\begin{center}
\begin{tabular}{ll }
\hline
+<<<<<<< HEAD
Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
- Network & N1 : $bw$=1Gbits/s, $lat$=5.10$^{-5}$ \\ %\hline
+ Network & N1 : $bw$=1Gbits/s, $lat$=5$\times$10$^{-5}$ \\ %\hline
\multirow{2}{*}{Matrix size} & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
& N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =170 $\times$ 170 $\times$ 170 \\ \hline
\end{tabular}
\caption{Test conditions: various grid configurations with the matrix sizes 150$^3$ or 170$^3$}
\LZK{Ce sont les caractéristiques du réseau intra ou inter clusters? Ce n'est pas précisé...}
+=======
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8\\ %\hline
+ Inter Network N2 & bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
+ Input matrix size & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
+ - & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =170 $\times$ 170 $\times$ 170 \\ \hline
+ \end{tabular}
+\caption{Test conditions: various grid configurations with the input matrix size N$_{x}$=N$_{y}$=N$_{z}$=150 or 170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}
+\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}. Idem dans le texte, les figures, etc.}}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\label{tab:01}
\end{center}
\end{table}
+<<<<<<< HEAD
In this section, we analyze the simulations conducted on various grid configurations presented in Table~\ref{tab:01}. Figure~\ref{fig:01} shows, for all grid configurations and a given matrix size, a non-variation in the number of iterations for the classical GMRES algorithm, which is not the case of the Krylov two-stage algorithm.
%% First, the results in Figure~\ref{fig:01}
%% show for all grid configurations the non-variation of the number of iterations of
%% multisplitting method.
\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
\RC{Les légendes ne sont pas explicites...}
+=======
+
+
+
+
+In this section, we analyze the performance of algorithms running on various
+grid configurations (2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8) and using an inter-network N2 defined in the test conditions in Table~\ref{tab:01}. First, the results in Figure~\ref{fig:01}
+show for all grid configurations the non-variation of the number of iterations of
+classical GMRES for a given input matrix size; it is not the case for the
+multisplitting method.
+
+%\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
+%\RC{Les légendes ne sont pas explicites...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\begin{figure} [ht!]
\begin{center}
\label{fig:01}
\end{figure}
+<<<<<<< HEAD
The execution times between the two algorithms is significant with different
grid architectures, even with the same number of processors (for example, 2x16
and 4x8). We can observe the low sensitivity of the Krylov multisplitting method
$40\%$ 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?}
\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exécution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?}
+=======
+
+Secondly, the execution times between the two algorithms is significant with different
+grid architectures, even with the same number of processors (for example, 2 $\times$ 16
+and 4 $\times$ 8). We can observ the 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 reduction of the execution time for GMRES (resp. Multisplitting) algorithm is around $40\%$ (resp. around $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors (grid 8 $\times$ 8) processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\subsubsection{Simulations for two different inter-clusters network speeds \\}
\begin{center}
\begin{tabular}{ll}
\hline
+<<<<<<< HEAD
Grid architecture & 2$\times$16, 4$\times$8\\ %\hline
- \multirow{2}{*}{Network} & N1: $bw$=1Gbs, $lat$=5.10$^{-5}$ \\ %\hline
- & N2: $bw$=10Gbs, $lat$=8.10$^{-6}$ \\
+ \multirow{2}{*}{Network} & N1: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
+ & N2: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
Matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
\end{tabular}
\caption{Test conditions: grid configurations 2$\times$16 and 4$\times$8 with networks N1 vs. N2}
+=======
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8\\ %\hline
+ Inter Networks & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
+ - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
+ Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
+ \end{tabular}
+\caption{Test conditions: grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\label{tab:02}
\end{center}
\end{table}
+<<<<<<< HEAD
These experiments compare the behavior of the algorithms running first on a
slow inter-cluster network (N1) and also on a more performant network (N2). \RC{Il faut définir cela avant...}
+=======
+In this section, the experiments compare the behavior of the algorithms running on a
+speeder inter-cluster network (N1) and also on a less performant network (N2) respectively defined in the test conditions Table~\ref{tab:02}. \RC{Il faut définir cela avant...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
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
+for both algorithms when using a grid architecture like 4 $\times$ 16 or 8 $\times$ 8: 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\%.
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Grid 2x16 and 4x8 with networks N1 vs N2
+\caption{Grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2
\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
\label{fig:02}
\end{figure}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline
Network & N1 : bw=1Gbs \\ %\hline
Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
\end{tabular}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline
Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
\end{tabular}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 4x8\\ %\hline
+ Grid Architecture & 4 $\times$ 8\\ %\hline
Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
Input matrix size & $N_{x}$ = From 40 to 200\\ \hline
\end{tabular}
These findings may help a lot end users to setup the best and the optimal
targeted environment for the application deployment when focusing on the problem
size scale up. It should be noticed that the same test has been done with the
-grid 2x16 leading to the same conclusion.
+grid 2 $\times$ 16 leading to the same conclusion.
\subsubsection{CPU Power impacts on performance}
\centering
\begin{tabular}{r c }
\hline
- Grid architecture & 2x16\\ %\hline
+ Grid architecture & 2 $\times$ 16\\ %\hline
Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline
\end{tabular}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2 $\times$ 50 totaling 100 processors\\ %\hline
Processors Power & 1 GFlops to 1.5 GFlops\\
Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
