\begin{tabular}{ll}
\hline
Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
-\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\
- & $N2$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ & $N2$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\
\multirow{2}{*}{Matrix size} & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
& $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
\end{tabular}
\end{center}
\end{table}
-\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
-\ \\
-% environment
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}
+
In this section, we analyze the simulations conducted on various grid
configurations and for different sizes of the 3D Poisson problem. The parameters
-of the network between clusters is fixed to $N1$ (see
-Table~\ref{tab:01}. Figure~\ref{fig:01} shows, for all grid configurations and a
+of the network between clusters is fixed to $N2$ (see
+Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and a
given matrix size 170$^3$ elements, a non-variation in the number of iterations
for the classical GMRES algorithm, which is not the case of the Krylov two-stage
algorithm. In fact, with multisplitting algorithms, the number of splitting (in
our case, it is the number of clusters) influences on the convergence speed. The
higher the number of splitting is, the slower the convergence of the algorithm
-is.
+is (see the output results obtained from configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs. 8$\times$8).
+The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16).
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\end{center}
+\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\LZK{CE, la légende de la Figure 3 est trop large. Remplacer les N$_x\times$N$_y\times$N$_z$ par $Mat1$=150$^3$ et $Mat2$=170$^3$ comme dans la Table 1}
+\label{fig:01}
+\end{figure}
+\subsubsection{Simulations for two different inter-clusters network speeds\\}
+In Figure~\ref{fig:02} we present the execution times of both algorithms to
+solve a 3D Poisson problem of size $150^3$ on two different simulated network
+$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
+this figure that the Krylov two-stage algorithm is sensitive to the number of
+clusters (i.e. it is better to have a small number of clusters). However, we can
+notice an interesting behavior of the Krylov two-stage algorithm. It is less
+sensitive to bad network bandwidth and latency for the inter-clusters links than
+the GMRES algorithms. This means that the multisplitting methods are more
+efficient for distributed systems with high latency networks.
+
+%% In this section, the experiments compare the behavior of the algorithms running on a
+%% speeder inter-cluster network (N2) and also on a less performant network (N1) respectively defined in the test conditions Table~\ref{tab:02}.
+%% %\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 4 $\times$ 16 or 8 $\times$ 8: the reduction factor is around $2$. 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}[t]
+\centering
+\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
+\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
+\label{fig:02}
+\end{figure}
-\begin{figure} [htbp]
- \begin{center}
- \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
- \end{center}
- \caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
-%\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}
-%\LZK{Pour quelle taille du problème sont calculés les nombres d'itérations? Que représente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?}
- %\RCE {Corrige}
- \RC{Idéalement dans la légende il faudrait insiquer Pb size=$150^3$ ou $170^3$ car pour l'instant Nx=150 ca n'indique rien concernant Ny et Nz}
- \label{fig:01}
-\end{figure}
-The execution times between both 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 observe a better 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 32 (grid 2 $\times$ 16) to 64 processors/cores (grid 8 $\times$ 8). Note that even with a grid 8 $\times$ 8 having the maximum number of clusters, the execution time of the multisplitting method is in average 32\% less compared to GMRES.
-%\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?}
-%\RCE{Remarquez que meme avec une grille 8x8, le multi est toujours plus performant}
-\subsubsection{Simulations for two different inter-clusters network speeds \\}
-\begin{table} [ht!]
-\begin{center}
-\begin{tabular}{ll}
- \hline
- Grid architecture & 2$\times$16, 4$\times$8\\ %\hline
- \multirow{2}{*}{Inter 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}
-\label{tab:02}
-\end{center}
-\end{table}
-In this section, the experiments compare the behavior of the algorithms running on a
-speeder inter-cluster network (N2) and also on a less performant network (N1) respectively defined in the test conditions Table~\ref{tab:02}.
-%\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 4 $\times$ 16 or 8 $\times$ 8: the reduction factor is around $2$. 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{wrapfigure}{l}{100mm}
-\begin{figure} [htbp]
-\centering
-\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Various grid configurations with networks N1 vs N2}
-%\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
-%\RCE{Corrige}
-\label{fig:02}
-\end{figure}
-%\end{wrapfigure}
-\subsubsection{Network latency impacts on performance}
-\ \\
+\subsubsection{Network latency impacts on performance\\}
+
\begin{table} [ht!]
\centering
\begin{tabular}{r c }
decreasing from $8.10^{-6}$ to $6.10^{-5}$ second.
-\subsubsection{Network bandwidth impacts on performance}
-\ \\
+\subsubsection{Network bandwidth impacts on performance\\}
+
\begin{table} [ht!]
\centering
\begin{tabular}{r c }
presents a better performance in the considered bandwidth interval with a gain
of $40\%$ which is only around $24\%$ for the classical GMRES.
-\subsubsection{Input matrix size impacts on performance}
-\ \\
+\subsubsection{Input matrix size impacts on performance\\}
+
\begin{table} [ht!]
\centering
\begin{tabular}{r c }
