\todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
\newcommand{\RCE}[2][inline]{%
\todo[color=yellow!10,#1]{\sffamily\textbf{RCE:} #2}\xspace}
+\newcommand{\DL}[2][inline]{%
+ \todo[color=pink!10,#1]{\sffamily\textbf{DL:} #2}\xspace}
\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}
accurate performance models. That is why another solution is to use a simulation
tool which allows us to change many parameters of the architecture (network
bandwidth, latency, number of processors) and to simulate the execution of such
-applications. We have decided to use SimGrid as it enables to benchmark MPI
-applications.
+applications. The main contribution of this paper is to show that the use of a
+simulation tool (here we have decided to use the SimGrid toolkit) can really
+help developpers to better tune their applications for a given multi-core
+architecture.
-In this paper, we focus our attention on two parallel iterative algorithms based
+In particular we focus our attention on two parallel iterative algorithms based
on the Multisplitting algorithm and we compare them to the GMRES algorithm.
-These algorithms are used to solve libear systems. Two different variants of
+These algorithms are used to solve linear systems. Two different variants of
the Multisplitting are studied: one using synchronoous iterations and another
-one with asynchronous iterations. For each algorithm we have tested different
-parameters to see their influence. We strongly recommend people interested
-by investing into a new expensive hardware architecture to benchmark
-their applications using a simulation tool before.
-
-
-
+one with asynchronous iterations. For each algorithm we have simulated
+different architecture parameters to evaluate their influence on the overall
+execution time. The obtain simulated results confirm the real results
+previously obtained on different real multi-core architectures and also confirm
+the efficiency of the asynchronous multisplitting algorithm compared to the
+synchronous GMRES method.
\end{abstract}
A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
\label{eq:03}
\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 ({\it Generalized Minimal RESidual})~\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}.
+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 ({\it Generalized Minimal RESidual})~\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{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{algorithm}[t]
In addition, the following arguments are given to the programs at runtime:
\begin{itemize}
- \item maximum number of inner and outer iterations;
- \item inner and outer precisions;
- \item maximum number of the GMRES restarts in the Arnorldi process;
- \item maximum number of iterations and the tolerance threshold in classical GMRES;
- \item tolerance threshold for outer and inner-iterations;
- \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on $x, y, z$ axis;
- \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments; \RC{CE tu vérifies, je dis ca de tête}
- \item matrix off-diagonal value;
- \item execution mode: synchronous or asynchronous;
- \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les arguments pour CGLS ci dessous} \RC{Vu que tu n'as pas fait varier ce paramètre, on peut ne pas en parler}
- \item Size of matrix S;
- \item Maximum number of iterations and tolerance threshold for CGLS.
+ \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
+ \item inner precision $\TOLG$ and outer precision $\TOLM$,
+ \item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
+ \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
+ \item matrix off-diagonal value is fixed to $-1.0$,
+ \item number of vectors in matrix $S$ (i.e. value of $s$),
+ \item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
+ \item maximum number of iterations and precision for the classical GMRES method,
+ \item maximum number of restarts for the Arnorldi process in GMRES method,
+ \item execution mode: synchronous or asynchronous.
\end{itemize}
+\LZK{CE pourrais tu vérifier et confirmer les valeurs des éléments diag et off-diag de la matrice?}
+\RCE{oui, les valeurs de diag et off-diag donnees sont ok}
It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
a grid environment}
When running a distributed application in a computational grid, many factors may
-have a strong impact on the performances. First of all, the architecture of the
+have a strong impact on the performance. First of all, the architecture of the
grid itself can obviously influence the performance results of the program. The
performance gain might be important theoretically when the number of clusters
and/or the number of nodes (processors/cores) in each individual cluster
increase.
-Another important factor impacting the overall performances of the application
+Another important factor impacting the overall performance of the application
is the network configuration. Two main network parameters can modify drastically
the program output results:
\begin{enumerate}
\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
In the scope of this paper, our first objective is to analyze when the Krylov
-Multisplitting method has better performances than the classical GMRES
-method. With an iterative method, better performances mean a smaller number of
-iterations and execution time before reaching the convergence. For a systematic
-study, the experiments should figure out that, for various grid parameters
-values, the simulator will confirm the targeted outcomes, particularly for poor
-and slow networks, focusing on the impact on the communication performance on
-the chosen class of algorithm.
+Multisplitting method has better performance than the classical GMRES
+method. With a synchronous iterative method, better performance mean a
+smaller number of iterations and execution time before reaching the convergence.
