X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/34053b2cbdb34bf90e60922c39c759d343ed375d..7c8fb94ad3ce704f81876b1dade93846a931a5a8:/paper.tex?ds=inline diff --git a/paper.tex b/paper.tex index 82777e3..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]} @@ -96,20 +98,21 @@ performed. With some applications, it is difficult, if not impossible, to build 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} @@ -343,19 +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. @@ -464,19 +466,19 @@ and between distant clusters. This parameter is application dependent. 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. +method. With a synchronous 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. 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 @@ -500,9 +502,9 @@ architecture and scaling up the input matrix size} 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 +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...} @@ -523,9 +525,9 @@ 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\%) less when running from 2x16=32 to 8x8=64 processors. +$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. -\subsubsection{Running on two different speed cluster inter-networks} +\subsubsection{Running on two different inter-clusters network speed} \ \\ \begin{figure} [ht!] @@ -535,7 +537,7 @@ in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs Grid & 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} \caption{Clusters x Nodes - Networks N1 x N2} \end{center} @@ -556,9 +558,10 @@ 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 in a factor of 2. The results depict also that when +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} \ \\ @@ -570,7 +573,7 @@ times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi 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 impact} +\caption{Network latency impacts} \end{figure} @@ -578,20 +581,20 @@ times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi \begin{figure} [ht!] \centering \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf} -\caption{Network latency impact on execution time} +\caption{Network latency impacts on execution time} \label{fig:03} \end{figure} -According the results in Figure~\ref{fig:03}, a 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. Krylov +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 +($lat=6.10^{-5 }$), 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$^{-6}$. +order of magnitude with a latency of $8.10^{-6}$. \subsubsection{Network bandwidth impacts on performance} \ \\ @@ -603,24 +606,22 @@ order of magnitude with a latency of 8.10$^{-6}$. 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 impact} +\caption{Network bandwidth impacts} \end{figure} \begin{figure} [ht!] \centering \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf} -\caption{Network bandwith impact on execution time} +\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 classical GMRES. +of $40\%$ which is only around $24\%$ for the classical GMRES. \subsubsection{Input matrix size impacts on performance} \ \\ @@ -629,30 +630,30 @@ of 40\% which is only around 24\% for classical GMRES. \begin{tabular}{r c } \hline Grid & 4x8\\ %\hline - Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ + 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 @@ -671,22 +672,26 @@ 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 @@ -710,7 +715,7 @@ 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 below: \\ \begin{figure} [ht!] \centering @@ -726,14 +731,14 @@ The test conditions are summarized in the table below : \\ \end{figure} 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{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 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}}