increasing complexity of these requested applications combined with a continuous increase of their sizes lead to write
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 \texttt{numerical iterative algorithms} executed in a distributed environment. As their name
+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
$X_{0}$ 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}.
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
-instance in the \textit{Asynchronous Iterations - Asynchronous Communications (AIAC)} model \cite{bcvc06:ij}, local
+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
synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to
As exposed in the introduction, parallel iterative methods are now widely used in many scientific domains. They can be
classified in three main classes depending on how iterations and communications are managed (for more details readers
-can refer to \cite{bcvc06:ij}). In the \textit{Synchronous Iterations - Synchronous Communications (SISC)} model data
+can refer to \cite{bcvc06:ij}). In the \textit{Synchronous Iterations~-- Synchronous Communications (SISC)} model data
are exchanged at the end of each iteration. All the processors must begin the same iteration at the same time and
-important idle times on processors are generated. The \textit{Synchronous Iterations - Asynchronous Communications
+important idle times on processors are generated. The \textit{Synchronous Iterations~-- Asynchronous Communications
(SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously
i.e. without stopping current computations. This technique allows to partially overlap communications by computations
but unfortunately, the overlapping is only partial and important idle times remain. It is clear that, in a grid
computing context, where the number of computational nodes is large, heterogeneous and widely distributed, the idle
times generated by synchronizations are very penalizing. One way to overcome this problem is to use the
-\textit{Asynchronous Iterations - Asynchronous Communications (AIAC)} model. Here, local computations do not need to
+\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
\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{AIAC.pdf}
- \caption{The Asynchronous Iterations - Asynchronous Communications model }
+ \caption{The Asynchronous Iterations~-- Asynchronous Communications model}
\label{fig:aiac}
\end{figure}
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 units in the SimGrid architecture. Second, the alignment of certain types of variables such as "long int" had
+shared memory used by threads simulating each computing units 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.
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
(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 2 Mbit/s.
+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
