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]
\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 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 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,
\ \\
% 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
+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}
\label{fig:01}
\end{figure}
in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs
$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors.
-\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 - 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).
+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}
+\caption{Grid 2x16 and 4x8 - 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}
\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}
+\label{tab:04}
+\end{table}
\begin{figure} [ht!]
\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 impacts}
-\end{figure}
+\caption{Test conditions: Input matrix size impacts}
+\label{tab:05}
+\end{table}
\begin{figure} [ht!]
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
+ \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.
\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 impacts}
-\end{figure}
+\caption{Test conditions: CPU Power impacts}
+\label{tab:06}
+\end{table}
\begin{figure} [ht!]
\centering
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} with 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
performance.
\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
-As stated before, the Simgrid simulator tool has been successfully used to show
+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 below : \\
+The test conditions are summarized in the table~\ref{tab:07}: \\
-\begin{figure} [ht!]
+\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{figure}
+\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 with different
-parametes as the CPU power, the network parameters (bandwidth and latency) in
-the simulator tool 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{table:01} are reported the best grid configurations
-allowing the multisplitting method to be more than 2.5 times faster than the
+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 throuth the internet.
+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{mytable}
%\end{table}
\caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
- \label{table:01}
+ \label{fig:07}
\end{figure}
