Ajout de la partie d adaptation du programme a Siimgrid

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index 2c7d65d7a6822eb285a1eab87f3fc93d882b4df9..3d861f3608b35fb40bd57a136053be10cad45b1a 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -40,11 +40,6 @@
 
 \newcommand{\MI}{\mathit{MaxIter}}
 
 
 \newcommand{\MI}{\mathit{MaxIter}}
 

-\usepackage{array}
-\usepackage{color, colortbl}
-\newcolumntype{M}[1]{>{\centering\arraybackslash}m{#1}}
-\newcolumntype{Z}[1]{>{\raggedleft}m{#1}}
-

 \begin{document}
 
 \title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
 \begin{document}
 
 \title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}


@@ -113,27 +108,27 @@ network, etc.) but also a non-negligible CPU execution time. We consider in this
 parallel algorithms called \texttt{numerical iterative algorithms} executed in a distributed environment. As their name
 suggests, these algorithm solves a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
 $X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods
 parallel algorithms called \texttt{numerical iterative algorithms} executed in a distributed environment. As their name
 suggests, these algorithm solves a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
 $X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods

-demonstrate the convergence of these algorithms \cite{}. 
+demonstrate the convergence of these algorithms \cite{BT89,Bahi07}. 

 
 

-Parallelization of such algorithms generally involved the division of the problem into several \emph{pieces} that will
+Parallelization of such algorithms generally involved the division of the problem into several \emph{blocks} that will

 be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new
 be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new

-iteration starts  until the approximate solution is reached. These parallel  computations can be performed either in
-\emph{synchronous} communication mode where a new iteration begin only when all nodes communications are completed,
+iteration starts and until the approximate solution is reached. These parallel  computations can be performed either in
+\emph{synchronous} mode where a new iteration begin only when all nodes communications are completed,

 either \emph{asynchronous} mode where processors can continue independently without or few synchronization points. For
 instance in the \textit{Asynchronous Iterations - Asynchronous   Communications (AIAC)} model \cite{bcvc06:ij}, local
 computations do not need to wait for required data. Processors can then perform their iterations with the data present
 at that time. Even if the number of iterations required before the convergence is generally greater than for the
 synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to
 either \emph{asynchronous} mode where processors can continue independently without or few synchronization points. For
 instance in the \textit{Asynchronous Iterations - Asynchronous   Communications (AIAC)} model \cite{bcvc06:ij}, local
 computations do not need to wait for required data. Processors can then perform their iterations with the data present
 at that time. Even if the number of iterations required before the convergence is generally greater than for the
 synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to

-synchronizations especially in a grid computing context (see \cite{bcvc06:ij} for more details).
+synchronizations especially in a grid computing context (see \cite{Bahi07} for more details).

 
 Parallel numerical applications (synchronous or asynchronous) may have different configuration and deployment
 
 Parallel numerical applications (synchronous or asynchronous) may have different configuration and deployment

-requirements.  Quantifying their performance of resource allocation policies and application scheduling algorithms in
-grid computing environments under varying load, CPU power and network speeds is very costly, labor intensive and time
-consuming \cite{BuRaCa}. The case of AIAC algorithms is even more problematic since they are very sensible to the
+requirements.  Quantifying their resource allocation policies and application scheduling algorithms in
+grid computing environments under varying load, CPU power and network speeds is very costly, very labor intensive and very time
+consuming \cite{Calheiros:2011:CTM:1951445.1951450}. The case of AIAC algorithms is even more problematic since they are very sensible to the

 execution environment context. For instance, variations in the network bandwith (intra and inter- clusters), in the
 number and the power of nodes, in the number of clusters... can lead to very different number of iterations and so to
 execution environment context. For instance, variations in the network bandwith (intra and inter- clusters), in the
 number and the power of nodes, in the number of clusters... can lead to very different number of iterations and so to

-very different execution times. In this context, it appears that the use of simulation tools to explore various platform
-scenarios and to run enormous numbers of experiments quickly can be very promising. In this way, the use of a simulation
+very different execution times. Then, it appears that the use of simulation tools to explore various platform
+scenarios and to run large numbers of experiments quickly can be very promising. In this way, the use of a simulation

 environment to execute parallel  iterative algorithms found some interests in reducing the highly cost of  access to
 computing resources: (1) for the applications development life  cycle and in code debugging (2) and in production to get
 results in a reasonable execution time with a simulated infrastructure not accessible  with physical resources. Indeed,
 environment to execute parallel  iterative algorithms found some interests in reducing the highly cost of  access to
 computing resources: (1) for the applications development life  cycle and in code debugging (2) and in production to get
 results in a reasonable execution time with a simulated infrastructure not accessible  with physical resources. Indeed,


@@ -142,15 +137,17 @@ environment challenges to find optimal configurations giving the best results wi
 best of execution time. 
 