+For a systematic study, the experiments should figure out that, for various
+grid parameters values, the simulator will confirm the targeted outcomes,
+particularly for poor and slow networks, focusing on the impact on the
+communication performance on the chosen class of algorithm.
The following paragraphs present the test conditions, the output results
and our comments.\\
-\subsubsection{Execution of the the algorithms on various computational grid
-architecture and scaling up the input matrix size}
+\subsubsection{Execution of the algorithms on various computational grid
+architectures and scaling up the input matrix size}
\ \\
% environment
-\begin{figure} [ht!]
+\begin{table} [ht!]
\begin{center}
\begin{tabular}{r c }
\hline
- Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
+ Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
- & N$_{x}$ x N$_{y}$ x N$_{z}$ =170 x 170 x 170 \\ \hline
\end{tabular}
-\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\caption{Test conditions: Various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\label{tab:01}
\end{center}
-\end{figure}
+\end{table}
%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
-In this section, we analyze the performences of algorithms running on various
-grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01}
-show for all grid configuration the non-variation of the number of iterations of
-classical GMRES for a given input matrix size; it is not the case for the
+In this section, we analyze the performance of algorithms running on various
+grid configurations (2x16, 4x8, 4x16 and 8x8). 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...}
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
- \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
+ \caption{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170}
\label{fig:01}
\end{figure}
-The execution time difference between the two algorithms is important when
-comparing between different grid architectures, even with the same number of
-processors (like 2x16 and 4x8 = 32 processors for example). The
-experiment concludes the low sensitivity of the 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\%) less when running from 2x16=32 to 8x8=64 processors.
+The execution times between the two algorithms is significant with different
+grid architectures, even with the same number of processors (for example, 2x16
+and 4x8). We can observ the low 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 2x16=32 to 8x8=64 processors.
-\textit{\\3.b Running on two different speed cluster inter-networks\\}
+\subsubsection{Running on two different inter-clusters network speeds \\}
-% environment
-\begin{footnotesize}
+\begin{table} [ht!]
+\begin{center}
\begin{tabular}{r c }
\hline
- Grid & 2x16, 4x8\\ %\hline
+ Grid Architecture & 2x16, 4x8\\ %\hline
Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
- & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
\end{tabular}
-Table 2 : Clusters x Nodes - Networks N1 x N2 \\
-
- \end{footnotesize}
+\caption{Test conditions: Grid 2x16 and 4x8 - Networks N1 vs N2}
+\label{tab:02}
+\end{center}
+\end{table}
+These experiments compare the behavior of the algorithms running first on a
+speed inter-cluster network (N1) and also on a less performant network (N2).
+Figure~\ref{fig:02} shows that end users will gain to reduce the execution time
+for both algorithms in using a grid architecture like 4x16 or 8x8: the
+performance was increased by a factor of $2$. 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\%.
+%\RC{c'est pas clair : la différence entre quoi et quoi?}
+%\DL{pas clair}
+%\RCE{Modifie}
%\begin{wrapfigure}{l}{100mm}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Cluster x Nodes N1 x N2}
-%\label{overflow}}
+\caption{Grid 2x16 and 4x8 - Networks N1 vs N2}
+\label{fig:02}
\end{figure}
%\end{wrapfigure}
-The experiments compare the behavior of the algorithms running first on
-a speed inter- cluster network (N1) and also on a less performant network (N2).
-Figure 4 shows that end users will gain to reduce the execution time
-for both algorithms in using a grid architecture like 4x16 or 8x8: the
-performance was increased in a factor of 2. The results depict also that
-when the network speed drops down (12.5\%), the difference between the execution
-times can reach more than 25\%.
-\textit{\\3.c Network latency impacts on performance\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Network latency impacts on performance}
+\ \\
+\begin{table} [ht!]
+\centering
\begin{tabular}{r c }
\hline
- Grid & 2x16\\ %\hline
+ Grid Architecture & 2x16\\ %\hline
Network & N1 : bw=1Gbs \\ %\hline
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline\\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
\end{tabular}
-Table 3 : Network latency impact \\
-
-\end{footnotesize}
+\caption{Test conditions: Network latency impacts}
+\label{tab:03}
+\end{table}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impact on execution time}
-%\label{overflow}}
+\caption{Network latency impacts on execution time}
+\label{fig:03}
\end{figure}
-According the results in figure 5, degradation of the network
-latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time
-increase more than 75\% (resp. 82\%) of the execution for the classical
-GMRES (resp. multisplitting) algorithm. In addition, it appears that the
-multisplitting method tolerates more the network latency variation with
-a less rate increase of the execution time. Consequently, in the worst case (lat=6.10$^{-5
-}$), the execution time for GMRES is almost the double of the time for
-the multisplitting, even though, the performance was on the same order
-of magnitude with a latency of 8.10$^{-6}$.
