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).
+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$16).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
-\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\caption{Various grid configurations with two matrix sizes: $150^3$ and $170^3$}
\label{fig:01}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}
\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
\RCE{ok}
\label{fig:02}
\end{figure}
\subsubsection{Network bandwidth impacts on performances\\}
-Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of 2$\times$16 processors interconnected by a network of latency $lat=50\mu$s to solve a 3D Poisson problem of size $150^3$. The results of increasing the network bandwidth from 1Gbs to 10Gbs show the performances improvement for both algorithms by reducing the execution times. However, the Krylov two-stage algorithm presents a better performance in the considered bandwidth interval with a gain of $40\%$ compared to only about $24\%$ for the classical GMRES algorithm.
+
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
+$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
+solve a 3D Poisson problem of size $150^3$. The results of increasing the
+network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
+both algorithms by reducing the execution times. However, the Krylov two-stage
+algorithm presents a better performance gain in the considered bandwidth
+interval with a gain of $40\%$ compared to only about $24\%$ for the classical
+GMRES algorithm.
\begin{figure}[ht]
\centering
\end{figure}
\subsubsection{Matrix size impacts on performances\\}
-In these experiments, the matrix size of the 3D Poisson problem is varied from $50^3$ to $190^3$ elements. The simulated computational grid is composed of 4 clusters of 8 processors each interconnected by the network $N2$ (see Table~\ref{tab:01}). Obviously, as shown in Figure~\ref{fig:05}, the execution times for both algorithms increase with increased matrix sizes. For all problem sizes, GMRES algorithm is always slower than the Krylov two-stage algorithm. Moreover, for this benchmark, it seems that the greater the problem size is, the bigger the ratio between execution times of both algorithms is. We can also observe that for some problem sizes, the convergence (and thus the execution time) of the Krylov two-stage algorithm varies quite a lot. %This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
+
+In these experiments, the matrix size of the 3D Poisson problem is varied from
+$50^3$ to $190^3$ elements. The simulated computational grid is composed of $4$
+clusters of $8$ processors each interconnected by the network $N2$ (see
+Table~\ref{tab:01}). As shown in Figure~\ref{fig:05}, the execution
+times for both algorithms increase with increased matrix sizes. For all problem
+sizes, the GMRES algorithm is always slower than the Krylov two-stage algorithm.
+Moreover, for this benchmark, it seems that the greater the problem size is, the
+bigger the ratio between execution times of both algorithms is. We can also
+observe that for some problem sizes, the convergence (and thus the execution
+time) of the Krylov two-stage algorithm varies quite a lot.
+%This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
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.
\begin{figure}[ht]
