-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.
-\CER{Voir ma remarque plus si nécessaire de décrire en détail le partitionnement 3D}
-\LZK{Je pense qu'il faut donner ici le type du problème traité (Poisson 3D). Le partitionnement 3D permet juste de définir le schéma de dépendances (1 proc a au max 6 voisins dans le cluster local ou dans les clusters distants)}
+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 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.
+
+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 reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
