\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.
-%\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
-
-\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} & $N1$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ & $N2$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\
+\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 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}
+configurations and for different sizes of the 3D Poisson problem. The parameters
+of the network between clusters is fixed to $N2$ (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 fact, with multisplitting algorithms, the number of splitting (in
+our case, it is the number of clusters) influences on the convergence speed. The
+higher the number of splitting is, the slower the convergence of the algorithm
+is (see the output results obtained from configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs. 8$\times$8).
+
+The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16).
-\begin{figure} [ht!]
- \begin{center}
- \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
- \end{center}
- \caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
-%\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}
-%\LZK{Pour quelle taille du problème sont calculés les nombres d'itérations? Que représente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?}
- %\RCE {Corrige}
- \RC{Idéalement dans la légende il faudrait insiquer Pb size=$150^3$ ou $170^3$ car pour l'instant Nx=150 ca n'indique rien concernant Ny et Nz}
- \label{fig:01}
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\end{center}
+\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\label{fig:01}
\end{figure}
-
-
-The execution times between the two algorithms is significant with different
-grid architectures, even with the same number of processors (for example, 2 $\times$ 16
-and 4 $\times 8$). We can observe a better sensitivity of the Krylov multisplitting method
-(compared with the classical GMRES) when scaling up the number of the processors
-in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs
-$40\%$ better (resp. $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors/cores (grid 8 $\times$ 8). Note that even with a grid 8 $\times$ 8 having the maximum number of clusters, the execution time of the multisplitting method is in average 32\% less compared to GMRES.
-\RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
-\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exécution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?}
-\RCE{Remarquez que meme avec une grille 8x8, le multi est toujours plus performant}
-
\subsubsection{Simulations for two different inter-clusters network speeds \\}
-\begin{table} [ht!]
-\begin{center}
-\begin{tabular}{ll}
- \hline
- Grid architecture & 2$\times$16, 4$\times$8\\ %\hline
- \multirow{2}{*}{Inter Network} & N1: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
- & N2: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
- Matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
- \end{tabular}
-\caption{Test conditions: grid configurations 2$\times$16 and 4$\times$8 with networks N1 vs. N2}
-\label{tab:02}
-\end{center}
-\end{table}
-
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...}
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\%.
-
-
-%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\begin{figure}[t]
\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. $N2$}
\label{fig:02}
\end{figure}
-%\end{wrapfigure}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
\subsubsection{Network latency impacts on performance}
\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} [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}
\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
size scale up. It should be noticed that the same test has been done with the
-grid 2 $\times$ 16 leading to the same conclusion.
+grid 4 $\times$ 8 leading to the same conclusion.
\subsubsection{CPU Power impacts on performance}
