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+++ b/hpcc.tex
@@ -1,3 +1,4 @@
+

 \documentclass[conference]{IEEEtran}
 
 \usepackage[T1]{fontenc}
 \documentclass[conference]{IEEEtran}
 
 \usepackage[T1]{fontenc}


@@ -102,50 +103,57 @@ suggests, these algorithms solve a given problem by successive iterations ($X_{n
 $X_{0}$ 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}.
 
 $X_{0}$ 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}.
 

-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
-\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
-synchronizations especially in a grid computing context (see~\cite{Bahi07} for more details).
-
-Parallel   (synchronous  or  asynchronous)   applications  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.
+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 \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 no
+synchronization points~\cite{bcvc06:ij}. In this case, 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,
+asynchronous  iterative 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 applications  based on a (synchronous or  asynchronous) iteration model
+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 asynchronous
+iterative 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
 
 To our knowledge,  there is no existing work on the  large-scale simulation of a

-real  AIAC application.   {\bf  The contribution  of  the present  paper can  be
-  summarised  in two  main  points}.  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  the
-asynchronous  multisplitting algorithm  by  comparing its  performance with  the
-synchronous GMRES (Generalized Minimal  Residual) \cite{ref1}.  Both these codes
-can be  used to  solve large linear  systems. In  this paper, we  focus on  a 3D
-Poisson  problem.  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.
+real asynchronous  iterative application.  {\bf The contribution  of the present
+  paper can be  summarised in two main points}.  First we  give a first approach
+of the simulation  of asynchronous iterative algorithms using  a simulation tool
+(i.e.    the   SimGrid   toolkit~\cite{SimGrid}).    Second,  we   confirm   the
+effectiveness  of the  asynchronous  multisplitting algorithm  by comparing  its
+performance   with  the   synchronous  GMRES   (Generalized   Minimal  Residual)
+\cite{ref1}.  Both  these codes can  be used to  solve large linear  systems. In
+this  paper, we  focus  on  a 3D  Poisson  problem.  We  show,  that with  minor
+modifications of the initial MPI code,  the SimGrid toolkit allows us to perform
+a  test campaign  of  a  real asynchronous  iterative  application on  different
+computing architectures.

 % The  simulated results  we
 %obtained are  in line with real  results exposed in  ??\AG[]{ref?}. 
 SimGrid  had  allowed us  to  launch the  application  from  a modest  computing
 % The  simulated results  we
 %obtained are  in line with real  results exposed in  ??\AG[]{ref?}. 
 SimGrid  had  allowed us  to  launch the  application  from  a modest  computing


@@ -167,56 +175,72 @@ based  on GMRES to  solve each  block obtained  of the  splitting. This  code is
 written with MPI  primitives and its adaptation to  SimGrid with SMPI (Simulated
 MPI) is  detailed in the next  section. At last, the  simulation results carried
 out will be presented before some concluding remarks and future works.
 written with MPI  primitives and its adaptation to  SimGrid with SMPI (Simulated
 MPI) is  detailed in the next  section. At last, the  simulation results carried
 out will be presented before some concluding remarks and future works.

+

  
 \section{Motivations and scientific context}
 
  
 \section{Motivations and scientific context}
 

-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
-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
-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 gray blocks represent the computation phases, the white spaces the idle
-times and the arrows the communications.
-\AG{There are no ``white spaces'' on the figure.}
-With this algorithmic model, the number of iterations required before the
-convergence is generally greater than for the two former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC
-algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
-in a grid computing context.\LZK{Répétition par rapport à l'intro}
+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 synchronous  iterations 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.  It is possible to use asynchronous communications, in this case, the
+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 asynchronous iterations 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.  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},
+asynchronous  iterative 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}
 
