X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/fd6a9d41eaf24ef22ee658eb5da80ca4ea0df577..7c8fb94ad3ce704f81876b1dade93846a931a5a8:/paper.tex?ds=sidebyside diff --git a/paper.tex b/paper.tex index 42f4b5d..46ecc39 100644 --- a/paper.tex +++ b/paper.tex @@ -45,6 +45,8 @@ \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]} @@ -344,19 +346,18 @@ nodes/processors for each cluster). 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?} 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. @@ -620,7 +621,7 @@ 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 classical GMRES. +of $40\%$ which is only around $24\%$ for the classical GMRES. \subsubsection{Input matrix size impacts on performance} \ \\ @@ -632,27 +633,27 @@ of 40\% which is only around 24\% for classical GMRES. 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} +\caption{Input matrix size impacts} \end{figure} \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} + \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 +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} @@ -661,7 +662,7 @@ 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. -\subsubsection{CPU Power impact on performance} +\subsubsection{CPU Power impacts on performance} \begin{figure} [ht!] \centering @@ -671,51 +672,53 @@ grid 2x16 leading to the same conclusion. 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} +\caption{CPU Power impacts} \end{figure} \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} 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} +As stated before, the 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: \\ + +\begin{figure} [ht!] +\centering \begin{tabular}{r c } \hline Grid & 2x50 totaling 100 processors\\ %\hline @@ -725,15 +728,17 @@ The test conditions are summarized in the table below : \\ Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\ \end{tabular} -\end{footnotesize} +\end{figure} -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{table:01} 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}} @@ -744,14 +749,12 @@ the classical GMRES execution and convergence time. The experimentation has demo \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 \\ @@ -772,7 +775,11 @@ the classical GMRES \\ & 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{table:01} +\end{figure} + \section{Conclusion} CONCLUSION