\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 $N2$ (see
\label{fig:01}
\end{figure}
-\subsubsection{Simulations for two different inter-clusters network speeds \\}
-
-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\%.
+\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.
\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}
+\subsubsection{Network latency impacts on performance\\}
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poison problem of size $150^3$. According to the results, a degradation of the network latency from $8\times 10^{-6}$ to $6\times 10^{-5}$ implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
+\begin{figure}[t]
+\centering
+\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impacts on execution times}
+\label{fig:03}
+\end{figure}
-\subsubsection{Network latency impacts on performance}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2 $\times$ 16\\ %\hline
- \multirow{2}{*}{Inter Network N1} & $bw$=1Gbs, \\ %\hline
- & $lat$= From 8$\times$10$^{-6}$ to $6.10^{-5}$ second \\
- Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
- \end{tabular}
-\caption{Test conditions: network latency impacts}
-\label{tab:03}
-\end{table}
-\begin{figure} [htbp]
-\centering
-\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impacts on execution time}
-%\AG{\np{E-6}}}
-\label{fig:03}
-\end{figure}
-In Table~\ref{tab:03}, parameters for the influence of the network latency are
-reported. 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. The execution time factor
-between the two algorithms varies from 2.2 to 1.5 times with a network latency
-decreasing from $8.10^{-6}$ to $6.10^{-5}$.
-\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 }
size scale up. It should be noticed that the same test has been done with the
grid 4 $\times$ 8 leading to the same conclusion.
-\subsubsection{CPU Power impacts on performance}
+\subsubsection{CPU Power impacts on performance\\}
+
\begin{table} [htbp]
\centering
in Table~\ref{tab:07}. In order to compare the execution times, this table
reports the relative gain between both algorithms. It is defined by the ratio
between the execution time of GMRES and the execution time of the
-multisplitting. The ration is greater than one because the asynchronous
+multisplitting. The ratio is greater than one because the asynchronous
multisplitting version is faster than GMRES.
