)

     & 1         & 1         & 1         & 1.5       & 1.5       & 1.5 \\
     \hline
     size
     & 62        & 62        & 62        & 100       & 100       & 110 \\
     \hline
     & 1         & 1         & 1         & 1.5       & 1.5       & 1.5 \\
     \hline
     size
     & 62        & 62        & 62        & 100       & 100       & 110 \\
     \hline

-    Prec/Eprec
+    Precision

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


@@ -535,156 +567,158 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
 
   \begin{mytable}{6}
     \hline
 
   \begin{mytable}{6}
     \hline

-    bandwidth
-    & 50        & 50        & 50        & 50        & 10        & 10 \\
+    bandwidth (Mbits/s)
+    & 50        & 50        & 50        & 50 \\ %       & 10        & 10 \\

     \hline
     \hline

-    latency
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.03      & 0.01 \\
+    latency (ms)
+    & 0.02      & 0.02      & 0.02      & 0.02 \\ %      & 0.03      & 0.01 \\

     \hline
     \hline

-    power
-    & 1.5       & 1.5       & 1.5       & 1.5       & 1         & 1.5 \\
+    Power (GFlops)
+    & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\

     \hline
     size
     \hline
     size

-    & 120       & 130       & 140       & 150       & 171       & 171 \\
+    & 120       & 130       & 140       & 150  \\ %     & 171       & 171 \\

     \hline
     \hline

-    Prec/Eprec
-    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5}  & \np{E-5} \\
+    Precision
+    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5}  & \np{E-5} \\

     \hline
     \hline
     Relative gain
     \hline
     \hline
     Relative gain

-    & 2.51     & 2.58     & 2.55     & 2.54     & 1.59      & 1.29 \\
+    & 2.51     & 2.58     & 2.55     & 2.54   \\ %  & 1.59      & 1.29 \\

     \hline
   \end{mytable}
 \end{table}
   
     \hline
   \end{mytable}
 \end{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.
-
-\begin{table}[!t]
-  \centering
-  \caption{3 clusters, each with 33 nodes}
-  \label{tab.cluster.3x33}
-
-  \begin{mytable}{6}
-    \hline
-    bandwidth
-    & 10       & 5        & 4        & 3        & 2        & 6 \\
-    \hline
-    latency
-    & 0.01     & 0.02     & 0.02     & 0.02     & 0.02     & 0.02 \\
-    \hline
-    power
-    & 1        & 1        & 1        & 1        & 1        & 1 \\
-    \hline
-    size
-    & 62       & 100      & 100      & 100      & 100      & 171 \\
-    \hline
-    Prec/Eprec
-    & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
-    \hline
-    \hline
-    Relative gain
-    & 1.003    & 1.01     & 1.08     & 1.19     & 1.28     & 1.01 \\
-    \hline
-  \end{mytable}
-\end{table}
-
-In a final step, results of an execution attempt to scale up the three clustered
-configuration but increasing by two hundreds hosts has been recorded in
-Table~\ref{tab.cluster.3x67}.
-
-\begin{table}[!t]
-  \centering
-  \caption{3 clusters, each with 66 nodes}
-  \label{tab.cluster.3x67}
-
-  \begin{mytable}{1}
-    \hline
-    bandwidth  & 1 \\
-    \hline
-    latency    & 0.02 \\
-    \hline
-    power      & 1 \\
-    \hline
-    size       & 62 \\
-    \hline
-    Prec/Eprec & \np{E-5} \\
-    \hline
-    \hline
-    Relative gain    & 1.11 \\
-    \hline
-  \end{mytable}
-\end{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.
+
+\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}
+%  \label{tab.cluster.3x33}
+%
+%  \begin{mytable}{6}
+%    \hline
+%    bandwidth 
+%    & 10       & 5        & 4        & 3        & 2        & 6 \\
+%    \hline
+%    latency
+%    & 0.01     & 0.02     & 0.02     & 0.02     & 0.02     & 0.02 \\
+%    \hline
+%    power
+%    & 1        & 1        & 1        & 1        & 1        & 1 \\
+%    \hline
+%    size
+%    & 62       & 100      & 100      & 100      & 100      & 171 \\
+%    \hline
+%    Prec/Eprec
+%    & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
+%    \hline
+%    \hline
+%    Relative gain
+%    & 1.003    & 1.01     & 1.08     & 1.19     & 1.28     & 1.01 \\
+%    \hline
+%  \end{mytable}
+%\end{table}
+
+%In a final step, results of an execution attempt to scale up the three clustered
+%configuration but increasing by two hundreds hosts has been recorded in
+%Table~\ref{tab.cluster.3x67}.
+
+%\begin{table}[!t]
+%  \centering
+%  \caption{3 clusters, each with 66 nodes}
+%  \label{tab.cluster.3x67}
+%
+%  \begin{mytable}{1}
+%    \hline
+%    bandwidth  & 1 \\
+%    \hline
+%    latency    & 0.02 \\
+%    \hline
+%    power      & 1 \\
+%    \hline
+%    size       & 62 \\
+%    \hline
+%    Prec/Eprec & \np{E-5} \\
+%    \hline
+%    \hline
+%    Relative gain    & 1.11 \\
+%    \hline
+%  \end{mytable}
+%\end{table}