 To our knowledge, there is no existing work on the large-scale simulation of a real AIAC application. The aim of this
 best of execution time. 
 
 To our knowledge, there is no existing work on the large-scale simulation of a real AIAC application. The aim of this

-paper is to give a first approach of the simulation of AIAC algorithms using the SimGrid toolkit \cite{SimGrid}. We had
-in the scope of this work implemented a  program for solving large non-symmetric linear system of equations by numerical
-method GMRES (Generalized Minimal Residual). 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. The simulated results we obtained are in line with real results exposed in ?? In addition, it has been
-permitted to show the effectiveness of asynchronous mode algorithm by comparing its performance with the synchronous
-mode time. With selected parameters on the network platforms (bandwidth, latency of inter  cluster  network) and on the
-clusters architecture (number, capacity calculation power) in the simulated environment, the experimental results have
-demonstrated not only the algorithm convergence within a reasonable time compared with the physical environment
+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 non-symmetric
+linear system of equations by numerical method GMRES (Generalized Minimal Residual) []. We show, that with minor
+modifications of the initial MPI code, the SimGrid toolkit allows us to perform a test campain of a real AIAC
+application on different computing architectures. The simulated results we obtained are in line with real results
+exposed in ??. 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. It has been
+permitted to show  With selected parameters on the network platforms (bandwidth, latency of inter  cluster network) and
+on the clusters architecture (number, capacity calculation power) 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.
 
 This article is structured as follows: after this introduction, the next  section will give a brief description of
 performance, but also a time saving of up to \np[\%]{40} in asynchronous mode.
 
 This article is structured as follows: after this introduction, the next  section will give a brief description of


@@ -163,7 +160,7 @@ carried out will be presented before some concluding remarks and future works.
 
 As exposed in the introduction, parallel iterative methods are now widely used in many scientific domains. They can be
 classified in three main classes depending on how iterations and communications are managed (for more details readers
 
 As exposed in the introduction, parallel iterative methods are now widely used in many scientific domains. They can be
 classified in three main classes depending on how iterations and communications are managed (for more details readers

-can refer to \cite{bcvc02:ip}). In the \textit{Synchronous Iterations - Synchronous Communications (SISC)} model data
+can refer to \cite{bcvc06:ij}). In the \textit{Synchronous Iterations - Synchronous Communications (SISC)} model data

 are exchanged at the end of each iteration. All the processors must begin the same iteration at the same time and
 important idle times on processors are generated. The \textit{Synchronous Iterations - Asynchronous Communications
 (SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously
 are exchanged at the end of each iteration. All the processors must begin the same iteration at the same time and
 important idle times on processors are generated. The \textit{Synchronous Iterations - Asynchronous Communications
 (SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously


@@ -208,16 +205,18 @@ iterations and so to very different execution times.
 SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} is a simulation
 framework to sudy the behavior of large-scale distributed systems.  As its name
 says, it emanates from the grid computing community, but is nowadays used to
 SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} is a simulation
 framework to sudy the behavior of large-scale distributed systems.  As its name
 says, it emanates from the grid computing community, but is nowadays used to

-study grids, clouds, HPC or peer-to-peer systems.
-%- open source, developped since 1999, one of the major solution in the field
-%
+study grids, clouds, HPC or peer-to-peer systems.  The early versions of SimGrid
+date from 1999, but it's still actively developped and distributed as an open
+source software.  Today, it's one of the major generic tools in the field of
+simulation for large-scale distributed systems.
+

 SimGrid provides several programming interfaces: MSG to simulate Concurrent
 Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
 run real applications written in MPI~\cite{MPI}.  Apart from the native C
 interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
 languages.  The SMPI interface supports applications written in C or Fortran,
 SimGrid provides several programming interfaces: MSG to simulate Concurrent
 Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
 run real applications written in MPI~\cite{MPI}.  Apart from the native C
 interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
 languages.  The SMPI interface supports applications written in C or Fortran,

-with little or no modifications.
-%- implements most of MPI-2 \cite{ref} standard [CHECK]
+with little or no modifications.  SMPI implements about \np[\%]{80} of the MPI
+2.0 standard~\cite{bedaride:hal-00919507}.

