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]
\item maximum number of restarts for the Arnorldi process in GMRES method,
\item execution mode: synchronous or asynchronous.
\end{itemize}
-\LZK{CE pourrais tu vérifier et confirmer les valeurs des éléments diag et off-diag de la matrice?}
-\RCE{oui, les valeurs de diag et off-diag donnees sont ok}
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 the scope of this paper, our first objective is to analyze when the Krylov
Multisplitting method has better performance than the classical GMRES
-method. With a synchronous iterative method, better performance mean a
+method. With a synchronous iterative method, better performance means a
smaller number of iterations and execution time before reaching the convergence.
For a systematic study, the experiments should figure out that, for various
grid parameters values, the simulator will confirm the targeted outcomes,
Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
- & N$_{x}$ x N$_{y}$ x N$_{z}$ =170 x 170 x 170 \\ \hline
\end{tabular}
-\caption{Test conditions: Various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\caption{Test conditions: various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}}
\label{tab:01}
\end{center}
\end{table}
-%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
-
In this section, we analyze the performance of algorithms running on various
grid configurations (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01}
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
- \caption{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170}
+ \caption{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170\RC{idem}}
\label{fig:01}
\end{figure}
and 4x8). We can observ the low sensitivity of the Krylov multisplitting method
(compared with the classical GMRES) when scaling up the number of the processors
in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs
-$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors.
+$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
\subsubsection{Running on two different inter-clusters network speeds \\}
- & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
\end{tabular}
-\caption{Test conditions: Grid 2x16 and 4x8 - Networks N1 vs N2}
+\caption{Test conditions: grid 2x16 and 4x8 with networks N1 vs N2}
\label{tab:02}
\end{center}
\end{table}
These experiments compare the behavior of the algorithms running first on a
-speed inter-cluster network (N1) and also on a less performant network (N2).
-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
+speed inter-cluster network (N1) and also on a less performant network (N2). \RC{Il faut définir cela avant...}
+Figure~\ref{fig:02} shows that end users will reduce the execution time
+for both algorithms when using a grid architecture like 4x16 or 8x8: the reduction is about $2$. 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\%.
%\RC{c'est pas clair : la différence entre quoi et quoi?}
%\DL{pas clair}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Grid 2x16 and 4x8 - Networks N1 vs N2}
+\caption{Grid 2x16 and 4x8 with networks N1 vs N2}
\label{fig:02}
\end{figure}
%\end{wrapfigure}
Network & N1 : bw=1Gbs \\ %\hline
Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
\end{tabular}
-\caption{Test conditions: Network latency impacts}
+\caption{Test conditions: network latency impacts}
\label{tab:03}
\end{table}
\end{figure}
-According to the results of Figure~\ref{fig:03}, a degradation of the network
-latency from $8.10^{-6}$ to $6.10^{-5}$ implies an absolute time increase of more
-than $75\%$ (resp. $82\%$) of the execution for the classical GMRES (resp. Krylov
-multisplitting) algorithm. In addition, it appears that the Krylov
-multisplitting method tolerates more the network latency variation with a less
-rate increase of the execution time. Consequently, in the worst case
-($lat=6.10^{-5 }$), the execution time for GMRES is almost the double than the
-time of the Krylov multisplitting, even though, the performance was on the same
-order of magnitude with a latency of $8.10^{-6}$.
+According to the results of Figure~\ref{fig:03}, a degradation of the network
+latency from $8.10^{-6}$ to $6.10^{-5}$ implies an absolute time increase of
+more than $75\%$ (resp. $82\%$) of the execution for the classical GMRES
+(resp. Krylov multisplitting) algorithm. In addition, it appears that the
+Krylov multisplitting method tolerates more the network latency variation with a
+less rate increase of the execution time.\RC{Les 2 précédentes phrases me
+ semblent en contradiction....} Consequently, in the worst case ($lat=6.10^{-5
+}$), the execution time for GMRES is almost the double than the time of the
+Krylov multisplitting, even though, the performance was on the same order of
+magnitude with a latency of $8.10^{-6}$.
\subsubsection{Network bandwidth impacts on performance}
\ \\
Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
\end{tabular}
-\caption{Test conditions: Network bandwidth impacts}
+\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
\label{tab:04}
\end{table}
time for both algorithms increases when the input matrix size also increases.
But the interesting results are:
\begin{enumerate}
- \item the drastic increase ($10$ times) \RC{Je ne vois pas cela sur la figure}
-\RCE{Corrige} of the number of iterations needed to reach the convergence for the classical
-GMRES algorithm when the matrix size go beyond $N_{x}=150$;
+ \item the drastic increase ($10$ times) of the number of iterations needed to
+ reach the convergence for the classical GMRES algorithm when the matrix size
+ go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itérations est 10 fois plus long mais la suite de la phrase ne veut rien dire}
\item the classical GMRES execution time is almost the double for $N_{x}=140$
compared with the Krylov multisplitting method.
\end{enumerate}
