\subsection{Comparison between GMRES and two-stage multisplitting algorithms in synchronous mode}
-In the scope of this paper, our first objective is to analyze when the synchronous Krylov two-stage method has better performance than the classical GMRES method. With a synchronous iterative method, better performance means a smaller number of iterations and execution time before reaching the convergence. In what follows, we will present the test conditions, the output results and our comments. For all simulations, we fix the network parameters of the intra-cluster links: the bandwidth $bw$=10Gbs and the latency $lat$=8$\times$10$^{-6}$.
+In the scope of this paper, our first objective is to analyze when the synchronous Krylov two-stage method has better performance than the classical GMRES method. With a synchronous iterative method, better performance means a smaller number of iterations and execution time before reaching the convergence.
-\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
-\ \\
-% environment
-
- The network of intra-clusters links has been
-designed to operate with a bandwidth equals to 10Gbits and a latency of 8$\times$10$^{-6}$ seconds. \\
-
-\RC{Je ne comprends plus rien CE : pourquoi dans 5.4.1 il y a 2 network et aussi dans 5.4.2. Quelle est la différence? Dans la figure 3 de la section 5.4.1 pourquoi il n'y a pas N1 et N2?}
+Table~\ref{tab:01} summarizes the parameters used in the different simulations: the grid architectures, the network of inter-clusters backbone links and the matrix sizes of the 3D Poisson problem. However, for all simulations we fix the network parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency $lat$=8$\times$10$^{-6}$. In what follows, we will present the test conditions, the output results and our comments.
\begin{table} [ht!]
\begin{center}
-\begin{tabular}{ll }
- \hline
- Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
- \multirow{2}{*}{Network} & Inter (N2): $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
- & Intra (N1): $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
- \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...}
-%\RCE{oui c est precise}
+\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}{*}{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}
+\caption{Parameters for the different simulations}
\label{tab:01}
\end{center}
\end{table}
+
+
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
+\ \\
+% environment
+
+Table~\ref{tab:01} summarizes the different parameters used in the simulations: the grid architectures, the network of inter-cluster backbone links and the matrix sizes of the 3D Poisson problem.
+
+
+
+
+
+
+
+
+
+
+
In this section, we analyze the simulations conducted on various grid
configurations presented in Table~\ref{tab:01}. It should be noticed that two
networks are considered: N1 is the network between clusters (inter-cluster) and
\hline
Grid Architecture & 4 $\times$ 8\\ %\hline
Inter Network & $bw$=1Gbs - $lat$=5.10$^{-5}$ \\
- Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 40$^{3}$ to 200$^{3}$\\ \hline
+ Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 50$^{3}$ to 190$^{3}$\\ \hline
\end{tabular}
\caption{Test conditions: Input matrix size impacts}
\label{tab:05}
\label{fig:05}
\end{figure}
- In these experiments, the input matrix size has been set from $N_{x} = N_{y}
- = N_{z} = 40$ to $200$ side elements that is from $40^{3} = 64.000$ to $200^{3}
- = 8,000,000$ points. Obviously, as shown in Figure~\ref{fig:05}, the execution
- time for both algorithms increases when the input matrix size also increases.
- But the interesting results are:
- \begin{enumerate}
- \item the important increase ($10$ times) of the number of iterations needed to
- reach the convergence for the classical GMRES algorithm particularly, 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}
- \RCE{Le nombre d'iterations augmente de 10 fois, cela surtout a partir de N=150}
-
- \item the classical GMRES execution time is almost the double for $N_{x}=140$
- compared with the Krylov multisplitting method.
- \end{enumerate}
+ In these experiments, the input matrix size has been set from $50^3$ to
+ $190^3$. Obviously, as shown in Figure~\ref{fig:05}, the execution time for both
+ algorithms increases when the input matrix size also increases. For all problem
+ sizes, GMRES is always slower than the Krylov multisplitting. Moreover, for this
+ benchmark, it seems that the greater the problem size is, the bigger the ratio
+ between both algorithm execution times is. We can also observ that for some
+ problem sizes, the Krylov multisplitting convergence varies quite a
+ lot. Consequently the execution times in that cases also varies.
+
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
