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analysis of simulated grid-enabled numerical iterative algorithms}
%\itshape{\journalnamelc}\footnotemark[2]}
-\author{ Charles Emile Ramamonjisoa and
- David Laiymani and
- Arnaud Giersch and
- Lilia Ziane Khodja and
- Raphaël Couturier
+\author{Charles Emile Ramamonjisoa\affil{1},
+ David Laiymani\affil{1},
+ Arnaud Giersch\affil{1},
+ Lilia Ziane Khodja\affil{2} and
+ Raphaël Couturier\affil{1}
}
\address{
- \centering
- Femto-ST Institute - DISC Department\\
- Université de Franche-Comté\\
- Belfort\\
- Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
+ \affilnum{1}%
+ Femto-ST Institute, DISC Department,
+ University of Franche-Comté,
+ Belfort, France.
+ Email:~\email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}\break
+ \affilnum{2}
+ Department of Aerospace \& Mechanical Engineering,
+ Non Linear Computational Mechanics,
+ University of Liege, Liege, Belgium.
+ Email:~\email{l.zianekhodja@ulg.ac.be}
}
-%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
-
\begin{abstract} The behavior of multi-core applications is always a challenge
to predict, especially with a new architecture for which no experiment has been
performed. With some applications, it is difficult, if not impossible, to build
given application for a given architecture. In this way and in order to reduce
the access cost to these computing resources it seems very interesting to use a
simulation environment. The advantages are numerous: development life cycle,
-code debugging, ability to obtain results quickly~\ldots. In counterpart, the simulation results need to be consistent with the real ones.
+code debugging, ability to obtain results quickly\dots{} In counterpart, the simulation results need to be consistent with the real ones.
In this paper we focus on a class of highly efficient parallel algorithms called
\emph{iterative algorithms}. The parallel scheme of iterative methods is quite
tune their application for a given multi-core architecture. To show the validity
of this approach we first compare the simulated execution of the multisplitting
algorithm with the GMRES (Generalized Minimal Residual)
-solver~\cite{saad86} in synchronous mode. The obtained results on different
+solver~\cite{saad86} in synchronous mode.
+
+\LZK{Pas trop convainquant comme argument pour valider l'approche de simulation. \\On peut dire par exemple: on a pu simuler différents algos itératifs à large échelle (le plus connu GMRES et deux variantes de multisplitting) et la simulation nous a permis (sans avoir le vrai matériel) de déterminer quelle serait la meilleure solution pour une telle configuration de l'archi ou vice versa.\\A revoir...}
+
+The obtained results on different
simulated multi-core architectures confirm the real results previously obtained
-on non simulated architectures. We also confirm the efficiency of the
-asynchronous multisplitting algorithm compared to the synchronous GMRES. In
+on non simulated architectures.
+
+\LZK{Il n y a pas dans la partie expé cette comparaison et confirmation des résultats entre la simulation et l'exécution réelle des algos sur les vrais clusters.\\ Sinon on pourrait ajouter dans la partie expé une référence vers le journal supercomput de krylov multi pour confirmer que cette méthode est meilleure que GMRES sur les clusters large échelle.}
+
+We also confirm the efficiency of the
+asynchronous multisplitting algorithm compared to the synchronous GMRES.
+
+\LZK{P.S.: Pour tout le papier, le principal objectif n'est pas de faire des comparaisons entre des méthodes itératives!!\\Sinon, les deux algorithmes Krylov multisplitting synchrone et multisplitting asynchrone sont plus efficaces que GMRES sur des clusters à large échelle.\\Et préciser, si c'est vraiment le cas, que le multisplitting asynchrone est plus efficace et adapté aux clusters distants par rapport aux deux autres algos (je n'ai pas encore lu la partie expé)}
+
+In
this way and with a simple computing architecture (a laptop) SimGrid allows us
to run a test campaign of a real parallel iterative applications on
different simulated multi-core architectures. To our knowledge, there is no
experimental results are presented in section~\ref{sec:expe} followed by some
concluding remarks and perspectives.