 
 %%% explain simulation
 %- simulated processes folded in one real process
 
 %%% explain simulation
 %- simulated processes folded in one real process


@@ -231,12 +230,10 @@ with little or no modifications.
 
 %%% validation + refs
 
 
 %%% validation + refs
 

-\AG{Décrire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} (Arnaud)}
-

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}
 %Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}
 %Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.

-Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors. In this case, we apply a row-by-row splitting without overlapping  
+Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping  

 \[
 \left(\begin{array}{ccc}
 A_{11} & \cdots & A_{1L} \\
 \[
 \left(\begin{array}{ccc}
 A_{11} & \cdots & A_{1L} \\


@@ -323,7 +320,20 @@ where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tole
 
 \LZK{Description du processus d'adaptation de l'algo multisplitting à SimGrid}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \LZK{Description du processus d'adaptation de l'algo multisplitting à SimGrid}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-
+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 the six neighbors of each point in a submatrix 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. 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 units 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\_Finalise 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 [?]. Once the code debugging and adaptation were complete, the next section shows our methodology and experimental 
+results.

 
 
 
 
 
 


@@ -333,31 +343,30 @@ where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tole
 
 \section{Experimental results}
 
 
 \section{Experimental results}
 

-When the \emph{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
-study that the results depend on the following parameters: (1) at the network
-level, we found that the most critical values are the bandwidth (bw) and the
-network latency (lat). (2) Hosts power (GFlops) can also influence on the
-results. And finally, (3) when submitting job batches for execution, the
-arguments values passed to the program like the maximum number of iterations or
-the \emph{external} precision are critical 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, in others words, in having a \emph{speedup} less than 1
-({speedup}${}={}${execution time in synchronous mode}${}/{}${execution time in
-asynchronous mode}).
-
-A priori, obtaining a speedup less than 1 would be difficult in a local area
+When the \emph{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
+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 (bw) and the network latency (lat).  
+\item Hosts power (GFlops) can also
+influence on the results.  
+\item Finally, when submitting job batches for execution, the arguments values passed to the
+program like the maximum number of iterations or the \emph{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 $1$).
+\end{itemize}
+
+A priori, obtaining a speedup less 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 \emph{penalize} the synchronous
 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 \emph{penalize} the synchronous

-mode allowing to get a speedup lower than 1. This action simulates the case of
+mode allowing to get a speedup lower than $1$. This action simulates the case of

 clusters linked with long distance network like Internet.
 
 As a first step, the algorithm was run on a network consisting of two clusters
 clusters linked with long distance network like Internet.
 
 As a first step, the algorithm was run on a network consisting of two clusters

-containing fifty hosts each, totaling one hundred hosts. Various combinations of
+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 $N_x = N_y = N_z =
 62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to $171^{3} =
 the above factors have providing the results shown in
 Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z =
 62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to $171^{3} =


@@ -365,48 +374,91 @@ Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z =
 
 \begin{table}[!t]
   \centering
 
 \begin{table}[!t]
   \centering

-  \caption{2 clusters, each with 50 nodes}
+  \caption{$2$ clusters, each with $50$ nodes}

   \label{tab.cluster.2x50}
   \label{tab.cluster.2x50}

-
- \tiny
- 
-\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|} 
- \hline 
- \bf bw & 5 &5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 & 10 & 10\\ 
- \hline
- \bf lat & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01\\ 
- \hline 
- \bf power & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5\\ \hline    \bf size & 62 & 62 & 62 & 100 & 100 & 110 & 120& 130 & 140 & 150 & 171 & 171\\ \hline
- \bf Prec/Eprec & 10$^{-5}$ & 10$^{-8}$ & 10$^{-9}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline 
- \bf speedup & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778\\ \hline 
- \end{tabular}
-\end{table} 
+  \renewcommand{\arraystretch}{1.3}
+
+  \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+    \hline
+    bw
+    & 5         & 5         & 5         & 5         & 5         & 50 \\
+    \hline
+    lat
+    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\
+    \hline
+    power
+    & 1         & 1         & 1         & 1.5       & 1.5       & 1.5 \\
+    \hline
+    size
+    & 62        & 62        & 62        & 100       & 100       & 110 \\
+    \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 \\
+    \hline
+  \end{tabular}
+
+  \smallskip
+
+  \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+    \hline
+    bw
+    & 50        & 50        & 50        & 50        & 10        & 10 \\
+    \hline
+    lat
+    & 0.02      & 0.02      & 0.02      & 0.02      & 0.03      & 0.01 \\
+    \hline
+    power
+    & 1.5       & 1.5       & 1.5       & 1.5       & 1         & 1.5 \\
+    \hline
+    size
+    & 120       & 130       & 140       & 150       & 171       & 171 \\
+    \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 \\
+    \hline
+  \end{tabular}
+\end{table}

   
 Then we have changed the network configuration using three clusters containing
   
 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
+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
 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.
+speedups less than $1$ with a matrix size from $62$ to $100$ elements.