 
 Note that the program was run with the following parameters:
 
 \paragraph*{SMPI parameters}
 
 ~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).}
 
 Note that the program was run with the following parameters:
 
 \paragraph*{SMPI parameters}
 
 ~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).}

+\CER {Précisions ajoutées}
+

 \begin{itemize}
 \begin{itemize}

-\item HOSTFILE: Hosts file description.
-\item PLATFORM: file description of the platform architecture : clusters (CPU
-  power, \dots{}), intra cluster network description, inter cluster network
-  (bandwidth, latency, \dots{}).
+\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 :
+
+       - Processor unit power : 1.5 GFlops;
+
+       - Intracluster network : bandwidth = 1,25 Gbits/s and latency = 5E-05 ms;
+
+       - Intercluster network : bandwidth = 5 Mbits/s and latency = 5E-03 ms;

 \end{itemize}
 
 
 \paragraph*{Arguments of the program}
 
 \begin{itemize}
 \end{itemize}
 
 
 \paragraph*{Arguments of the program}
 
 \begin{itemize}

-       \item Description of the cluster architecture;
-       \item Maximum number of internal and external iterations;
-       \item Internal and external precisions;
+       \item Description of the cluster architecture matching the format <Number of cluster> <Number of hosts in cluster\_1> <Number of hosts in cluster\_2>;
+       \item Maximum number of iterations;
+       \item 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{6.0};
-       \item Matrix off-diagonal value: \np{-1.0};
-       \item Execution Mode: synchronous or asynchronous.
+       \item Matrix diagonal value: \np{1.0}   (See (3));
+       \item Matrix off-diagonal value: $-\frac{1}{6}$         (See(3));
+       \item Communication mode: Asynchronous.

 \end{itemize}
 
 \paragraph*{Interpretations and comments}
 
 \end{itemize}
 
 \paragraph*{Interpretations and comments}
 

-After analyzing the outputs, generally, for the configuration with two or three
-clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50}
-and~\ref{tab.cluster.3x33}), some combinations of the used parameters affecting
+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.
 
 the results have given a relative gain more than 2.5, showing the effectiveness of the
 asynchronous performance compared to the synchronous mode.
 

-In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows
-that with a deterioration of inter cluster network set with \np[Mbit/s]{5} of
-bandwidth, a latency in order of a hundredth of a millisecond and a system power
-of one GFlops, an efficiency of about \np[\%]{40} in asynchronous mode is
-obtained for a matrix size of 62 elements. It is noticed that the result remains
-stable even if we vary the external precision from \np{E-5} to \np{E-9}. By
+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
+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
 increasing the matrix size up to 100 elements, it was necessary to increase the

-CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a convergence of the algorithm
-with the same order of asynchronous mode efficiency.  Maintaining such a system
-power but this time, increasing network throughput inter cluster up to
-\np[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5\AG[]{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\AG[]{2.5 ?} is obtained with

 high external precision of \np{E-11} for a matrix size from 110 to 150 side
 elements.
 
 high external precision of \np{E-11} for a matrix size from 110 to 150 side
 elements.
 

-For the 3 clusters architecture including a total of 100 hosts,
-Table~\ref{tab.cluster.3x33} shows that it was difficult to have a combination
-which gives a relative gain of asynchronous mode more than 1.2. Indeed, for a
-matrix size of 62 elements, equality between the performance of the two modes
-(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}.
+%For the 3 clusters architecture including a total of 100 hosts,
+%Table~\ref{tab.cluster.3x33} shows that it was difficult to have a combination
+%which gives a relative gain of asynchronous mode more than 1.2. Indeed, for a
+%matrix size of 62 elements, equality between the performance of the two modes
+%(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
-obtained with a bandwidth of \np[Mbit/s]{1} as shown in
-Table~\ref{tab.cluster.3x67}.
+%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
+%obtained with a bandwidth of \np[Mbit/s]{1} as shown in
+%Table~\ref{tab.cluster.3x67}.

 
 \RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
 \RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
 \LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
 
 \RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
 \RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
 \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 
 \section{Conclusion}
 The experimental results on executing a parallel iterative algorithm in 
 asynchronous mode on an environment simulating a large scale of virtual