tune their application for a given multi-core architecture. To show the validity
of this approach we first compare the simulated execution of the multisplitting
algorithm with the GMRES (Generalized Minimal Residual)
-solver~\cite{saad86} in synchronous mode. The obtained results on different
+solver~\cite{saad86} in synchronous mode.
+
+\LZK{Pas trop convainquant comme argument pour valider l'approche de simulation. \\On peut dire par exemple: on a pu simuler différents algos itératifs à large échelle (le plus connu GMRES et deux variantes de multisplitting) et la simulation nous a permis (sans avoir le vrai matériel) de déterminer quelle serait la meilleure solution pour une telle configuration de l'archi ou vice versa.\\A revoir...}
+
+The obtained results on different
simulated multi-core architectures confirm the real results previously obtained
-on non simulated architectures. We also confirm the efficiency of the
-asynchronous multisplitting algorithm compared to the synchronous GMRES. In
+on non simulated architectures.
+
+\LZK{Il n y a pas dans la partie expé cette comparaison et confirmation des résultats entre la simulation et l'exécution réelle des algos sur les vrais clusters.\\ Sinon on pourrait ajouter dans la partie expé une référence vers le journal supercomput de krylov multi pour confirmer que cette méthode est meilleure que GMRES sur les clusters large échelle.}
+
+We also confirm the efficiency of the
+asynchronous multisplitting algorithm compared to the synchronous GMRES.
+
+\LZK{P.S.: Pour tout le papier, le principal objectif n'est pas de faire des comparaisons entre des méthodes itératives!!\\Sinon, les deux algorithmes Krylov multisplitting synchrone et multisplitting asynchrone sont plus efficaces que GMRES sur des clusters à large échelle.\\Et préciser, si c'est vraiment le cas, que le multisplitting asynchrone est plus efficace et adapté aux clusters distants par rapport aux deux autres algos (je n'ai pas encore lu la partie expé)}
+
+In
this way and with a simple computing architecture (a laptop) SimGrid allows us
to run a test campaign of a real parallel iterative applications on
different simulated multi-core architectures. To our knowledge, there is no
experimental results are presented in section~\ref{sec:expe} followed by some
concluding remarks and perspectives.
+\LZK{Proposition d'un titre pour le papier: Grid-enabled simulation of large-scale linear iterative solvers.}
-\section{The asynchronous iteration model}
+
+\section{The asynchronous iteration model and the motivations of our work}
\label{sec:asynchro}
Asynchronous iterative methods have been studied for many years theoritecally and
it must not synchronize all the processors. Interested readers can
consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
+The number of iterations required to reach the convergence is generally greater
+for the asynchronous scheme (this number depends depends on the delay of the
+messages). Note that, it is not the case in the synchronous mode where the
+number of iterations is the same than in the sequential mode. In this way, the
+set of the parameters of the platform (number of nodes, power of nodes,
+inter and intra clusters bandwidth and latency \ldots) and of the
+application can drastically change the number of iterations required to get the
+convergence. It follows that asynchronous iterative algorithms are difficult to
+optimize since the financial and deployment costs on large scale multi-core
+architecture are often very important. So, prior to delpoyment and tests it
+seems very promising to be able to simulate the behavior of asynchronous
+iterative algorithms. The problematic is then to show that the results produce
+by simulation are in accordance with reality i.e. of the same order of
+magnitude. To our knowledge, there is no study on this problematic.
+
\section{SimGrid}
\label{sec:simgrid}
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{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 ({\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}.
\begin{figure}[t]
%\begin{algorithm}[t]
in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs
$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors.
-\subsubsection{Running on two different inter-clusters network speeds \\}
+\subsubsection{Running on two different inter-clusters network speeds \\}
\begin{table} [ht!]
\begin{center}
Figure~\ref{fig:02} shows that end users will gain to reduce the execution time
for both algorithms in using a grid architecture like 4x16 or 8x8: the
performance was increased by a factor of $2$. The results depict also that when
-the network speed drops down (variation of 12.5\%), the difference between the two Multisplitting algorithms execution times can reach more than 25\%.
+the network speed drops down (variation of 12.5\%), the difference between the two Multisplitting algorithms execution times can reach more than 25\%.
%\RC{c'est pas clair : la différence entre quoi et quoi?}
%\DL{pas clair}
%\RCE{Modifie}