 \begin{figure}[!t]
   \centering
     \includegraphics[width=8cm]{AIAC.pdf}

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

   \label{fig:aiac}
 \end{figure}
 
 
   \label{fig:aiac}
 \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\dots{}
+%% 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{}
+
+%% \AG{Several works\dots{} what?\\
+%  Le paragraphe suivant se trouve déjà dans l'intro ?}
+In the context of asynchronous algorithms, the number of iterations to reach the
+convergence depends on  the delay of messages. With  synchronous iterations, the
+number of  iterations is exactly  the same than  in the sequential mode  (if the
+parallelization process does  not change the algorithm). So  the difficulty with
+asynchronous iteratie algorithms comes from the fact it is necessary to run the algorithm
+with real data. In fact, from an execution to another the order of messages will
+change and the  number of iterations to reach the  convergence will also change.
+According  to all  the parameters  of the  platform (number  of nodes,  power of
+nodes,  inter  and  intra clusrters  bandwith  and  latency,  ....) and  of  the
+algorithm  (number   of  splitting  with  the   multisplitting  algorithm),  the
+multisplitting code  will obtain the solution  more or less  quickly. Or course,
+the GMRES method also depends of the same parameters. As it is difficult to have
+access to  many clusters,  grids or supercomputers  with many  different network
+parameters,  it  is  interesting  to  be  able  to  simulate  the  behaviors  of
+asynchronous iterative algoritms before being able to runs real experiments.
+

 
 

-\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.

 
 
 
 
 
 


@@ -384,25 +408,61 @@ where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the
 tolerance threshold of the error computed between two successive local solution
 $X_\ell^k$ and $X_\ell^{k+1}$.
 
 tolerance threshold of the error computed between two successive local solution
 $X_\ell^k$ and $X_\ell^{k+1}$.
 

+
+
+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. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
+\begin{equation}
+\begin{array}{l}
+u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
+%u(x,y,z)= & \frac{1}{6}\times [u(x-1,y,z) + u(x+1,y,z) + \\
+ %         & u(x,y-1,z) + u(x,y+1,z) + \\
+  %        & u(x,y,z-1) + u(x,y,z+1) - \\ & h^2f(x,y,z)],
+\end{array}
+\label{eq:03}
+\end{equation} 
+where $h$ is the distance between two adjacent elements in the spatial discretization scheme and the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite. 
+
+The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries. 
+
+\begin{figure}[!t]
+\centering
+  \includegraphics[width=80mm,keepaspectratio]{partition}
+\caption{Example of the 3D data partitioning between two clusters of processors.}
+\label{fig:4.2}
+\end{figure}
+
+
+
+

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code 
 debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. In synchronous 
 mode, the execution of the program raised no particular issue but in asynchronous mode, the review of the sequence of MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions
 and with the addition of the primitive MPI\_Test was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code 
 debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. In synchronous 
 mode, the execution of the program raised no particular issue but in asynchronous mode, the review of the sequence of MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions
 and with the addition of the primitive MPI\_Test was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.

-\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}
+%\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} 
+%\CER{Le problème majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a été le plantage en SMPI de Waitall après un Isend et Irecv. J'avais proposé un workaround en utilisant un MPI\_wait séparé pour chaque échange a la place d'un waitall unique pour TOUTES les échanges, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais après, tu as modifié le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de détection de la convergence et du coup, l'échange global avec waitall a aussi fonctionné.}

 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.
 Note here that the use of SMPI functions optimizer for memory footprint and CPU usage is not recommended knowing that one wants to get real results by simulation.

-As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. First, all declared 
-global variables have been moved to local variables for each subroutine. In fact, global variables generate side effects arising from the concurrent access of 
-shared memory used by threads simulating each computing unit in the SimGrid architecture. Second, the alignment of certain types of variables such as ``long int'' had
-also to be reviewed.
-\AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.}
- Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
+As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. The scope of all declared 
+global variables have been moved to local to subroutine. Indeed, global variables generate side effects arising from the concurrent access of 
+shared memory used by threads simulating each computing unit in the SimGrid architecture. 
+%Second, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
+%\AG{compilation or run-time error?}

 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 successfully executed the code in synchronous mode using parallel GMRES algorithm compared with our multisplitting algorithm in asynchronous mode after few modifications. 
 
 
 
 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 successfully executed the code in synchronous mode using parallel GMRES algorithm compared with our multisplitting algorithm in asynchronous mode after few modifications. 
 
 
 

-\section{Experimental results}
+\section{Simulation results}

 
 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
 
 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


@@ -410,73 +470,32 @@ 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 and the network latency.
 \begin{itemize}
 \item At the network level, we found that the most critical values are the
   bandwidth and the network latency.

-\item Hosts power (GFlops) can also influence on the results.
+\item Hosts processors power (GFlops) can also influence on the results.

