distributed and parallel algorithms requiring significant hardware resources (grid computing, clusters, broadband
network, etc.) but also a non-negligible CPU execution time. We consider in this paper a class of highly efficient
parallel algorithms called \emph{numerical iterative algorithms} executed in a distributed environment. As their name
-suggests, these algorithm solves a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
+suggests, these algorithms solve a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
$X_{0}$ 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 involved the division of the problem into several \emph{blocks} that will
+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 begin only when all nodes communications are completed,
-either \emph{asynchronous} mode where processors can continue independently without or few synchronization points. For
+\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
linear system of equations by numerical method GMRES (Generalized Minimal Residual) []. 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 ??. 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. It has been
-permitted to show With selected parameters on the network platforms (bandwidth, latency of inter cluster network) and
+exposed in ??\AG[]{??}. 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.
