\end{equation}
where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
-\begin{figure}[t]
+\begin{figure}[htpb]
%\begin{algorithm}[t]
%\caption{Block Jacobi two-stage multisplitting method}
\begin{algorithmic}[1]
\end{equation}
The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
-\begin{figure}[t]
+\begin{figure}[htbp]
%\begin{algorithm}[t]
%\caption{Krylov two-stage method using block Jacobi multisplitting}
\begin{algorithmic}[1]
inter-cluster communications. In the following, these parameters are described:
\begin{itemize}
- \item hostfile: hosts description file.
+ \item hostfile: hosts description file,
\item platform: file describing the platform architecture: clusters (CPU power,
\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
-latency $lat$, \dots{}).
+latency $lat$, \dots{}),
\item archi : grid computational description (number of clusters, number of
nodes/processors in each cluster).
\end{itemize}
\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
on the one hand the algorithm execution mode (synchronous and asynchronous)
and on the other hand the execution time and the number of iterations to reach the convergence. \\
-\textbf{Step 4 }: Set up the different grid testbed environments that will be
+\textbf{Step 4}: Set up the different grid testbed environments that will be
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!]
%\RC{Les légendes ne sont pas explicites...}
%\RCE{Corrige}
-\begin{figure} [ht!]
+\begin{figure} [htbp]
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\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}
+\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}
\label{tab:03}
\end{table}
-\begin{figure} [ht!]
+\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}}}
+\caption{Network latency impacts on execution time}
+%\AG{\np{E-6}}}
\label{fig:03}
\end{figure}
-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 which means that the GMRES seems tolerate more the network latency variation with a less rate increase of the execution time. However, 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 $6.10^{-5}$.
+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 $6.10^{-5}$.
-\RC{Les 2 précédentes phrases me semblent en contradiction....}
-\RCE{Reformule}
\subsubsection{Network bandwidth impacts on performance}
\ \\
& $lat$= 5.10$^{-5}$ second \\
Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
\end{tabular}
-\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
-\RCE{C est le bw}
+\caption{Test conditions: Network bandwidth impacts}
+% \RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}
+%\RCE{C est le bw}
\label{tab:04}
\end{table}
-\begin{figure} [ht!]
+\begin{figure} [htbp]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
-\caption{Network bandwith impacts on execution time
-\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}
-\RCE{Corrige}
+\caption{Network bandwith impacts on execution time}
+%\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}
+%\RCE{Corrige}
\label{fig:04}
\end{figure}
\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}
\end{table}
-\begin{figure} [ht!]
+\begin{figure} [htbp]
\centering
\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
\caption{Problem size impacts on execution time}
\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
\subsubsection{CPU Power impacts on performance}
-\begin{table} [ht!]
+\begin{table} [htbp]
\centering
\begin{tabular}{r c }
\hline
theoretically reduce the overall execution time and can improve the algorithm
performance.
-\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
-\RCE{C est la description du dernier test sync/async avec l'introduction de la notion de relative gain}
-In this section, Simgrid simulator tool has been successfully used to show
-the efficiency of the multisplitting in asynchronous mode and to find the best
-combination of the grid resources (CPU, Network, input matrix size, \ldots ) to
-get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ /
-exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
+In this section, the Simgrid simulator is used to compare the behavior of the
+multisplitting in asynchronous mode with GMRES in synchronous mode. Several
+benchmarks have been performed with various combination of the grid resources
+(CPU, Network, input matrix size, \ldots ). The test conditions are summarized
+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 version is faster than GMRES.
-The test conditions are summarized in the table~\ref{tab:07}: \\
-\begin{table} [ht!]
+\begin{table} [htbp]
\centering
\begin{tabular}{r c }
\hline
power (GFlops)
& 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\
\hline
- size (N)
+ size ($N^3$)
& 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\
\hline
Precision
