Correct value.

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index 77f7248e7ec920f5441e4c9e81178c6f34d6eb34..ab350c8423436ed9661f2b46edb98a4a1aa5fadb 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -31,6 +31,8 @@
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}

+\newcommand{\CER}[2][inline]{%
+  \todo[color=pink!10,#1]{\sffamily\textbf{CER:} #2}\xspace}

 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}


@@ -70,6 +72,7 @@
 
 \RC{Ordre des auteurs pas définitif.}
 \begin{abstract}
 
 \RC{Ordre des auteurs pas définitif.}
 \begin{abstract}

+\AG{L'abstract est AMHA incompréhensible et ne donne pas envie de lire la suite.}

 In recent years, the scalability of large-scale implementation in a 
 distributed environment of algorithms becoming more and more complex has 
 always been hampered by the limits of physical computing resources 
 In recent years, the scalability of large-scale implementation in a 
 distributed environment of algorithms becoming more and more complex has 
 always been hampered by the limits of physical computing resources 


@@ -149,11 +152,11 @@ 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
 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

-non-symmetric linear system of equations by numerical method GMRES (Generalized
-Minimal Residual) []\AG[]{[]?}.\LZK{Problème traité dans le papier est symétrique ou asymétrique? (Poisson 3D symétrique?)} We show, that with minor modifications of the
+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
 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[]{??}.
+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.  With selected
 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.  With selected


@@ -163,6 +166,9 @@ in the simulated environment, the experimental results have demonstrated not
 only the algorithm convergence within a reasonable time compared with the
 physical environment performance, but also a time saving of up to \np[\%]{40} in
 asynchronous mode.
 only the algorithm convergence within a reasonable time compared with the
 physical environment performance, but also a time saving of up to \np[\%]{40} in
 asynchronous 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!}

 
 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
 
 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


@@ -185,7 +191,9 @@ times generated by synchronizations are very penalizing. One way to overcome thi
 \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model. Here, local computations do not need to
 wait for required data. Processors can then perform their iterations with the data present at that time. Figure~\ref{fig:aiac}
 illustrates this model where the gray blocks represent the computation phases, the white spaces the idle
 \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model. Here, local computations do not need to
 wait for required data. Processors can then perform their iterations with the data present at that time. Figure~\ref{fig:aiac}
 illustrates this model where the gray blocks represent the computation phases, the white spaces the idle

-times and the arrows the communications. With this algorithmic model, the number of iterations required before the
+times and the arrows the communications.
+\AG{There are no ``white spaces'' on the figure.}
+With this algorithmic model, the number of iterations required before the

 convergence is generally greater than for the two former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC
 algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
 in a grid computing context.
 convergence is generally greater than for the two former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC
 algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
 in a grid computing context.


@@ -240,17 +248,27 @@ 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:hal-00919507}, and supports applications written in C or
 Fortran, with little or no modifications.
 

-With SimGrid, the execution of a distributed application is simulated on a
+Within SimGrid, the execution of a distributed application is simulated on a

 single machine.  The application code is really executed, but some operations
 single machine.  The application code is really executed, but some operations

-like the communications are intercepted to be simulated 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 node
-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.
-
-\AG{Faut-il ajouter quelque-chose ?}
+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 node 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.
+
+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
+simulated application, SimGrid/SMPI allows to skip long lasting computations and
+to only take their duration into account.  When the real computations cannot be
+skipped, but the results have no importance for the simulation results, there is
+also the possibility to share dynamically allocated data structures between
+several simulated processes, and thus to reduce the whole memory consumption.
+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}


@@ -319,19 +337,7 @@ is solved independently by a cluster and communications are required to update t
 \label{algo:01}
 \end{figure}
 
