\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]
\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}
-
-
-%\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
+\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.
-\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
+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 $N1$ (see Table~\ref{tab:01}. Figure~\ref{fig:01} shows, for all grid configurations and a given matrix size 170$^3$ elements, a non-variation in the number of iterations for the classical GMRES algorithm, which is not the case of the Krylov two-stage algorithm.
-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
-N2 is the network inside a cluster (intra-cluster). Figure~\ref{fig:01} shows,
-for all grid configurations and a given matrix size, a non-variation in the
-number of iterations for the classical GMRES algorithm, which is not the case of
-the Krylov two-stage algorithm.
-%% First, the results in Figure~\ref{fig:01}
-%% show for all grid configurations the non-variation of the number of iterations of
-%% classical GMRES for a given input matrix size; it is not the case for the
-%% multisplitting method.
-%\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
-%\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