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

-  \caption{3 clusters, each with 33 nodes}
+  \caption{$3$ clusters, each with $33$ nodes}

   \label{tab.cluster.3x33}
   \label{tab.cluster.3x33}

-
- \tiny
- 
-\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|} 
- \hline 
- \bf bw & 10 &5 & 4 & 3 & 2 & 6\\ \hline
- \bf lat & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02\\ 
- \hline 
- \bf power & 1 & 1 & 1 & 1 & 1 & 1\\ \hline    
- \bf size & 62 & 100 & 100 & 100 & 100 & 171\\ \hline
- \bf Prec/Eprec & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline 
- \bf speedup & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99\\ 
- \hline 
- \end{tabular}
-\end{table} 
+  \renewcommand{\arraystretch}{1.3}
+
+  \begin{tabular}{|>{\bfseries}r|*{6}{c|}}
+    \hline
+    bw
+    & 10       & 5        & 4        & 3        & 2        & 6 \\
+    \hline
+    lat
+    & 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
+    speedup
+    & 0.997    & 0.99     & 0.93     & 0.84     & 0.78     & 0.99 \\
+    \hline
+  \end{tabular}
+\end{table}

 
 
 In a final step, results of an execution attempt to scale up the three clustered
 
 
 In a final step, results of an execution attempt to scale up the three clustered


@@ -417,23 +469,24 @@ Table~\ref{tab.cluster.3x67}.
   \centering
   \caption{3 clusters, each with 66 nodes}
   \label{tab.cluster.3x67}
   \centering
   \caption{3 clusters, each with 66 nodes}
   \label{tab.cluster.3x67}

-
- \tiny
-\begin{tabular}{|M{0.55cm}|M{0.25cm}|} 
- \hline 
- \bf bw & 1\\ \hline
- \bf lat & 0.02\\ 
- \hline 
- \bf power & 1\\ 
- \hline    
- \bf size & 62\\ 
- \hline
- \bf Prec/Eprec & 10$^{-5}$\\ 
- \hline 
- \bf speedup & 0.9\\ 
- \hline 
+  \renewcommand{\arraystretch}{1.3}
+
+  \begin{tabular}{|>{\bfseries}r|c|}
+    \hline
+    bw         & 1 \\
+    \hline
+    lat        & 0.02 \\
+    \hline
+    power      & 1 \\
+    \hline
+    size       & 62 \\
+    \hline
+    Prec/Eprec & \np{E-5} \\
+    \hline
+    speedup    & 0.9 \\
+    \hline

  \end{tabular}
  \end{tabular}

-\end{table} 
+\end{table}

 
 Note that the program was run with the following parameters:
 
 
 Note that the program was run with the following parameters:
 


@@ -472,21 +525,21 @@ 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 problem 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[Mbits/s]{50}, the result of efficiency of about \np[\%]{40} is obtained with
 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[Mbits/s]{50}, the result of efficiency of about \np[\%]{40} is obtained with

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

 elements.
 
 elements.
 

-For the 3 clusters architecture including a total of 100 hosts,
+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
 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

-matrix size of 62 elements, equality between the performance of the two modes
+matrix size of $62$ elements, equality between the performance of the two modes

 (synchronous and asynchronous) is achieved with an inter cluster of
 \np[Mbits/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency by
 (synchronous and asynchronous) is achieved with an inter cluster of
 \np[Mbits/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[\%]{78} with a matrix size of $100$ points, it was necessary to degrade the

 inter cluster network bandwidth from 5 to 2 Mbit/s.
 
 A last attempt was made for a configuration of three clusters but more powerful
 inter cluster network bandwidth from 5 to 2 Mbit/s.
 
 A last attempt was made for a configuration of three clusters but more powerful


@@ -526,7 +579,7 @@ mode in a grid architecture.
 
 \section*{Acknowledgment}
 
 
 \section*{Acknowledgment}
 

-
+This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).

 The authors would like to thank\dots{}
 
 
 The authors would like to thank\dots{}