+\LZK{Proposition d'un titre pour le papier: Grid-enabled simulation of large-scale linear iterative solvers.}
+
-\section{The asynchronous iteration model}
+\section{The asynchronous iteration model and the motivations of our work}
\label{sec:asynchro}
Asynchronous iterative methods have been studied for many years theoritecally and
it must not synchronize all the processors. Interested readers can
consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
+The number of iterations required to reach the convergence is generally greater
+for the asynchronous scheme (this number depends depends on the delay of the
+messages). Note that, it is not the case in the synchronous mode where the
+number of iterations is the same than in the sequential mode. In this way, the
+set of the parameters of the platform (number of nodes, power of nodes,
+inter and intra clusters bandwidth and latency, \ldots) and of the
+application can drastically change the number of iterations required to get the
+convergence. It follows that asynchronous iterative algorithms are difficult to
+optimize since the financial and deployment costs on large scale multi-core
+architecture are often very important. So, prior to delpoyment and tests it
+seems very promising to be able to simulate the behavior of asynchronous
+iterative algorithms. The problematic is then to show that the results produce
+by simulation are in accordance with reality i.e. of the same order of
+magnitude. To our knowledge, there is no study on this problematic.
+
\section{SimGrid}
- \label{sec:simgrid}
+\label{sec:simgrid}
+SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile} is a discrete event simulation framework to study the behavior of large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds and High Performance Computation systems. It is widely used to simulate and evaluate heuristics, prototype applications or even assess legacy MPI applications. It is still actively developed by the scientific community and distributed as an open source software.
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
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}
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 a synchronous iterative method, better performances mean a
+Multisplitting 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.
For a systematic study, the experiments should figure out that, for various
grid parameters values, the simulator will confirm the targeted outcomes,
\ \\
% 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{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}}
+\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
+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
\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\RC{idem}}
\label{fig:01}
\end{figure}
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.
+$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
-\subsubsection{Running on two different inter-clusters network speed}
-\ \\
+\subsubsection{Running on two different inter-clusters network speeds \\}
-\begin{figure} [ht!]
+\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
\end{tabular}
-\caption{Clusters x Nodes - Networks N1 x N2}
+\caption{Test conditions: grid 2x16 and 4x8 with networks N1 vs N2}
+\label{tab:02}
\end{center}
-\end{figure}
+\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). \RC{Il faut définir cela avant...}
+Figure~\ref{fig:02} shows that end users will reduce the execution time
+for both algorithms when using a grid architecture like 4x16 or 8x8: the reduction is about $2$. 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}
+\caption{Grid 2x16 and 4x8 with networks N1 vs N2}
\label{fig:02}
\end{figure}
%\end{wrapfigure}
-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 (12.5\%), the difference between the execution
-times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi et quoi?}
-\DL{pas clair}
\subsubsection{Network latency impacts on performance}
\ \\
-\begin{figure} [ht!]
+\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
\end{tabular}
-\caption{Network latency impacts}
-\end{figure}
+\caption{Test conditions: network latency impacts}
+\label{tab:03}
+\end{table}
\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. 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 $8.10^{-6}$.
+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.\RC{Les 2 précédentes phrases me
+ semblent en contradiction....} 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 $8.10^{-6}$.
\subsubsection{Network bandwidth impacts on performance}
\ \\
-\begin{figure} [ht!]
+\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}
-\caption{Network bandwidth impacts}
-\end{figure}
+\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
+\label{tab:04}
+\end{table}
\begin{figure} [ht!]
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 classical GMRES.
+of $40\%$ which is only around $24\%$ for the classical GMRES.
\subsubsection{Input matrix size impacts on performance}
\ \\
-\begin{figure} [ht!]
+\begin{table} [ht!]
\centering
\begin{tabular}{r c }
\hline
- Grid & 4x8\\ %\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}
-\caption{Input matrix size impact}
-\end{figure}
+\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{Problem size impact on execution time}
+\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
+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 (300 times) \RC{Je ne vois pas cela sur la figure}
-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
+ \item the drastic increase ($10$ times) of the number of iterations needed to
+ reach the convergence for the classical GMRES algorithm 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}
+\item the classical GMRES execution time is almost the double for $N_{x}=140$
compared with the Krylov multisplitting method.
\end{enumerate}
size scale up. It should be noticed that the same test has been done with the
grid 2x16 leading to the same conclusion.
-\subsubsection{CPU Power impact on performance}
+\subsubsection{CPU Power impacts on performance}
-\begin{figure} [ht!]
+\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}
-\caption{CPU Power impact}
-\end{figure}
+\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}
+\caption{CPU Power impacts on execution time}
\label{fig:06}
\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 Figure~\ref{fig:06} confirm the
-performance gain, around 95\% for both of the two methods, after adding more
+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. 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 : \\
+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}.
-% environment
-\begin{footnotesize}
+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
-\section*{Acknowledgment}
-
+%\section*{Acknowledgment}
+\ack
This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
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