From: ziane Date: Fri, 8 May 2015 08:41:52 +0000 (+0200) Subject: petites corrections X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/commitdiff_plain/96e726f9524ffdb8eaead14a943d80870ea1da65?hp=976b793822811b4335b31747dc0e5542a41fc62b petites corrections Merge branch 'master' of ssh://bilbo.iut-bm.univ-fcomte.fr/rce2015 --- diff --git a/paper.tex b/paper.tex index 0e69066..c3cfdbb 100644 --- a/paper.tex +++ b/paper.tex @@ -321,7 +321,7 @@ A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L, \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~\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}. -\begin{figure}[t] +\begin{figure}[htpb] %\begin{algorithm}[t] %\caption{Block Jacobi two-stage multisplitting method} \begin{algorithmic}[1] @@ -359,7 +359,7 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde \end{equation} The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}). -\begin{figure}[t] +\begin{figure}[htbp] %\begin{algorithm}[t] %\caption{Krylov two-stage method using block Jacobi multisplitting} \begin{algorithmic}[1] @@ -591,7 +591,7 @@ the Krylov two-stage algorithm. %\RC{Les légendes ne sont pas explicites...} %\RCE{Corrige} -\begin{figure} [ht!] +\begin{figure} [htbp] \begin{center} \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf} \end{center} @@ -641,7 +641,7 @@ the network speed drops down (variation of 12.5\%), the difference between t %\begin{wrapfigure}{l}{100mm} -\begin{figure} [ht!] +\begin{figure} [htbp] \centering \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf} \caption{Various grid configurations with networks N1 vs N2 @@ -667,7 +667,7 @@ the network speed drops down (variation of 12.5\%), the difference between t \label{tab:03} \end{table} -\begin{figure} [ht!] +\begin{figure} [htbp] \centering \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf} \caption{Network latency impacts on execution time @@ -700,12 +700,12 @@ more than $75\%$ (resp. $82\%$) of the execution for the classical GMRES \end{table} -\begin{figure} [ht!] +\begin{figure} [htbp] \centering \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf} -\caption{Network bandwith impacts on execution time -\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.} -\RCE{Corrige} +\caption{Network bandwith impacts on execution time} +%\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.} +%\RCE{Corrige} \label{fig:04} \end{figure} @@ -730,7 +730,7 @@ of $40\%$ which is only around $24\%$ for the classical GMRES. \end{table} -\begin{figure} [ht!] +\begin{figure} [htbp] \centering \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf} \caption{Problem size impacts on execution time} @@ -759,7 +759,7 @@ grid 2 $\times$ 16 leading to the same conclusion. \subsubsection{CPU Power impacts on performance} -\begin{table} [ht!] +\begin{table} [htbp] \centering \begin{tabular}{r c } \hline @@ -811,18 +811,19 @@ 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} -\RCE{C est la description du dernier test sync/async avec l'introduction de la notion de relative gain} -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. +In this section, the Simgrid simulator is used to compare the behavior of the +multisplitting in asynchronous mode with GMRES in synchronous mode. Several +benchmarks have been performed with various combination of the grid resources +(CPU, Network, input matrix size, \ldots ). The test conditions are summarized +in Table~\ref{tab:07}. In order to compare the execution times, this table +reports the relative gain between both algorithms. It is defined by the ratio +between the execution time of GMRES and the execution time of the +multisplitting. The ration is greater than one because the asynchronous +multisplitting version is faster than GMRES. -The test conditions are summarized in the table~\ref{tab:07}: \\ -\begin{table} [ht!] +\begin{table} [htbp] \centering \begin{tabular}{r c } \hline @@ -872,7 +873,7 @@ geographically distant clusters through the internet. power (GFlops) & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\ \hline - size (N) + size ($N^3$) & 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline Precision