 \label{algo:01}
 \end{figure}
 

-Algorithm on Figure~\ref{algo:01} shows the main key points of the
-multisplitting method to solve a large sparse linear system. This algorithm is
-based on an outer-inner iteration method where the parallel synchronous GMRES
-method is used to solve the inner iteration. It is executed in parallel by each
-cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors
-with the subscript $l$ represent the local data for cluster $l$, while
-$\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and
-$\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with
-neighboring clusters. At every outer iteration $k$, asynchronous communications
-are performed between processors of the local cluster and those of distant
-clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in
-Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are
-exchanged by message passing using MPI non-blocking communication routines.
+Algorithm on Figure~\ref{algo:01} shows the main key points of the multisplitting method to solve a large sparse linear system. This algorithm is based on an outer-inner iteration method where the parallel synchronous GMRES method is used to solve the inner iteration. It is executed in parallel by each cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors with the subscript $l$ represent the local data for cluster $l$, while $\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with neighboring clusters. At every outer iteration $k$, asynchronous communications are performed between processors of the local cluster and those of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines.

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


@@ -349,7 +355,7 @@ 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 $i$
 sets the token to \textit{True} if the local convergence is achieved or to
 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 $i$
 sets the token to \textit{True} if the local convergence is achieved or to

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


@@ -358,25 +364,23 @@ condition is satisfied
 \begin{equation*}
   (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)
 \end{equation*}
 \begin{equation*}
   (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)
 \end{equation*}

-where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$. 
+where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the
+tolerance threshold of the error computed between two successive local solution
+$X_l^k$ and $X_l^{k+1}$.

 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SIMGRI\LZK[]{SimGrid} unless some code 
-debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between the six neighbors of each point in a submatrix within a cluster or 
-between clusters, \LZK{Il faut expliquer pourquoi 6 points voisins (7-point stencil problem)}
-the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. In synchronous 
+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.
 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.

-\LZK{Peut-être mettre plus de précisions sur les difficultés rencontrées dans la version async et les adaptaions effectuées pour SimGrid}
+\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}

 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. 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 
 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. 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 tested in synchronous mode with a simulated platform starting from a modest 2 or 3 clusters grid to a larger configuration like simulating 
-Grid5000 with more than 1500 hosts with 5000 cores~\cite{bolze2006grid}. Once the code debugging and adaptation were complete, the next section shows our methodology and experimental results.\LZK{Dernière phrase peut être supprimée}
-
+environment. We have successfully executed the code in synchronous mode using GMRES algorithm compared with a multisplitting method in asynchrnous mode after few modification. 

 
 
 \section{Experimental results}
 
 
 \section{Experimental results}


@@ -393,26 +397,57 @@ study that the results depend on the following parameters:
   \textit{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
   \textit{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 (i.e. speed-up less than 1).
+  less than in synchronous mode. The ratio between the execution time of asynchronous compared to the synchronous mode is defined as the "relative gain". So, our objective running the algorithm in SimGrid is to obtain a relative gain greater than 1.

 \end{itemize}
 \end{itemize}

-\LZK{Propositions pour changer le terme ``speedup'': acceleration ratio ou relative gain}
-
-A priori, obtaining a speedup less than 1 would be difficult in a local area
+\LZK{Propositions pour remplacer le terme ``speedup'': acceleration ratio ou relative gain}
+\CER{C'est fait. En conséquence, les tableaux et les commentaires ont été aussi modifiés}
+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
 configuration, degrading the inter-cluster network performance will
 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
 configuration, degrading the inter-cluster network performance will

-\textit{penalize} the synchronous mode allowing to get a speedup lower than 1.
-This action simulates the case of clusters linked with long distance network
+\textit{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.
 
 like Internet.
 

+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 providing 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} =
 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 providing 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{5211000}}$ entries.
-\LZK{Donner le type et la description du problème traité (problème symétrique Poisson 3D) et préciser peut être aussi qu'on a utilisé un partitionnement 3D}
+\text{\np{5000211}}$ entries.

 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}


@@ -444,8 +479,8 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
     Prec/Eprec
     & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} \\
     \hline
     Prec/Eprec
     & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} \\
     \hline

-    speedup
-    & 0.396     & 0.392     & 0.396     & 0.391     & 0.393     & 0.395 \\
+    Relative gain
+    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54     & 2.53 \\

     \hline
   \end{mytable}
 
     \hline
   \end{mytable}
 


@@ -468,8 +503,8 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
     Prec/Eprec
     & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5}  & \np{E-5} \\
     \hline
     Prec/Eprec
     & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5}  & \np{E-5} \\
     \hline

-    speedup
-    & 0.398     & 0.388     & 0.393     & 0.394     & 0.63      & 0.778 \\
+    Relative gain
+    & 2.51     & 2.58     & 2.55     & 2.54     & 1.59      & 1.29 \\

     \hline
   \end{mytable}
 \end{table}
     \hline
   \end{mytable}
 \end{table}


@@ -478,7 +513,7 @@ 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
 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

-speedups less than 1 with a matrix size from 62 to 100 elements.
+relative gains greater than 1 with a matrix size from 62 to 100 elements.

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


@@ -502,8 +537,8 @@ speedups less than 1 with a matrix size from 62 to 100 elements.
     Prec/Eprec
     & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
     \hline
     Prec/Eprec
     & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
     \hline

-    speedup
-    & 0.997    & 0.99     & 0.93     & 0.84     & 0.78     & 0.99 \\
+    Relative gain
+    & 1.003    & 1.01     & 1.08     & 0.19     & 1.28     & 1.01 \\

     \hline
   \end{mytable}
 \end{table}
     \hline
   \end{mytable}
 \end{table}


@@ -529,7 +564,7 @@ Table~\ref{tab.cluster.3x67}.
     \hline
     Prec/Eprec & \np{E-5} \\
     \hline
     \hline
     Prec/Eprec & \np{E-5} \\
     \hline

-    speedup    & 0.9 \\
+    Relative gain    & 1.11 \\

     \hline
   \end{mytable}
 \end{table}
     \hline
   \end{mytable}
 \end{table}


@@ -553,7 +588,8 @@ lat latency, \dots{}).
        \item Maximum number of internal and external iterations;
        \item Internal and external precisions;
        \item Matrix size $N_x$, $N_y$ and $N_z$;
        \item Maximum number of internal and external iterations;
        \item Internal and external precisions;
        \item Matrix size $N_x$, $N_y$ and $N_z$;

-       \item Matrix diagonal value: \np{6.0}, \LZK{Off-diagonal values? (-1.0?)}
+       \item Matrix diagonal value: \np{6.0};
+       \item Matrix Off-diagonal value: \np{-1.0};

        \item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 
        \item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 


@@ -562,7 +598,7 @@ lat latency, \dots{}).
 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 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

-the results have given a speedup less than 1, showing the effectiveness of the
+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
 asynchronous performance compared to the synchronous mode.
 
 In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows


@@ -571,29 +607,29 @@ 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
 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

-increasing the problem 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
 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 of about \np[\%]{40} is obtained with
+\np[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5 is obtained with

 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
 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 an efficiency of asynchronous below \np[\%]{80}. Indeed, for a
+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
 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 by
-\np[\%]{78} with a matrix size of 100 points, it was necessary to degrade the
+\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}.
 
 A last attempt was made for a configuration of three clusters but more powerful
 inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
 
 A last attempt was made for a configuration of three clusters but more powerful

-with 200 nodes in total. The convergence with a speedup of \np[\%]{90} was
+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}.
 
 \LZK{Dans le papier, on compare les deux versions synchrone et asycnhrone du multisplitting. Y a t il des résultats pour comparer gmres parallèle classique avec multisplitting asynchrone? Ca permettra de montrer l'intérêt du multisplitting asynchrone sur des clusters distants}
 obtained with a bandwidth of \np[Mbit/s]{1} as shown in
 Table~\ref{tab.cluster.3x67}.
 
 \LZK{Dans le papier, on compare les deux versions synchrone et asycnhrone du multisplitting. Y a t il des résultats pour comparer gmres parallèle classique avec multisplitting asynchrone? Ca permettra de montrer l'intérêt du multisplitting asynchrone sur des clusters distants}

+\CER{En fait, les résultats ont été obtenus en comparant les temps d'exécution entre l'algo classique GMRES en mode synchrone avec le multisplitting en mode asynchrone, le tout sur un environnement de clusters distants}

 
 \section{Conclusion}
 The experimental results on executing a parallel iterative algorithm in 
 
 \section{Conclusion}
 The experimental results on executing a parallel iterative algorithm in