24-04-2014b

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index c3a0b34827d65b520a9cb513aff9848fdfec0307..29eca38179aabe085bd84945c74473f1a1c1a17c 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -31,6 +31,8 @@
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}

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

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


@@ -121,34 +123,48 @@ at that time. Even if the number of iterations required before the convergence i
 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).
 
 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).
 

-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 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
-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,
-the launch of distributed iterative  asynchronous algorithms to solve a given problem on a large-scale  simulated
-environment challenges to find optimal configurations giving the best results with a lowest residual error and in the
-best of execution time. 
-
-To our knowledge, there is no existing work on the large-scale simulation of a real AIAC application. The aim of this
-paper is twofold. First we give a first approach of the simulation of AIAC algorithms using a simulation tool (i.e. the
-SimGrid toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of asynchronous mode algorithms by comparing their
-performance with the synchronous mode. More precisely, we had implemented a program for solving large 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 campaign of a real AIAC
-application on different computing architectures. The simulated results we obtained are in line with real results
-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.
+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 bandwidth
+(intra and inter-clusters), in the number and the power of nodes, in the number
+of clusters\dots{} 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
+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, the launch of distributed
+iterative asynchronous algorithms to solve a given problem on a large-scale
+simulated environment challenges to find optimal configurations giving the best
+results with a lowest residual error and in the best of execution time.
+
+To our knowledge, there is no existing work on the large-scale simulation of a
+real AIAC application. The aim of this paper is twofold. First we give a first
+approach of the simulation of AIAC algorithms using a simulation tool (i.e. the
+SimGrid toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of
+asynchronous mode algorithms by comparing their performance with the synchronous
+mode. More precisely, we had implemented a program for solving large
+non-symmetric linear system of equations by numerical method GMRES (Generalized
+Minimal Residual) []\AG[]{[]?}\LZK[]{\cite{ref1}}.\LZK{Problème traité dans le papier est symétrique ou asymétrique? (Poisson 3D symétrique?)} We show, that with minor modifications of the
+initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a
+real AIAC application on different computing architectures. The simulated
+results we obtained are in line with real results exposed in ??\AG[]{??}.
+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
 
 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


@@ -184,18 +200,25 @@ in a grid computing context.
 \end{figure}
 
 
 \end{figure}
 
 

-It is very challenging to develop efficient applications for large scale, heterogeneous and distributed platforms such
-as computing grids. Researchers and engineers have to develop techniques for maximizing application performance of these
-multi-cluster platforms, by redesigning the applications and/or by using novel algorithms that can account for the
-composite and heterogeneous nature of the platform. Unfortunately, the deployment of such applications on these very
-large scale systems is very costly, labor intensive and time consuming. In this context, it appears that the use of
-simulation tools to explore various platform scenarios at will and to run enormous numbers of experiments quickly can be
-very promising. Several works...
+It is very challenging to develop efficient applications for large scale,
+heterogeneous and distributed platforms such as computing grids. Researchers and
+engineers have to develop techniques for maximizing application performance of
+these multi-cluster platforms, by redesigning the applications and/or by using
+novel algorithms that can account for the composite and heterogeneous nature of
+the platform. Unfortunately, the deployment of such applications on these very
+large scale systems is very costly, labor intensive and time consuming. In this
+context, it appears that the use of simulation tools to explore various platform
+scenarios at will and to run enormous numbers of experiments quickly can be very
+promising. Several works\dots{}

 
 

-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 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.
+\AG{Several works\dots{} what?\\
+  Le paragraphe suivant se trouve déjà dans l'intro ?}
+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 bandwidth (intra
+and inter-clusters), in the number and the power of nodes, in the number of
+clusters\dots{} can lead to very different number of iterations and so to very
+different execution times.