 \item Finally, when submitting job batches for execution, the arguments values
 \item Finally, when submitting job batches for execution, the arguments values

-  passed to the program like the maximum number of iterations or the 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. The ratio between the execution time of asynchronous
-  compared to the synchronous mode is defined as the \emph{relative gain}. So,
+  passed to the program like the maximum number of iterations or the precision are critical. They allow us to ensure not only the convergence of the
+  algorithm but also to get the main objective in getting an execution time in asynchronous communication less than in
+  synchronous mode. The ratio between the simulated execution time of synchronous GMRES algorithm
+  compared to the asynchronous multisplitting algorithm ($t_\text{GMRES} / t_\text{Multisplitting}$) is defined as the \emph{relative gain}. So,

   our objective running the algorithm in SimGrid is to obtain a relative gain
   greater than 1.
   our objective running the algorithm in SimGrid is to obtain a relative gain
   greater than 1.

-  \AG{$t_\text{async} / t_\text{sync} > 1$, l'objectif est donc que ça dure plus
-    longtemps (que ça aille moins vite) en asynchrone qu'en synchrone ?
-    Ce n'est pas plutôt l'inverse ?}

 \end{itemize}
 
 A priori, obtaining a relative gain greater than 1 would be difficult in a local
 area network configuration where the synchronous mode will take advantage on the
 rapid exchange of information on such high-speed links. Thus, the methodology
 \end{itemize}
 
 A priori, obtaining a relative gain greater than 1 would be difficult in a local
 area network configuration where the synchronous mode will take advantage on the
 rapid exchange of information on such high-speed links. Thus, the methodology

-adopted was to launch the application on clustered network. In this last
+adopted was to launch the application on a clustered network. In this

 configuration, degrading the inter-cluster network performance will penalize the
 synchronous mode allowing to get a relative gain greater than 1.  This action
 configuration, degrading the inter-cluster network performance will penalize the
 synchronous mode allowing to get a relative gain greater than 1.  This action

-simulates the case of distant clusters linked with long distance network like
-Internet.
-
-\AG{Cette partie sur le poisson 3D
-  % on sait donc que ce n'est pas une plie ou une sole (/me fatigué)
-  n'est pas à sa place.  Elle devrait être placée plus tôt.}
-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. 
+simulates the case of distant clusters linked with long distance network as in grid computing context.

 
 

-The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries. 

 
 

-\begin{figure}[!t]
-\centering
-  \includegraphics[width=80mm,keepaspectratio]{partition}
-\caption{Example of the 3D data partitioning between two clusters of processors.}
-\label{fig:4.2}
-\end{figure}
-
-
-As a first step, the algorithm was run on a network consisting of two clusters
-containing 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50} with a
-matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from
-$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
-\text{\np{5000211}}$ entries.
-\AG{Expliquer comment lire les tableaux.}

 
 

+Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
+factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem  ranges from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
+$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
+\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one. 
+%\AG{Expliquer comment lire les tableaux.}
+%\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}

 % 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
 % 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


@@ -490,183 +509,184 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
   \caption{2 clusters, each with 50 nodes}
   \label{tab.cluster.2x50}
 
   \caption{2 clusters, each with 50 nodes}
   \label{tab.cluster.2x50}
 

-  \begin{mytable}{6}
+  \begin{mytable}{5}

     \hline
     \hline

-    bandwidth
-    & 5         & 5         & 5         & 5         & 5         & 50 \\
+    bandwidth (Mbit/s)
+    & 5         & 5         & 5         & 5         & 5         \\

     \hline
     \hline

-    latency
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\
+    latency (ms)
+    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      \\

     \hline
     \hline

-    power
-    & 1         & 1         & 1         & 1.5       & 1.5       & 1.5 \\
+    power (GFlops)
+    & 1         & 1         & 1         & 1.5       & 1.5       \\

     \hline
     \hline

-    size
-    & 62        & 62        & 62        & 100       & 100       & 110 \\
+    size $(n^3)$
+    & 62        & 62        & 62        & 100       & 100       \\

     \hline
     \hline

-    Prec/Eprec
-    & \np{E-5}   & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} \\
+    Precision
+    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} \\

     \hline
     \hline
     Relative gain
     \hline
     \hline
     Relative gain

-    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54     & 2.53 \\
+    & 2.52      & 2.55      & 2.52      & 2.57      & 2.54      \\

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

-  \begin{mytable}{6}
+  \begin{mytable}{5}

     \hline
     \hline

-    bandwidth
-    & 50        & 50        & 50        & 50        & 10        & 10 \\
+    bandwidth (Mbit/s)
+    & 50        & 50        & 50        & 50        & 50 \\ %       & 10        & 10 \\

     \hline
     \hline

-    latency
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.03      & 0.01 \\
+    latency (ms)
+    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\ %      & 0.03      & 0.01 \\

