Use \ell instead of l in equations.

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index d37619469442bee9c8f51c604bb526d60a114ab8..e780bbdbcee4303ed1683158d297dbe278c226d7 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -41,6 +41,7 @@
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newcommand{\MI}{\mathit{MaxIter}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newcommand{\MI}{\mathit{MaxIter}}

+\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}}

 
 \begin{document}
 
 
 \begin{document}
 


@@ -97,6 +98,8 @@ a residual precision up to \np{E-11}. Such successful results open
 perspectives on experimentations for running the algorithm on a 
 simulated large scale growing environment and with larger problem size. 
 
 perspectives on experimentations for running the algorithm on a 
 simulated large scale growing environment and with larger problem size. 
 

+\LZK{Long\ldots}
+

 % no keywords for IEEE conferences
 % Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
 \end{abstract}
 % no keywords for IEEE conferences
 % Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
 \end{abstract}


@@ -172,7 +175,7 @@ asynchronous mode.
 
 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

-distributed architectures. The algorithm of  the multisplitting method used by GMRES written with MPI primitives and
+distributed architectures. The algorithm of  the multisplitting method used by GMRES \LZK{??? GMRES n'utilise pas la méthode de multisplitting! Sinon ne doit on pas expliquer le choix d'une méthode de multisplitting?} 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.
  
 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.
  


@@ -196,7 +199,7 @@ times and the arrows the communications.
 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
 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.
+in a grid computing context.\LZK{Répétition par rapport à l'intro}

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


@@ -248,21 +251,27 @@ 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.
-
-%%% TODO: add some words+refs about SimGrid's accuracy and scalability.}
-
-\AG{Faut-il ajouter quelque-chose ?} 
-\CER{Comme tu as décrit la plateforme d'exécution, on peut ajouter éventuellement le fichier XML contenant des hosts dans les clusters formant la grille
-  \AG{Bof.}}
+like the communications are intercepted, and their running time is computed
+according to the characteristics of the simulated execution platform.  The
+description of this target platform is given as an input for the execution, by
+the mean of an XML file.  It describes the properties of the platform, such as
+the computing nodes with their computing power, the interconnection links with
+their bandwidth and latency, and the routing strategy.  The simulated running
+time of the application is computed according to these properties.
+
+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}


@@ -287,51 +296,75 @@ Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, whe
       B_L
     \end{array} \right)
 \end{equation*}
       B_L
     \end{array} \right)
 \end{equation*}

-in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$.
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$
+are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$ $A_{\ell
+  m}$ is a rectangular block of $A$ of size $n_\ell\times n_m$, $X_\ell$ and
+$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each and
+$\sum_{\ell} n_\ell=\sum_{m} n_m=n$.

 
 The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
 \begin{equation}
   \label{eq:4.1}
   \left\{
     \begin{array}{l}
 
 The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
 \begin{equation}
   \label{eq:4.1}
   \left\{
     \begin{array}{l}

-      A_{ll}X_l = Y_l \text{, such that}\\
-      Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
+      A_{\ell\ell}X_\ell = Y_\ell \text{, such that}\\
+      Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\ m\neq \ell}}^{L}A_{\ell m}X_m

     \end{array}
   \right.
 \end{equation}
     \end{array}
   \right.
 \end{equation}

-is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers. 
+is solved independently by a cluster and communications are required to update
+the right-hand side sub-vector $Y_\ell$, such that the sub-vectors $X_m$
+represent the data dependencies between the clusters. As each sub-system
+(\ref{eq:4.1}) is solved in parallel by a cluster of processors, our
+multisplitting method uses an iterative method as an inner solver which is
+easier to parallelize and more scalable than a direct method. In this work, we
+use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most
+used iterative method by many researchers.

 
 \begin{figure}[!t]
   %%% IEEE instructions forbid to use an algorithm environment here, use figure
   %%% instead
 \begin{algorithmic}[1]
 
 \begin{figure}[!t]
   %%% IEEE instructions forbid to use an algorithm environment here, use figure
   %%% instead
 \begin{algorithmic}[1]

-\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
-\Output $X_l$ (solution sub-vector)\vspace{0.2cm}
-\State Load $A_l$, $B_l$
+\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
+\Output $X_\ell$ (solution sub-vector)\medskip
+
+\State Load $A_\ell$, $B_\ell$

 \State Set the initial guess $x^0$
 \For {$k=0,1,2,\ldots$ until the global convergence}
 \State Restart outer iteration with $x^0=x^k$
 \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
 \State Set the initial guess $x^0$
 \For {$k=0,1,2,\ldots$ until the global convergence}
 \State Restart outer iteration with $x^0=x^k$
 \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}

-\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters
-\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
+\State\label{algo:01:send} Send shared elements of $X_\ell^{k+1}$ to neighboring clusters
+\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq \ell}$

 \EndFor
 
 \Statex
 
 \Function {InnerSolver}{$x^0$, $k$}
 \EndFor
 
 \Statex
 
 \Function {InnerSolver}{$x^0$, $k$}

-\State Compute local right-hand side $Y_l$:
+\State Compute local right-hand side $Y_\ell$:

        \begin{equation*}
        \begin{equation*}

-         Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0
+         Y_\ell = B_\ell - \sum\nolimits^L_{\substack{m=1\\ m\neq \ell}}A_{\ell m}X_m^0

        \end{equation*}
        \end{equation*}

-\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method
-\State \Return $X_l^k$
+\State Solving sub-system $A_{\ell\ell}X_\ell^k=Y_\ell$ with the parallel GMRES method
+\State \Return $X_\ell^k$

 \EndFunction
 \end{algorithmic}
 \caption{A multisplitting solver with GMRES method}
 \label{algo:01}
 \end{figure}
 
 \EndFunction
 \end{algorithmic}
 \caption{A multisplitting solver with GMRES method}
 \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 $\ell,m\in\{1,\ldots,L\}$, the matrices and
+vectors with the subscript $\ell$ represent the local data for cluster $\ell$,
+while $\{A_{\ell m}\}_{m\neq \ell}$ are off-diagonal matrices of sparse matrix
+$A$ and $\{X_m\}_{m\neq \ell}$ 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


@@ -353,14 +386,14 @@ sets the token to \textit{True} if the local convergence is achieved or to
 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,

-the local convergence on each cluster $l$ is detected when the following
+the local convergence on each cluster $\ell$ is detected when the following

 condition is satisfied
 \begin{equation*}
 condition is satisfied
 \begin{equation*}

-  (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)
+  (k\leq \MI) \text{ or } (\|X_\ell^k - X_\ell^{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
 \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 $X_l^{k+1}$.
+$X_\ell^k$ and $X_\ell^{k+1}$.

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


@@ -372,10 +405,11 @@ Note here that the use of SMPI functions optimizer for memory footprint and CPU
 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
 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.
+also to be reviewed.
+\AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.}
+ 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 
 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}.
+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. 

 
 
 
 
 
 


@@ -393,17 +427,16 @@ 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 remplacer le terme ``speedup'': acceleration ratio ou relative gain}

 
 

-A priori, obtaining a speedup less than 1 would be difficult in a local area
+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.
 
 In this paper, we solve the 3D Poisson problem whose the mathematical model is 
 like Internet.
 
 In this paper, we solve the 3D Poisson problem whose the mathematical model is 


@@ -443,7 +476,8 @@ 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} =
 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.
+\text{\np{5000211}}$ entries.
+\AG{Expliquer comment lire les tableaux.}

 
 % 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}}


@@ -475,8 +509,8 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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}
 


@@ -499,8 +533,8 @@ $\text{62}^\text{3} = \text{\np{238328}}$ 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}


@@ -509,7 +543,7 @@ 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


@@ -533,8 +567,8 @@ 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     & 1.19     & 1.28     & 1.01 \\

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


@@ -560,7 +594,7 @@ 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}


@@ -569,6 +603,7 @@ Note that the program was run with the following parameters:
 
 \paragraph*{SMPI parameters}
 
 
 \paragraph*{SMPI parameters}
 

+~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).}

 \begin{itemize}
        \item HOSTFILE: Hosts file description.
        \item PLATFORM: file description of the platform architecture : clusters (CPU power,
 \begin{itemize}
        \item HOSTFILE: Hosts file description.
        \item PLATFORM: file description of the platform architecture : clusters (CPU power,


@@ -584,11 +619,8 @@ 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 $N_z$;

-%<<<<<<< HEAD

        \item Matrix diagonal value: \np{6.0};
        \item Matrix diagonal value: \np{6.0};

-       \item Matrix Off-diagonal value: \np{-1.0};
-%=======
-%>>>>>>> 5fb6769d88c1720b6480a28521119ef010462fa6
+       \item Matrix off-diagonal value: \np{-1.0};

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


@@ -597,7 +629,7 @@ 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


@@ -606,29 +638,30 @@ 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}.
 
 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}
+\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 bw et lat sont ceux inter-clusters ou pour les deux inter et intra cluster??}

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