 
 
 
 
 
 


@@ -214,21 +237,23 @@ 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
 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.  SMPI implements about \np[\%]{80} of the MPI
-2.0 standard~\cite{bedaride:hal-00919507}.
-
-%%% explain simulation
-%- simulated processes folded in one real process
-%- simulates interactions on the network, fluid model
-%- able to skip long-lasting computations
-%- traces + visu?
-
-%%% platforms
-%- describe resources and their interconnection, with their properties
-%- XML files
-
-%%% validation + refs
+languages.  SMPI is the interface that has been used for the work exposed in
+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.
+
+With SimGrid, the execution of a distributed application is simulated on a
+single machine.  The application code is really executed, but some operations
+like the communications are intercepted to be simulated according to the
+characteristics of the simulated execution platform.  The description of this
+target platform is given as an input for the execution, by the mean of an XML
+file.  It describes the properties of the platform, such as the computing node
+with their computing power, the interconnection links with their bandwidth and
+latency, and the routing strategy.  The simulated running time of the
+application is computed according to these properties.
+
+\AG{Faut-il ajouter quelque-chose ?} 
+\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} 

 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}


@@ -297,19 +322,7 @@ is solved independently by a cluster and communications are required to update t
 \label{algo:01}
 \end{figure}
 
 \label{algo:01}
 \end{figure}
 

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

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


@@ -318,52 +331,34 @@ exchanged by message passing using MPI non-blocking communication routines.
 \label{fig:4.1}
 \end{figure}
 
 \label{fig:4.1}
 \end{figure}
 

-The global convergence of the asynchronous multisplitting solver is detected
-when the clusters of processors have all converged locally. We implemented the
-global convergence detection process as follows. On each cluster a master
-processor is designated (for example the processor with rank 1) and masters of
-all clusters are interconnected by a virtual unidirectional ring network (see
-Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around
-the virtual ring from a master processor to another until the global convergence
-is achieved. So starting from the cluster with rank 1, each master processor $i$
-sets the token to \textit{True} if the local convergence is achieved or to
-\text\it{False} otherwise, and sends it to master processor $i+1$. Finally, the
-global convergence is detected when the master of cluster 1 receives from the
-master of cluster $L$ a token set to \textit{True}. In this case, the master of
-cluster 1 broadcasts a stop message to masters of other clusters. In this work,
-the local convergence on each cluster $l$ is detected when the following
-condition is satisfied
+The global convergence of the asynchronous multisplitting solver is detected when the clusters of processors have all converged locally. We implemented the global convergence detection process as follows. On each cluster a master processor is designated (for example the processor with rank 1) and masters of all clusters are interconnected by a virtual unidirectional ring network (see Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around the virtual ring from a master processor to another until the global convergence is achieved. So starting from the cluster with rank 1, each master processor $i$ sets the token to {\it True} if the local convergence is achieved or to {\it False} otherwise, and sends it to master processor $i+1$. Finally, the global convergence is detected when the master of cluster 1 receives from the master of cluster $L$ a token set to {\it 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 condition is satisfied

 \begin{equation*}
   (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)
 \end{equation*}
 where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$. 
 
 \begin{equation*}
   (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)
 \end{equation*}
 where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$. 
 

-\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 
+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 (left,right,front,behind,top,down) in a cubic partitionned submatrix within a cluster or between clusters, \CER{J'ai rajouté quelques précisions mais serait-il nécessaire de décrire a ce niveau la discrétisation 3D ?}
+\LZK{Non ce n'est pas nécessaire. A ce niveau, on décrit l'algorithme général de multisplitting. Donc, je pense qu'il est préférable de ne pas préciser le schéma de communication qui peut changer selon le type de problème. \\ {\bf Par exemple: Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters}}
+the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. In synchronous 

 mode, the execution of the program raised no particular issue but in asynchronous mode, the review of the sequence of MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions
 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.
+and with the addition of the primitive MPI\_Test was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
+\CER{On voulait en fait montrer la simplicité de l'adaptation de l'algo a SimGrid. Les problèmes rencontrés décrits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais préféré avoir un peu plus de détails sur l'adaptation de la version async}
+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 
 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
+shared memory used by threads simulating each computing unit in the SimGrid architecture. Second, the alignment of certain types of variables such as ``long int'' had

 also to be reviewed. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
 In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real 
 environment. We have tested in synchronous mode with a simulated platform starting from a modest 2 or 3 clusters grid to a larger configuration like simulating 
 also to be reviewed. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
 In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real 
 environment. We have tested in synchronous mode with a simulated platform starting from a modest 2 or 3 clusters grid to a larger configuration like simulating 

-Grid5000 with more than 1500 hosts with 5000 cores~\cite{bolze2006grid}. Once the code debugging and adaptation were complete, the next section shows our methodology and experimental 
-results.
-
-
-
-
+Grid5000 with more than 1500 hosts with 5000 cores~\cite{bolze2006grid}.