     \hline
     \hline

-    power
-    & 1.5       & 1.5       & 1.5       & 1.5       & 1         & 1.5 \\
+    Power (GFlops)
+    & 1.5       & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\

     \hline
     \hline

-    size
-    & 120       & 130       & 140       & 150       & 171       & 171 \\
+    size $(n^3)$
+    & 110       & 120       & 130       & 140       & 150  \\ %     & 171       & 171 \\

     \hline
     \hline

-    Prec/Eprec
-    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5}  & \np{E-5} \\
+    Precision
+    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5}  & \np{E-5} \\

     \hline
     \hline
     Relative gain
     \hline
     \hline
     Relative gain

-    & 2.51     & 2.58     & 2.55     & 2.54     & 1.59      & 1.29 \\
+    & 2.53      & 2.51     & 2.58     & 2.55     & 2.54   \\ %  & 1.59      & 1.29 \\

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

-Then we have changed the network configuration using three clusters containing
-respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
-clusters. In the same way as above, a judicious choice of key parameters has
-permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the
-relative gains greater than 1 with a matrix size from 62 to 100 elements.
-
-\begin{table}[!t]
-  \centering
-  \caption{3 clusters, each with 33 nodes}
-  \label{tab.cluster.3x33}
-
-  \begin{mytable}{6}
-    \hline
-    bandwidth
-    & 10       & 5        & 4        & 3        & 2        & 6 \\
-    \hline
-    latency
-    & 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
-    \hline
-    Relative gain
-    & 1.003    & 1.01     & 1.08     & 1.19     & 1.28     & 1.01 \\
-    \hline
-  \end{mytable}
-\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}.
-
-\begin{table}[!t]
-  \centering
-  \caption{3 clusters, each with 66 nodes}
-  \label{tab.cluster.3x67}
-
-  \begin{mytable}{1}
-    \hline
-    bandwidth  & 1 \\
-    \hline
-    latency    & 0.02 \\
-    \hline
-    power      & 1 \\
-    \hline
-    size       & 62 \\
-    \hline
-    Prec/Eprec & \np{E-5} \\
-    \hline
-    \hline
-    Relative gain    & 1.11 \\
-    \hline
-  \end{mytable}
-\end{table}
+%Then we have changed the network configuration using three clusters containing
+%respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
+%clusters. In the same way as above, a judicious choice of key parameters has
+%permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the
+%relative gains greater than 1 with a matrix size from 62 to 100 elements.
+
+\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}
+%\begin{table}[!t]
+%  \centering
+%  \caption{3 clusters, each with 33 nodes}
+%  \label{tab.cluster.3x33}
+%
+%  \begin{mytable}{6}
+%    \hline
+%    bandwidth 
+%    & 10       & 5        & 4        & 3        & 2        & 6 \\
+%    \hline
+%    latency
+%    & 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
+%    \hline
+%    Relative gain
+%    & 1.003    & 1.01     & 1.08     & 1.19     & 1.28     & 1.01 \\
+%    \hline
+%  \end{mytable}
+%\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}.
+
+%\begin{table}[!t]
+%  \centering
+%  \caption{3 clusters, each with 66 nodes}
+%  \label{tab.cluster.3x67}
+%
+%  \begin{mytable}{1}
+%    \hline
+%    bandwidth  & 1 \\
+%    \hline
+%    latency    & 0.02 \\
+%    \hline
+%    power      & 1 \\
+%    \hline
+%    size       & 62 \\
+%    \hline
+%    Prec/Eprec & \np{E-5} \\
+%    \hline
+%    \hline
+%    Relative gain    & 1.11 \\
+%    \hline
+%  \end{mytable}
+%\end{table}

 
 Note that the program was run with the following parameters:
 
 \paragraph*{SMPI parameters}
 
 
 Note that the program was run with the following parameters:
 
 \paragraph*{SMPI parameters}
 

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

 \begin{itemize}
 \begin{itemize}

-\item HOSTFILE: Hosts file description.
-\item PLATFORM: file description of the platform architecture : clusters (CPU
-  power, \dots{}), intra cluster network description, inter cluster network
-  (bandwidth, latency, \dots{}).
+\item HOSTFILE: Text file containing the list of the processors units name. Here 100 hosts;
+\item PLATFORM: XML file description of the platform architecture : two clusters (cluster1 and cluster2) with the following characteristics :
+  \begin{itemize}
+  \item Processor unit power: \np[GFlops]{1.5};
+  \item Intracluster network bandwidth: \np[Gbit/s]{1.25} and latency:
+    \np[$\mu$s]{0.05};
+  \item Intercluster network bandwidth: \np[Mbit/s]{5} and latency:
+    \np[$\mu$s]{5};
+  \end{itemize}

 \end{itemize}
 
 
 \paragraph*{Arguments of the program}
 
 \begin{itemize}
 \end{itemize}
 
 
 \paragraph*{Arguments of the program}
 
 \begin{itemize}

-       \item Description of the cluster architecture;
-       \item Maximum number of internal and external iterations;
-       \item Internal and external precisions;
-       \item Matrix size $N_x$, $N_y$ and $N_z$;
-       \item Matrix diagonal value: \np{6.0};
-       \item Matrix off-diagonal value: \np{-1.0};
-       \item Execution Mode: synchronous or asynchronous.
+\item Description of the cluster architecture matching the format <Number of
+  cluster> <Number of hosts in cluster1> <Number of hosts in cluster2>;
+\item Maximum number of iterations;
+\item Precisions on the residual error;
+\item Matrix size $N_x$, $N_y$ and $N_z$;
+\item Matrix diagonal value: $6$ (See~(\ref{eq:03}));
+\item Matrix off-diagonal value: $-1$;
+\item Communication mode: asynchronous.

 \end{itemize}
 
 \paragraph*{Interpretations and comments}
 
 \end{itemize}
 
 \paragraph*{Interpretations and comments}
 

-After analyzing the outputs, generally, for the configuration with two or three
-clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50}
-and~\ref{tab.cluster.3x33}), some combinations of the used parameters affecting
+After analyzing the outputs, generally, for the two clusters including one hundred hosts configuration (Tables~\ref{tab.cluster.2x50}), some combinations of parameters affecting

 the results have given a relative gain more than 2.5, showing the effectiveness of the
 asynchronous performance compared to the synchronous mode.
 
 the results have given a relative gain more than 2.5, showing the effectiveness of the
 asynchronous performance compared to the synchronous mode.
 

-In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows
-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
-stable even if we vary the external precision from \np{E-5} to \np{E-9}. By
+With these settings, Table~\ref{tab.cluster.2x50} shows
+that after a deterioration of inter cluster network with a bandwidth of \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
+of one GFlops, an efficiency of about \np[\%]{40} is
+obtained in asynchronous mode for a matrix size of 62 elements. It is noticed that the result remains
+stable even we vary the residual error precision from \np{E-5} to \np{E-9}. By

 increasing the matrix size up to 100 elements, it was necessary to increase the
 increasing the matrix size up to 100 elements, it was necessary to increase the

-CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a convergence of the algorithm
-with the same order of asynchronous mode efficiency.  Maintaining such a system
-power but this time, increasing network throughput inter cluster up to
-\np[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5\AG[]{2.5 ?} is obtained with
+CPU power of \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency.  Maintaining such processor power but increasing network throughput inter cluster up to
+\np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5 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.
 

-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 a relative gain of asynchronous mode more than 1.2. Indeed, for a
-matrix size of 62 elements, equality between the performance of the two modes
-(synchronous and asynchronous) is achieved with an inter cluster of
-\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency greater than 1.2 with a matrix size of 100 points, it was necessary to degrade the
-inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
+%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 a relative gain of asynchronous mode more than 1.2. Indeed, for a
+%matrix size of 62 elements, equality between the performance of the two modes
+%(synchronous and asynchronous) is achieved with an inter cluster of
+%\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency greater than 1.2 with a matrix %size of 100 points, it was necessary to degrade the
+%inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.

 \AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
   Quelle est la perte de perfs en faisant ça ?}
 
 \AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
   Quelle est la perte de perfs en faisant ça ?}
 

-A last attempt was made for a configuration of three clusters but more powerful
-with 200 nodes in total. The convergence with a relative gain around 1.1 was
-obtained with a bandwidth of \np[Mbit/s]{1} as shown in
-Table~\ref{tab.cluster.3x67}.
-
-\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
-\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
-\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
+%A last attempt was made for a configuration of three clusters but more powerful
+%with 200 nodes in total. The convergence with a relative gain around 1.1 was
+%obtained with a bandwidth of \np[Mbit/s]{1} as shown in
+%Table~\ref{tab.cluster.3x67}.

 
 

+%\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
+%\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
+%\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
+%\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}

 \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