-
-\textit{\\3.d Network bandwidth impacts on performance\\}
+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. In addition, it appears that the Krylov
+multisplitting method tolerates more the network latency variation with a less
+rate increase of the execution time. Consequently, in the worst case
+($lat=6.10^{-5 }$), the execution time for GMRES is almost the double than the
+time of the Krylov multisplitting, even though, the performance was on the same
+order of magnitude with a latency of $8.10^{-6}$.
-% environment
-\begin{footnotesize}
+\subsubsection{Network bandwidth impacts on performance}
+\ \\
+\begin{table} [ht!]
+\centering
\begin{tabular}{r c }
\hline
- Grid & 2x16\\ %\hline
+ Grid Architecture & 2x16\\ %\hline
Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
\end{tabular}
-Table 4 : Network bandwidth impact \\
-
-\end{footnotesize}
+\caption{Test conditions: Network bandwidth impacts}
+\label{tab:04}
+\end{table}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
-\caption{Network bandwith impact on execution time}
-%\label{overflow}
+\caption{Network bandwith impacts on execution time}
+\label{fig:04}
\end{figure}
+The results of increasing the network bandwidth show the improvement of the
+performance for both algorithms by reducing the execution time (see
+Figure~\ref{fig:04}). However, in this case, the Krylov multisplitting method
+presents a better performance in the considered bandwidth interval with a gain
+of $40\%$ which is only around $24\%$ for the classical GMRES.
-
-The results of increasing the network bandwidth show the improvement
-of the performance for both of the two algorithms by reducing the execution time (Figure 6). However, and again in this case, the multisplitting method presents a better performance in the considered bandwidth interval with a gain of 40\% which is only around 24\% for classical GMRES.
-
-\textit{\\3.e Input matrix size impacts on performance\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Input matrix size impacts on performance}
+\ \\
+\begin{table} [ht!]
+\centering
\begin{tabular}{r c }
\hline
- Grid & 4x8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\
+ Grid Architecture & 4x8\\ %\hline
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
+ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
\end{tabular}
-Table 5 : Input matrix size impact\\
-
-\end{footnotesize}
+\caption{Test conditions: Input matrix size impacts}
+\label{tab:05}
+\end{table}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
-\caption{Pb size impact on execution time}
-%\label{overflow}}
+\caption{Problem size impacts on execution time}
+\label{fig:05}
\end{figure}
-In this experimentation, 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 the figure 7,
-the execution time for the two algorithms convergence increases with the
-iinput matrix size. But the interesting results here direct on (i) the
-drastic increase (300 times) of the number of iterations needed before
-the convergence for the classical GMRES algorithm when the matrix size
-go beyond N$_{x}$=150; (ii) the classical GMRES execution time also almost
-the double from N$_{x}$=140 compared with the convergence time of the
-multisplitting method. 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. Note that the
-same test has been done with the grid 2x16 getting the same conclusion.
-
-\textit{\\3.f CPU Power impact on performance\\}
+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 drastic increase ($10$ times) \RC{Je ne vois pas cela sur la figure}
+\RCE{Corrige} of the number of iterations needed to reach the convergence for the classical
+GMRES algorithm when the matrix size go beyond $N_{x}=150$;
+\item the classical GMRES execution time is almost the double for $N_{x}=140$
+ compared with the Krylov multisplitting method.
+\end{enumerate}
+
+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 2x16 leading to the same conclusion.
-% environment
-\begin{footnotesize}
+\subsubsection{CPU Power impacts on performance}
+
+\begin{table} [ht!]
+\centering
\begin{tabular}{r c }
\hline
- Grid & 2x16\\ %\hline
+ Grid architecture & 2x16\\ %\hline
Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
\end{tabular}
-Table 6 : CPU Power impact \\
-
-\end{footnotesize}
-
+\caption{Test conditions: CPU Power impacts}
+\label{tab:06}
+\end{table}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
-\caption{CPU Power impact on execution time}
-%\label{overflow}}
-s\end{figure}
-
-Using the Simgrid simulator flexibility, we have tried to determine the
-impact on the algorithms performance in varying the CPU power of the
-clusters nodes from 1 to 19 GFlops. The outputs depicted in the figure 6
-confirm the performance gain, around 95\% for both of the two methods,
-after adding more powerful CPU.