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

-When the \emph{real} application runs in the simulation environment and produces the expected results, varying the input
+When the \textit{real} application runs in the simulation environment and produces the expected results, varying the input

 parameters and the program arguments allows us to compare outputs from the code execution. We have noticed from this
 study that the results depend on the following parameters:  
 \begin{itemize}
 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}


@@ -372,21 +367,54 @@ study that the results depend on the following parameters:
 \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
 \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
+  \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).
 \end{itemize}
   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}

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

+In this paper, we solve the 3D Poisson problem whose the mathematical model is 
+\begin{equation}
+\left\{
+\begin{array}{l}
+\nabla^2 u = f \text{~in~} \Omega \\
+u =0 \text{~on~} \Gamma =\partial\Omega
+\end{array}
+\right.
+\label{eq:02}
+\end{equation}
+where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. The general iteration scheme of our multisplitting method in a 3D domain using a seven point stencil could be written as 
+\begin{equation}
+\begin{array}{ll}
+u^{k+1}(x,y,z)= & u^k(x,y,z) - \frac{1}{6}\times\\
+               & (u^k(x-1,y,z) + u^k(x+1,y,z) + \\
+               & u^k(x,y-1,z) + u^k(x,y+1,z) + \\
+               & u^k(x,y,z-1) + u^k(x,y,z+1)),
+\end{array}
+\label{eq:03}
+\end{equation} 
+where the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite. 
+
+The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster of in distant clusters with which it shares data at sub-domain boundaries. 
+
+\begin{figure}[!t]
+\centering
+  \includegraphics[width=80mm,keepaspectratio]{partition}
+\caption{Example of the 3D data partitioning between two clusters of processors.}
+\label{fig:4.2}
+\end{figure}
+
+

 As a first step, the algorithm was run on a network consisting of two clusters
 containing 50 hosts each, totaling 100 hosts. Various combinations of the above
 factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a
 As a first step, the algorithm was run on a network consisting of two clusters
 containing 50 hosts each, totaling 100 hosts. Various combinations of the above
 factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a


@@ -511,7 +539,7 @@ Table~\ref{tab.cluster.3x67}.
     \hline
     speedup    & 0.9 \\
     \hline
     \hline
     speedup    & 0.9 \\
     \hline

- \end{mytable}
+  \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:


@@ -533,7 +561,11 @@ 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 Execution Mode: synchronous or asynchronous.
 \end{itemize}
 
        \item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 


@@ -573,6 +605,8 @@ with 200 nodes in total. The convergence with a speedup of \np[\%]{90} 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}
+

 \section{Conclusion}
 The experimental results on executing a parallel iterative algorithm in 
 asynchronous mode on an environment simulating a large scale of virtual 
 \section{Conclusion}
 The experimental results on executing a parallel iterative algorithm in 
 asynchronous mode on an environment simulating a large scale of virtual 


@@ -601,12 +635,13 @@ demonstrated an original solution to optimize the use of a simulation
 tool to run efficiently an iterative parallel algorithm in asynchronous 
 mode in a grid architecture. 
 
 tool to run efficiently an iterative parallel algorithm in asynchronous 
 mode in a grid architecture. 
 

+\LZK{Perspectives???}
+

 \section*{Acknowledgment}
 
 This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 \todo[inline]{The authors would like to thank\dots{}}
 
 \section*{Acknowledgment}
 
 This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 \todo[inline]{The authors would like to thank\dots{}}
 

-

 % trigger a \newpage just before the given reference
 % number - used to balance the columns on the last page
 % adjust value as needed - may need to be readjusted if
 % trigger a \newpage just before the given reference
 % number - used to balance the columns on the last page
 % adjust value as needed - may need to be readjusted if


@@ -614,6 +649,8 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 \bibliographystyle{IEEEtran}
 \bibliography{IEEEabrv,hpccBib}
 
 \bibliographystyle{IEEEtran}
 \bibliography{IEEEabrv,hpccBib}
 

+
+

 \end{document}
 
 %%% Local Variables:
 \end{document}
 
 %%% Local Variables: