A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
\label{eq:03}
\end{equation}
-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
\begin{figure}[t]
%\begin{algorithm}[t]
\subsection{Simulation of the two-stage methods using SimGrid toolkit}
\label{sec:04.02}
-One of our objectives when simulating the application in Simgrid is, as in real
+One of our objectives when simulating the application in SimGrid is, as in real
life, to get accurate results (solutions of the problem) but also to ensure the
test reproducibility under the same conditions. According to our experience,
-very few modifications are required to adapt a MPI program for the Simgrid
+very few modifications are required to adapt a MPI program for the SimGrid
simulator using SMPI (Simulator MPI). The first modification is to include SMPI
-libraries and related header files (smpi.h). The second modification is to
+libraries and related header files (\verb+smpi.h+). The second modification is to
suppress all global variables by replacing them with local variables or using a
-Simgrid selector called "runtime automatic switching"
+SimGrid selector called "runtime automatic switching"
(smpi/privatize\_global\_variables). Indeed, global variables can generate side
effects on runtime between the threads running in the same process and generated by
-Simgrid to simulate the grid environment.
+SimGrid to simulate the grid environment.
-%\RC{On vire cette phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The
-%last modification on the MPI program pointed out for some cases, the review of
-%the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which
-%might cause an infinite loop.
-
-
-\paragraph{Simgrid Simulator parameters}
-\ \\ \noindent Before running a Simgrid benchmark, many parameters for the
+\paragraph{Parameters of the simulation in SimGrid}
+\ \\ \noindent Before running a SimGrid benchmark, many parameters for the
computation platform must be defined. For our experiments, we consider platforms
in which several clusters are geographically distant, so there are intra and
inter-cluster communications. In the following, these parameters are described:
\begin{itemize}
\item hostfile: hosts description file.
\item platform: file describing the platform architecture: clusters (CPU power,
-\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
-latency lat, \dots{}).
+\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
+latency $lat$, \dots{}).
\item archi : grid computational description (number of clusters, number of
-nodes/processors for each cluster).
+nodes/processors in each cluster).
\end{itemize}
\noindent
In addition, the following arguments are given to the programs at runtime:
\begin{itemize}
\item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
\item inner precision $\TOLG$ and outer precision $\TOLM$,
- \item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
- \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
+ \item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+ \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,
\item matrix off-diagonal value is fixed to $-1.0$,
\item number of vectors in matrix $S$ (i.e. value of $s$),
\item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
\item execution mode: synchronous or asynchronous.
\end{itemize}
-It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
+It should also be noticed that both solvers have been executed with the SimGrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
\subsection{The 3D Poisson problem}
+\label{3dpoisson}
We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