-
-\subsection{Comparing GMRES in native synchronous mode and
-Multisplitting algorithms in asynchronous mode}
-
-The previous paragraphs put in evidence the interests to simulate the
-behavior of the application before any deployment in a real environment.
-We have focused the study on analyzing the performance in varying the
-key factors impacting the results. The study compares
-the performance of the two proposed algorithms both in \textit{synchronous mode
-}. In this section, following the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textit{
-asynchronous mode} compared with the classical GMRES staying in
-\textit{synchronous mode}.
-
-Note that the interest of using the asynchronous mode for data exchange
-is mainly, in opposite of the synchronous mode, the non-wait aspects of
-the current computation after a communication operation like sending
-some data between nodes. Each processor can continue their local
-calculation without waiting for the end of the communication. Thus, the
-asynchronous may theoretically reduce the overall execution time and can
-improve the algorithm performance.
-
-As stated supra, Simgrid simulator tool has been used to prove 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.
-
-
-The test conditions are summarized in the table below : \\
+\caption{CPU Power impacts on execution time}
+\label{fig:06}
+\end{figure}
-% environment
-\begin{footnotesize}
+Using the Simgrid simulator flexibility, we have tried to determine the impact
+on the algorithms performance in varying the CPU power of the clusters nodes
+from $1$ to $19$ GFlops. The outputs depicted in Figure~\ref{fig:06} confirm the
+performance gain, around $95\%$ for both of the two methods, after adding more
+powerful CPU.
+
+\DL{il faut une conclusion sur ces tests : ils confirment les résultats déjà
+obtenus en grandeur réelle. Donc c'est une aide précieuse pour les dev. Pas
+besoin de déployer sur une archi réelle}
+
+
+\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
+
+The previous paragraphs put in evidence the interests to simulate the behavior
+of the application before any deployment in a real environment. In this
+section, following the same previous methodology, our goal is to compare the
+efficiency of the multisplitting method in \textit{ asynchronous mode} compared with the
+classical GMRES in \textit{synchronous mode}.
+
+The interest of using an asynchronous algorithm is that there is no more
+synchronization. With geographically distant clusters, this may be essential.
+In this case, each processor can compute its iteration freely without any
+synchronization with the other processors. Thus, the asynchronous may
+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}
+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.
+
+
+The test conditions are summarized in the table~\ref{tab:07}: \\
+
+\begin{table} [ht!]
+\centering
\begin{tabular}{r c }
\hline
- Grid & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2x50 totaling 100 processors\\ %\hline
Processors Power & 1 GFlops to 1.5 GFlops\\
Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
\end{tabular}
-\end{footnotesize}
+\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
+\label{tab:07}
+\end{table}
-Again, comprehensive and extensive tests have been conducted varying the
-CPU power and the network parameters (bandwidth and latency) in the
-simulator tool with different problem size. The relative gains greater
-than 1 between the two algorithms have been captured after each step of
-the test. Table 7 below has recorded the best grid configurations
-allowing the multisplitting method execution time more performant 2.5 times than
-the classical GMRES execution and convergence time. The experimentation has demonstrated the relative multisplitting algorithm tolerance when using a low speed network that we encounter usually with distant clusters thru the internet.
+Again, comprehensive and extensive tests have been conducted with different
+parameters as the CPU power, the network parameters (bandwidth and latency)
+and with different problem size. The relative gains greater than $1$ between the
+two algorithms have been captured after each step of the test. In
+Figure~\ref{fig:07} are reported the best grid configurations allowing
+the multisplitting method to be more than $2.5$ times faster than the
+classical GMRES. These experiments also show the relative tolerance of the
+multisplitting algorithm when using a low speed network as usually observed with
+geographically distant clusters through the internet.
% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
\end{tabular}}
-\begin{table}[!t]
- \centering
+\begin{figure}[!t]
+\centering
+%\begin{table}
% \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
% \label{"Table 7"}
-Table 7. Relative gain of the multisplitting algorithm compared with
-the classical GMRES \\
-
- \begin{mytable}{11}
+ \begin{mytable}{11}
\hline
bandwidth (Mbit/s)
& 5 & 5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 \\
& 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 & 2.51 & 2.58 & 2.55 & 2.54 \\
\hline
\end{mytable}
-\end{table}
+%\end{table}
+ \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+ \label{fig:07}
+\end{figure}
+
\section{Conclusion}
CONCLUSION
