Use labels to reference line numbers of algorithm.

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index 7cb74fa528d3929115637c95f4bb0573f190ee13..3dd67ef8614cc072037465711a8576e58d6263e2 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -68,7 +68,7 @@
 
 \maketitle
 
 
 \maketitle
 

-\RC{Ordre des autheurs pas définitif.}
+\RC{Ordre des auteurs pas définitif.}

 \begin{abstract}
 In recent years, the scalability of large-scale implementation in a 
 distributed environment of algorithms becoming more and more complex has 
 \begin{abstract}
 In recent years, the scalability of large-scale implementation in a 
 distributed environment of algorithms becoming more and more complex has 


@@ -105,27 +105,27 @@ problems raised by  researchers on various scientific disciplines but also by in
 increasing complexity of these requested  applications combined with a continuous increase of their sizes lead to  write
 distributed and parallel algorithms requiring significant hardware  resources (grid computing, clusters, broadband
 network, etc.) but also a non-negligible CPU execution time. We consider in this paper a class of highly efficient
 increasing complexity of these requested  applications combined with a continuous increase of their sizes lead to  write
 distributed and parallel algorithms requiring significant hardware  resources (grid computing, clusters, broadband
 network, etc.) but also a non-negligible CPU execution time. We consider in this paper a class of highly efficient

-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
+parallel algorithms called \emph{numerical iterative algorithms} executed in a distributed environment. As their name
+suggests, these algorithms solve 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
 $X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods

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

 
 

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

 be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new
 iteration starts and until the approximate solution is reached. These parallel  computations can be performed either in
 be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new
 iteration starts and until the approximate solution is reached. These 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
+\emph{synchronous} mode where a new iteration begins only when all nodes communications are completed,
+or in \emph{asynchronous} mode where processors can continue independently with few or no 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
 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{Bahi07} 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
 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
 
 Parallel numerical applications (synchronous or asynchronous) may have different configuration and deployment
 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
+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 bandwidth (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. 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
 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. 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


@@ -138,21 +138,21 @@ 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 twofold. First we give a first approach of the simulation of AIAC algorithms using a simulation tool (i.e. the
 
 To our knowledge, there is no existing work on the large-scale simulation of a real AIAC application. The aim of this
 paper is twofold. First we give a first approach of the simulation of AIAC algorithms using a simulation tool (i.e. the

-SimGrid toolkit \cite{SimGrid}). Second, we confirm the effectiveness of asynchronous mode algorithms by comparing their
+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
 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
+modifications of the initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a real AIAC

 application on different computing architectures. The simulated results we obtained are in line with real results
 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
+exposed in ??\AG[]{??}. 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 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
 iterative asynchronous model.  Then, the simulation framework SimGrid is presented with the settings to create various
 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
 iterative asynchronous model.  Then, the simulation framework SimGrid is presented with the settings to create various

-distributed architectures. The algorithm of  the multi-splitting method used by GMRES written with MPI primitives and
+distributed architectures. The algorithm of  the multisplitting method used by GMRES 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.
  


@@ -160,26 +160,26 @@ 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{bcvc06:ij}). 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
 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
+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
 i.e.  without stopping current computations. This technique allows to partially overlap communications by computations
 but unfortunately, the overlapping is only partial and important idle times remain.  It is clear that, in a grid
 computing context, where the number of computational nodes is large, heterogeneous and widely distributed, the idle
 times generated by synchronizations are very penalizing. One way to overcome this problem is to use the
 (SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously
 i.e.  without stopping current computations. This technique allows to partially overlap communications by computations
 but unfortunately, the overlapping is only partial and important idle times remain.  It is clear that, in a grid
 computing context, where the number of computational nodes is large, heterogeneous and widely distributed, the idle
 times generated by synchronizations are very penalizing. One way to overcome this problem is to use the

-\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 grey 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. 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
+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.
 
 \begin{figure}[!t]
   \centering
     \includegraphics[width=8cm]{AIAC.pdf}
 algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
 in a grid computing context.
 
 \begin{figure}[!t]
   \centering
     \includegraphics[width=8cm]{AIAC.pdf}

-  \caption{The Asynchronous Iterations - Asynchronous Communications model } 
+  \caption{The Asynchronous Iterations~-- Asynchronous Communications model}

   \label{fig:aiac}
 \end{figure}
 
   \label{fig:aiac}
 \end{figure}
 


@@ -193,7 +193,7 @@ simulation tools to explore various platform scenarios at will and to run enormo
 very promising. Several works...
 
 In the context of AIAC algorithms, the use of simulation tools is even more relevant. Indeed, this class of applications
 very promising. Several works...
 
 In the context of AIAC algorithms, the use of simulation tools is even more relevant. Indeed, this class of applications

-is very sensible to the execution environment context. For instance, variations in the network bandwith (intra and
+is very sensible to the execution environment context. For instance, variations in the network bandwidth (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.
 
 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.
 


@@ -202,11 +202,11 @@ iterations and so to very different execution times.
 
 \section{SimGrid}
 
 
 \section{SimGrid}
 

-SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} is a simulation
-framework to sudy the behavior of large-scale distributed systems.  As its name
+SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid} is a simulation
+framework to study the behavior of large-scale distributed systems.  As its name

 says, it emanates from the grid computing community, but is nowadays used to
 study grids, clouds, HPC or peer-to-peer systems.  The early versions of SimGrid
 says, it emanates from the grid computing community, but is nowadays used to
 study grids, clouds, HPC or peer-to-peer systems.  The early versions of SimGrid

-date from 1999, but it's still actively developped and distributed as an open
+date from 1999, but it's still actively developed and distributed as an open

 source software.  Today, it's one of the major generic tools in the field of
 simulation for large-scale distributed systems.
 
 source software.  Today, it's one of the major generic tools in the field of
 simulation for large-scale distributed systems.
 


@@ -277,8 +277,8 @@ is solved independently by a cluster and communications are required to update t
 \For {$k=0,1,2,\ldots$ until the global convergence}
 \State Restart outer iteration with $x^0=x^k$
 \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
 \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 Send shared elements of $X_l^{k+1}$ to neighboring clusters
-\State Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
+\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 $\{X_m^{k+1}\}_{m\neq l}$

 \EndFor
 
 \Statex
 \EndFor
 
 \Statex


@@ -303,9 +303,9 @@ $\{A_{lm}\}_{m\neq l}$ 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
 $\{X_m\}_{m\neq l}$ 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 $6$ and $7$ in Figure~\ref{algo:01}). The shared vector
-elements of the solution $x$ are exchanged by message passing using MPI
-non-blocking communication routines.
+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


@@ -323,13 +323,13 @@ where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tole
 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 
 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 
+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 
 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.
+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\_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 
 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 


@@ -372,13 +372,20 @@ 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 $171^{3} =
 \np{5211000}$ entries.
 
 62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to $171^{3} =
 \np{5211000}$ entries.
 

+% use the same column width for the following three tables
+\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
+\newenvironment{mytable}[1]{% #1: number of columns for data
+  \renewcommand{\arraystretch}{1.3}%
+  \begin{tabular}{|>{\bfseries}r%
+                  |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
+    \end{tabular}}
+

 \begin{table}[!t]
   \centering
   \caption{$2$ clusters, each with $50$ nodes}
   \label{tab.cluster.2x50}
 \begin{table}[!t]
   \centering
   \caption{$2$ clusters, each with $50$ nodes}
   \label{tab.cluster.2x50}

-  \renewcommand{\arraystretch}{1.3}

 
 

-  \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+  \begin{mytable}{6}

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


@@ -398,11 +405,11 @@ Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z =
     speedup
     & 0.396     & 0.392     & 0.396     & 0.391     & 0.393     & 0.395 \\
     \hline
     speedup
     & 0.396     & 0.392     & 0.396     & 0.391     & 0.393     & 0.395 \\
     \hline

-  \end{tabular}
+  \end{mytable}

 
   \smallskip
 
 
   \smallskip
 

-  \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+  \begin{mytable}{6}

     \hline
     bw
     & 50        & 50        & 50        & 50        & 10        & 10 \\
     \hline
     bw
     & 50        & 50        & 50        & 50        & 10        & 10 \\


@@ -422,7 +429,7 @@ Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z =
     speedup
     & 0.398     & 0.388     & 0.393     & 0.394     & 0.63      & 0.778 \\
     \hline
     speedup
     & 0.398     & 0.388     & 0.393     & 0.394     & 0.63      & 0.778 \\
     \hline

-  \end{tabular}
+  \end{mytable}

 \end{table}
   
 Then we have changed the network configuration using three clusters containing
 \end{table}
   
 Then we have changed the network configuration using three clusters containing


@@ -435,9 +442,8 @@ speedups less than $1$ with a matrix size from $62$ to $100$ elements.
   \centering
   \caption{$3$ clusters, each with $33$ nodes}
   \label{tab.cluster.3x33}
   \centering
   \caption{$3$ clusters, each with $33$ nodes}
   \label{tab.cluster.3x33}

-  \renewcommand{\arraystretch}{1.3}

 
 

-  \begin{tabular}{|>{\bfseries}r|*{6}{c|}}
+  \begin{mytable}{6}

     \hline
     bw
     & 10       & 5        & 4        & 3        & 2        & 6 \\
     \hline
     bw
     & 10       & 5        & 4        & 3        & 2        & 6 \\


@@ -457,10 +463,9 @@ speedups less than $1$ with a matrix size from $62$ to $100$ elements.
     speedup
     & 0.997    & 0.99     & 0.93     & 0.84     & 0.78     & 0.99 \\
     \hline
     speedup
     & 0.997    & 0.99     & 0.93     & 0.84     & 0.78     & 0.99 \\
     \hline

-  \end{tabular}
+  \end{mytable}

 \end{table}
 
 \end{table}
 

-

 In a final step, results of an execution attempt to scale up the three clustered
 configuration but increasing by two hundreds hosts has been recorded in
 Table~\ref{tab.cluster.3x67}.
 In a final step, results of an execution attempt to scale up the three clustered
 configuration but increasing by two hundreds hosts has been recorded in
 Table~\ref{tab.cluster.3x67}.


@@ -469,9 +474,8 @@ 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}

-  \renewcommand{\arraystretch}{1.3}

 
 

-  \begin{tabular}{|>{\bfseries}r|c|}
+  \begin{mytable}{1}

     \hline
     bw         & 1 \\
     \hline
     \hline
     bw         & 1 \\
     \hline


@@ -485,7 +489,7 @@ Table~\ref{tab.cluster.3x67}.
     \hline
     speedup    & 0.9 \\
     \hline
     \hline
     speedup    & 0.9 \\
     \hline

- \end{tabular}
+ \end{mytable}

 \end{table}
 
 Note that the program was run with the following parameters:
 \end{table}
 
 Note that the program was run with the following parameters:


@@ -520,7 +524,7 @@ the results have given a speedup less than 1, 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

-that with a deterioration of inter cluster network set with \np[Mbits/s]{5} of
+that with a deterioration of inter cluster network set with \np[Mbit/s]{5} of

 bandwidth, a latency in order of a hundredth of a millisecond and a system power
 of one GFlops, an efficiency of about \np[\%]{40} in asynchronous mode is
 obtained for a matrix size of 62 elements. It is noticed that the result remains
 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


@@ -529,7 +533,7 @@ increasing the problem size up to $100$ elements, it was necessary to increase t
 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[Mbits/s]{50}, the result of efficiency of about \np[\%]{40} is obtained with
+\np[Mbit/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
 elements.
 
 high external precision of \np{E-11} for a matrix size from $110$ to $150$ side
 elements.
 


@@ -538,13 +542,13 @@ 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
 (synchronous and asynchronous) is achieved with an inter cluster of
 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
 (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[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[\%]{78} with a matrix size of $100$ points, it was necessary to degrade the

-inter cluster network bandwidth from 5 to 2 Mbit/s.
+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
 
 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

-obtained with a bandwidth of \np[Mbits/s]{1} as shown in
+obtained with a bandwidth of \np[Mbit/s]{1} as shown in

 Table~\ref{tab.cluster.3x67}.
 
 \section{Conclusion}
 Table~\ref{tab.cluster.3x67}.
 
 \section{Conclusion}


@@ -554,17 +558,15 @@ computers organized with interconnected clusters have been presented.
 Our work has demonstrated that using such a simulation tool allow us to 
 reach the following three objectives: 
 
 Our work has demonstrated that using such a simulation tool allow us to 
 reach the following three objectives: 
 

-\newcounter{numberedCntD}

 \begin{enumerate}
 \item To have a flexible configurable execution platform resolving the 
 hard exercise to access to very limited but so solicited physical 
 resources;
 \begin{enumerate}
 \item To have a flexible configurable execution platform resolving the 
 hard exercise to access to very limited but so solicited physical 
 resources;

-\item to ensure the algorithm convergence with a raisonnable time and 
+\item to ensure the algorithm convergence with a reasonable time and

 iteration number ;
 \item and finally and more importantly, to find the correct combination 
 of the cluster and network specifications permitting to save time in 
 executing the algorithm in asynchronous mode.
 iteration number ;
 \item and finally and more importantly, to find the correct combination 
 of the cluster and network specifications permitting to save time in 
 executing the algorithm in asynchronous mode.

-\setcounter{numberedCntD}{\theenumi}

 \end{enumerate}
 Our results have shown that in certain conditions, asynchronous mode is 
 speeder up to \np[\%]{40} than executing the algorithm in synchronous mode
 \end{enumerate}
 Our results have shown that in certain conditions, asynchronous mode is 
 speeder up to \np[\%]{40} than executing the algorithm in synchronous mode


@@ -580,7 +582,7 @@ mode in a grid architecture.
 \section*{Acknowledgment}
 
 This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 \section*{Acknowledgment}
 
 This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).

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

 
 
 % trigger a \newpage just before the given reference
 
 
 % trigger a \newpage just before the given reference


@@ -598,3 +600,10 @@ The authors would like to thank\dots{}
 %%% fill-column: 80
 %%% ispell-local-dictionary: "american"
 %%% End:
 %%% fill-column: 80
 %%% ispell-local-dictionary: "american"
 %%% End:

+
+% LocalWords:  Ramamonjisoa Laiymani Arnaud Giersch Ziane Khodja Raphaël Femto
+% LocalWords:  Université Franche Comté IUT Montbéliard Maréchal Juin Inria Sud
+% LocalWords:  Ouest Vieille Talence cedex scalability experimentations HPC MPI
+% LocalWords:  Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua
+% LocalWords:  Fortran GFlops priori Mbit de du fcomte multisplitting scalable
+% LocalWords:  SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib