X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/5fb6769d88c1720b6480a28521119ef010462fa6..20552ea673ac9a1d972c077d6926781e47dc5911:/hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index 77f7248..6f7f7ca 100644
--- 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}
+\newcommand{\CER}[2][inline]{%
+  \todo[color=pink!10,#1]{\sffamily\textbf{CER:} #2}\xspace}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
@@ -39,6 +41,7 @@
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newcommand{\MI}{\mathit{MaxIter}}
+\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}}
 
 \begin{document}
 
@@ -70,6 +73,7 @@
 
 \RC{Ordre des auteurs pas dÃ©finitif.}
 \begin{abstract}
+\AG{L'abstract est AMHA incomprÃ©hensible et ne donne pas envie de lire la suite.}
 In recent years, the scalability of large-scale implementation in a 
 distributed environment of algorithms becoming more and more complex has 
 always been hampered by the limits of physical computing resources 
@@ -94,6 +98,8 @@ a residual precision up to \np{E-11}. Such successful results open
 perspectives on experimentations for running the algorithm on a 
 simulated large scale growing environment and with larger problem size. 
 
+\LZK{Long\ldots}
+
 % no keywords for IEEE conferences
 % Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
 \end{abstract}
@@ -149,24 +155,25 @@ 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{ProblÃ¨me traitÃ© dans le papier est symÃ©trique ou asymÃ©trique? (Poisson 3D symÃ©trique?)} We show, that with minor modifications of the
+linear system of equations by numerical method GMRES (Generalized
+Minimal Residual) \cite{ref1}. 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[]{??}.
+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
 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.
+clusters nodes interconnected by variable speed networks. In the simulated environment, after setting appropriate 
+network and cluster parameters like the network bandwidth, latency or the processors power, 
+the experimental results have demonstrated a asynchronous execution time saving up to \np[\%]{40} in
+compared to the synchronous mode.
+\AG{Il faudrait revoir la phrase prÃ©cÃ©dente (couper en deux?).  LÃ , on peut
+  avoir l'impression que le gain de \np[\%]{40} est entre une exÃ©cution rÃ©elle
+  et une exÃ©cution simulÃ©e!}
+\CER{La phrase a Ã©tÃ© modifiÃ©e}
 
 This article is structured as follows: after this introduction, the next  section will give a brief description of
 iterative asynchronous model.  Then, the simulation framework SimGrid is presented with the settings to create various
-distributed architectures. The algorithm of  the multisplitting method used by GMRES written with MPI primitives and
+distributed architectures. The algorithm of  the multisplitting method based on GMRES \LZK{??? GMRES n'utilise pas la mÃ©thode de multisplitting! Sinon ne doit on pas expliquer le choix d'une mÃ©thode de multisplitting?} \CER{La phrase a Ã©tÃ© corrigÃ©e} 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.
  
@@ -185,10 +192,12 @@ times generated by synchronizations are very penalizing. One way to overcome thi
 \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.
+\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.
+in a grid computing context.\LZK{RÃ©pÃ©tition par rapport Ã  l'intro}
 
 \begin{figure}[!t]
   \centering
@@ -240,17 +249,27 @@ this paper.  The SMPI interface implements about \np[\%]{80} of the MPI 2.0
 standard~\cite{bedaride:hal-00919507}, and supports applications written in C or
 Fortran, with little or no modifications.
 
-With SimGrid, the execution of a distributed application is simulated on a
+Within SimGrid, the execution of a distributed application is simulated on a
 single machine.  The application code is really executed, but some operations
-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 ?}
+like the communications are intercepted, and their running time is computed
+according to the characteristics of the simulated execution platform.  The
+description of this target platform is given as an input for the execution, by
+the mean of an XML file.  It describes the properties of the platform, such as
+the computing nodes with their computing power, the interconnection links with
+their bandwidth and latency, and the routing strategy.  The simulated running
+time of the application is computed according to these properties.
+
+To compute the durations of the operations in the simulated world, and to take
+into account resource sharing (e.g. bandwidth sharing between competing
+communications), SimGrid uses a fluid model.  This allows to run relatively fast
+simulations, while still keeping accurate
+results~\cite{bedaride:hal-00919507,tomacs13}.  Moreover, depending on the
+simulated application, SimGrid/SMPI allows to skip long lasting computations and
+to only take their duration into account.  When the real computations cannot be
+skipped, but the results have no importance for the simulation results, there is
+also the possibility to share dynamically allocated data structures between
+several simulated processes, and thus to reduce the whole memory consumption.
+These two techniques can help to run simulations at a very large scale.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}
@@ -275,44 +294,56 @@ Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, whe
       B_L
     \end{array} \right)
 \end{equation*}
-in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$.
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$
+are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$, $A_{\ell
+  m}$ is a rectangular block of $A$ of size $n_\ell\times n_m$, $X_\ell$ and
+$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each,
+and $\sum_{\ell} n_\ell=\sum_{m} n_m=n$.
 
 The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
 \begin{equation}
   \label{eq:4.1}
   \left\{
     \begin{array}{l}
-      A_{ll}X_l = Y_l \text{, such that}\\
-      Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
+      A_{\ell\ell}X_\ell = Y_\ell \text{, such that}\\
+      Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\ m\neq \ell}}^{L}A_{\ell m}X_m
     \end{array}
   \right.
 \end{equation}
-is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers. 
+is solved independently by a cluster and communications are required to update
+the right-hand side sub-vector $Y_\ell$, such that the sub-vectors $X_m$
+represent the data dependencies between the clusters. As each sub-system
+(\ref{eq:4.1}) is solved in parallel by a cluster of processors, our
+multisplitting method uses an iterative method as an inner solver which is
+easier to parallelize and more scalable than a direct method. In this work, we
+use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most
+used iterative method by many researchers.
 
 \begin{figure}[!t]
   %%% IEEE instructions forbid to use an algorithm environment here, use figure
   %%% instead
 \begin{algorithmic}[1]
-\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
-\Output $X_l$ (solution sub-vector)\vspace{0.2cm}
-\State Load $A_l$, $B_l$
+\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
+\Output $X_\ell$ (solution sub-vector)\medskip
+
+\State Load $A_\ell$, $B_\ell$
 \State Set the initial guess $x^0$
 \For {$k=0,1,2,\ldots$ until the global convergence}
 \State Restart outer iteration with $x^0=x^k$
 \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
-\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters
-\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
+\State\label{algo:01:send} Send shared elements of $X_\ell^{k+1}$ to neighboring clusters
+\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq \ell}$
 \EndFor
 
 \Statex
 
 \Function {InnerSolver}{$x^0$, $k$}
-\State Compute local right-hand side $Y_l$:
+\State Compute local right-hand side $Y_\ell$:
        \begin{equation*}
-         Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0
+         Y_\ell = B_\ell - \sum\nolimits^L_{\substack{m=1\\ m\neq \ell}}A_{\ell m}X_m^0
        \end{equation*}
-\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method
-\State \Return $X_l^k$
+\State Solving sub-system $A_{\ell\ell}X_\ell^k=Y_\ell$ with the parallel GMRES method
+\State \Return $X_\ell^k$
 \EndFunction
 \end{algorithmic}
 \caption{A multisplitting solver with GMRES method}
@@ -323,13 +354,13 @@ 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
+cluster of processors. For all $\ell,m\in\{1,\ldots,L\}$, the matrices and
+vectors with the subscript $\ell$ represent the local data for cluster $\ell$,
+while $\{A_{\ell m}\}_{m\neq \ell}$ are off-diagonal matrices of sparse matrix
+$A$ and $\{X_m\}_{m\neq \ell}$ contain vector elements of solution $x$ shared
+with neighboring clusters. At every outer iteration $k$, asynchronous
+communications are performed between processors of the local cluster and those
+of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in
 Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are
 exchanged by message passing using MPI non-blocking communication routines.
 
@@ -349,33 +380,36 @@ 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
+\textit{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
+the local convergence on each cluster $\ell$ 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)
+  (k\leq \MI) \text{ or } (\|X_\ell^k - X_\ell^{k+1}\|_{\infty}\leq\epsilon)
 \end{equation*}
-where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$. 
+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 $X_\ell^{k+1}$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SIMGRI\LZK[]{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, \LZK{Il faut expliquer pourquoi 6 points voisins (7-point stencil problem)}
-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 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.
-\LZK{Peut-Ãªtre mettre plus de prÃ©cisions sur les difficultÃ©s rencontrÃ©es dans la version async et les adaptaions effectuÃ©es pour SimGrid}
+\CER{On voulait en fait montrer la simplicitÃ© de l'adaptation de l'algo a SimGrid. Les problÃ¨mes rencontrÃ©s dÃ©crits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais prÃ©fÃ©rÃ© avoir un peu plus de dÃ©tails sur l'adaptation de la version async} 
+\CER{Le problÃ¨me majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a Ã©tÃ© le plantage en SMPI de Waitall aprÃ¨s un Isend et Irecv. J'avais proposÃ© un workaround en utilisant un MPI\_wait sÃ©parÃ© pour chaque Ã©change a la place d'un waitall unique pour TOUTES les Ã©changes, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais aprÃ¨s, tu as modifiÃ© le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de dÃ©tection de la convergence et du coup, l'Ã©change global avec waitall a aussi fonctionnÃ©.}
 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. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
+also to be reviewed.
+\AG{Ã propos de ces problÃ¨mes d'alignement, en dire plus si Ã§a a un intÃ©rÃªt, ou l'enlever.}
+\CER{Ce problÃ¨me fait partie des modifications que j'ai dÃ» faire dans l'adaptation du programme MPI vers SMPI. IL dÃ©coule de la diffÃ©rence de la taille des mots en mÃ©moire : en 32 bits, pour les variables declarees en long int, on garde dans les instructions de sortie (printf, sprintf, ...) le format \%lu sinon en 64 bits, on le substitue par \%llu.} 
+ 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.\LZK{DerniÃ¨re phrase peut Ãªtre supprimÃ©e}
+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. 
 
 
 
@@ -386,34 +420,72 @@ parameters and the program arguments allows us to compare outputs from the code
 study that the results depend on the following parameters:  
 \begin{itemize}
 \item At the network level, we found that the most critical values are the
-  bandwidth (bw) and the network latency (lat).
-\item Hosts power (GFlops) can also influence on the results.
+  bandwidth and the network latency.
+\item Hosts processors 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
-  \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).
+  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 execution time of synchronous
+  compared to the asynchronous mode ($t_\text{sync} / t_\text{async}$) is defined as the \emph{relative gain}. So,
+  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 Ã§a dure plus
+    longtemps (que Ã§a aille moins vite) en asynchrone qu'en synchrone ?
+    Ce n'est pas plutÃ´t l'inverse ?}
+  \CER{J'ai modifie la phrase.}
 \end{itemize}
-\LZK{Propositions pour changer 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
+A priori, obtaining a relative gain greater than 1 would be difficult in a local
+area network configuration where the synchronous mode will take advantage on the
 rapid exchange of information on such high-speed links. Thus, the methodology
-adopted was to launch the application on clustered network. In this last
-configuration, degrading the inter-cluster network performance will
-\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.
-
-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
-matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from
-$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
-\text{\np{5211000}}$ entries.
-\LZK{Donner le type et la description du problÃ¨me traitÃ© (problÃ¨me symÃ©trique Poisson 3D) et prÃ©ciser peut Ãªtre aussi qu'on a utilisÃ© un partitionnement 3D}
+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
+simulates the case of distant clusters linked with long distance network as in grid computing context.
+
+\AG{Cette partie sur le poisson 3D
+  % on sait donc que ce n'est pas une plie ou une sole (/me fatiguÃ©)
+  n'est pas Ã  sa place.  Elle devrait Ãªtre placÃ©e plus tÃ´t.}
+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 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 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 algorithm convergence with a 3D
+matrix size ranging 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), is obtained in asynchronous in average 2.5 times speeder than the synchronous mode. 
+\AG{Expliquer comment lire les tableaux.}
+\CER{J'ai reformulÃ© la phrase par la lecture du tableau. Plus de dÃ©tails seront lus dans la partie InterprÃ©tations 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
@@ -429,172 +501,183 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
 
   \begin{mytable}{6}
     \hline
-    bw
+    bandwidth (Mbits/s)
     & 5         & 5         & 5         & 5         & 5         & 50 \\
     \hline
-    lat
+    latency (ms)
     & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\
     \hline
-    power
+    power (GFlops)
     & 1         & 1         & 1         & 1.5       & 1.5       & 1.5 \\
     \hline
     size
     & 62        & 62        & 62        & 100       & 100       & 110 \\
     \hline
-    Prec/Eprec
-    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} \\
-    \hline
-    speedup
-    & 0.396     & 0.392     & 0.396     & 0.391     & 0.393     & 0.395 \\
+    Precision
+    & \np{E-5}   & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} \\
     \hline
-  \end{mytable}
-
-  \smallskip
-
-  \begin{mytable}{6}
-    \hline
-    bw
-    & 50        & 50        & 50        & 50        & 10        & 10 \\
-    \hline
-    lat
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.03      & 0.01 \\
     \hline
-    power
-    & 1.5       & 1.5       & 1.5       & 1.5       & 1         & 1.5 \\
-    \hline
-    size
-    & 120       & 130       & 140       & 150       & 171       & 171 \\
-    \hline
-    Prec/Eprec
-    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5}  & \np{E-5} \\
-    \hline
-    speedup
-    & 0.398     & 0.388     & 0.393     & 0.394     & 0.63      & 0.778 \\
+    Relative gain
+    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54     & 2.53 \\
     \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
-speedups less 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}
+  \bigskip
 
   \begin{mytable}{6}
     \hline
-    bw
-    & 10       & 5        & 4        & 3        & 2        & 6 \\
+    bandwidth (Mbits/s)
+    & 50        & 50        & 50        & 50 \\ %       & 10        & 10 \\
     \hline
-    lat
-    & 0.01     & 0.02     & 0.02     & 0.02     & 0.02     & 0.02 \\
+    latency (ms)
+    & 0.02      & 0.02      & 0.02      & 0.02 \\ %      & 0.03      & 0.01 \\
     \hline
-    power
-    & 1        & 1        & 1        & 1        & 1        & 1 \\
+    Power (GFlops)
+    & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\
     \hline
     size
-    & 62       & 100      & 100      & 100      & 100      & 171 \\
-    \hline
-    Prec/Eprec
-    & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
-    \hline
-    speedup
-    & 0.997    & 0.99     & 0.93     & 0.84     & 0.78     & 0.99 \\
-    \hline
-  \end{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}
+    & 120       & 130       & 140       & 150  \\ %     & 171       & 171 \\
     \hline
-    bw         & 1 \\
+    Precision
+    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5}  & \np{E-5} \\
     \hline
-    lat        & 0.02 \\
     \hline
-    power      & 1 \\
-    \hline
-    size       & 62 \\
-    \hline
-    Prec/Eprec & \np{E-5} \\
-    \hline
-    speedup    & 0.9 \\
+    Relative gain
+    & 2.51     & 2.58     & 2.55     & 2.54   \\ %  & 1.59      & 1.29 \\
     \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 enlevÃ© les tableaux 2 et 3 sachant que les rÃ©sultats obtenus sont limites. De mÃªme, on a enlevÃ© aussi les deux derniÃ¨res 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}
 
+~\\{}\AG{Donner un peu plus de prÃ©cisions (plateforme en particulier).}
+\CER {PrÃ©cisions ajoutÃ©es}
+
 \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 bw,
-lat 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 :
+
+	- Processor unit power : 1.5 GFlops;
+
+	- Intracluster network : bandwidth = 1,25 Gbits/s and latency = 5E-05 ms;
+
+	- Intercluster network : bandwidth = 5 Mbits/s and latency = 5E-03 ms;
 \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 Description of the cluster architecture matching the format <Number of cluster> <Number of hosts in cluster\_1> <Number of hosts in cluster\_2>;
+	\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: \np{6.0}, \LZK{Off-diagonal values? (-1.0?)}
-	\item Execution Mode: synchronous or asynchronous.
+	\item Matrix diagonal value: \np{1.0} 	(See (3));
+	\item Matrix off-diagonal value: $-\frac{1}{6}$ 	(See(3));
+	\item Communication mode: Asynchronous.
 \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
-the results have given a speedup less than 1, showing the effectiveness of the
+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.
 
-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
-increasing the problem size up to 100 elements, it was necessary to increase the
-CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a convergence of the algorithm
-with the same order of asynchronous mode efficiency.  Maintaining such a system
-power but this time, increasing network throughput inter cluster up to
-\np[Mbit/s]{50}, the result of efficiency of about \np[\%]{40} is obtained with
+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
+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\AG[]{2.5 ?} is obtained with
 high external precision of \np{E-11} for a matrix size from 110 to 150 side
 elements.
 
-For the 3 clusters architecture including a total of 100 hosts,
-Table~\ref{tab.cluster.3x33} shows that it was difficult to have a combination
-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[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
-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
-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 rÃ©sultats pour comparer gmres parallÃ¨le classique avec multisplitting asynchrone? Ca permettra de montrer l'intÃ©rÃªt du multisplitting asynchrone sur des clusters distants}
-
+%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 Ã§a ?}
+
+%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 diffÃ©rence entre les rÃ©sultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas trÃ¨s bons... Je me demande s'il ne faut pas les enlever...}
+\RC{En fait je pense avoir la rÃ©ponse Ã  ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne prÃ©cision. 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{DÃ©finitivement, les paramÃ¨tres rÃ©seaux variables ici se rapportent au rÃ©seau 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 
@@ -654,3 +737,5 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 % 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
+% LocalWords:  intra durations nonsingular Waitall discretization discretized
+% LocalWords:  InnerSolver Isend Irecv
