$X_{0}$ 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
+Parallelization of such algorithms generally involves 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
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
+asynchronous iterative 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,
+nodes, inter and intra clusrters bandwith and latency, etc.) and of the
+algorithm (number of splittings with the multisplitting algorithm), the
+multisplitting code will obtain the solution more or less quickly. Of 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
framework to study the behavior of large-scale distributed systems. As its name
says, it emanates from the grid computing community, but is nowadays used to
study grids, clouds, HPC or peer-to-peer systems. The early versions of SimGrid
-date from 1999, but it's still actively developed and distributed as an open
-source software. Today, it's one of the major generic tools in the field of
+date from 1999, but it is still actively developed and distributed as an open
+source software. Today, it is one of the major generic tools in the field of
simulation for large-scale distributed systems.
SimGrid provides several programming interfaces: MSG to simulate Concurrent
\begin{figure}[!t]
\centering
- \includegraphics[width=60mm,keepaspectratio]{clustering2}
-\caption{Example of two distant clusters of processors.}
+ \includegraphics[width=60mm,keepaspectratio]{clustering}
+\caption{Example of three distant clusters of processors.}
\label{fig:4.1}
\end{figure}
\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
+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 differences 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),
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method , the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
+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. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method, the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
and with the addition of the primitive \texttt{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{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.}
