\subsection{The 3D Poisson problem}
\label{3dpoisson}
-
-
We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
\begin{equation}
\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
simulated in the simulator tool to run the program. The following architectures
have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number
represents the number of clusters in the grid and the second number represents
- the number of hosts (processors/cores) in each cluster. The network has been
- designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a
- latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links
- (resp. inter-clusters backbone links). \\
-
- %\LZK{Il me semble que le bw et lat des deux réseaux varient dans les expés d'une simu à l'autre. On vire la dernière phrase?}
- %\RC{il me semble qu'on peut laisser ca}
+ the number of hosts (processors/cores) in each cluster. \\
\textbf{Step 5}: Conduct an extensive and comprehensive testings
within these configurations by varying the key parameters, especially
a lower speed. The network between distant clusters might be a bottleneck
for the global performance of the application.
- \subsection{Comparison of GMRES and Krylov two-stage algorithms in synchronous mode}
-
- In the scope of this paper, our first objective is to analyze when the 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.
- %%RAPH : on vire ca, c'est pas clair et pas important
- %For a systematic study, the experiments should figure out that, for various
- %grid parameters values, the simulator will confirm Multisplitting method better performance compared to classical GMRES, particularly on poor and slow networks.
- %\LZK{Pas du tout claire la dernière phrase (For a systematic...)!!}
- %\RCE { Reformule autrement}
+ \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}$.
-
-
- %\subsubsection{Execution of the algorithms on various computational grid architectures and scaling up the input matrix size}
\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
- \ \\
+ \ \\
% environment
+ The network of intra-clusters links has been
+ designed to operate with a bandwidth equals to 10Gbits and a latency of 8$\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?}
\begin{table} [ht!]
\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
