From d6d5ef6890f9b888cf59c00e8eacc5f2863b1458 Mon Sep 17 00:00:00 2001 From: raphael couturier Date: Thu, 9 Oct 2014 21:09:08 +0200 Subject: [PATCH 1/1] new --- paper.tex | 56 +++++++++++++++++++++++++++++-------------------------- 1 file changed, 30 insertions(+), 26 deletions(-) diff --git a/paper.tex b/paper.tex index 9f8ded7..4f9f60e 100644 --- a/paper.tex +++ b/paper.tex @@ -241,7 +241,7 @@ % quality. -%\usepackage{eqparbox} +\usepackage{eqparbox} % Also of notable interest is Scott Pakin's eqparbox package for creating % (automatically sized) equal width boxes - aka "natural width parboxes". % Available at: @@ -369,7 +369,7 @@ % % paper title % can use linebreaks \\ within to get better formatting as desired -\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization to solve large sparse linear systems} +\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems} %où %\title{A two-stage algorithm with error minimization to solve large sparse linear systems} %où @@ -646,12 +646,12 @@ appropriate than a single direct method in a parallel context. \begin{algorithm}[t] -\caption{TSARM} +\caption{TSIRM} \begin{algorithmic}[1] \Input $A$ (sparse matrix), $b$ (right-hand side) \Output $x$ (solution vector)\vspace{0.2cm} \State Set the initial guess $x^0$ - \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsarm}$)} \label{algo:conv} + \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv} \State $x^k=Solve(A,b,x^{k-1},max\_iter_{kryl})$ \label{algo:solve} \State retrieve error \State $S_{k \mod s}=x^k$ \label{algo:store} @@ -670,17 +670,17 @@ iteration is inside the for loop. Line~\ref{algo:solve}, the Krylov method is called for a maximum of $max\_iter_{kryl}$ iterations. In practice, we suggest to set this parameter equals to the restart number of the GMRES-like method. Moreover, a tolerance threshold must be specified for the solver. In practice, this threshold must be -much smaller than the convergence threshold of the TSARM algorithm (\emph{i.e.} -$\epsilon_{tsarm}$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the +much smaller than the convergence threshold of the TSIRM algorithm (\emph{i.e.} +$\epsilon_{tsirm}$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the solution $x_k$ into the column $k~ mod~ s$ of the matrix $S$. After the minimization, the matrix $S$ is reused with the new values of the residuals. To solve the minimization problem, an iterative method is used. Two parameters are required for that: the maximum number of iteration and the threshold to stop the method. -Let us summarize the most important parameters of TSARM: +Let us summarize the most important parameters of TSIRM: \begin{itemize} -\item $\epsilon_{tsarm}$: the threshold to stop the TSARM method; +\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method; \item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method; \item $s$: the number of outer iterations before applying the minimization step; \item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-square method; @@ -688,7 +688,7 @@ Let us summarize the most important parameters of TSARM: \end{itemize} -The parallelisation of TSARM relies on the parallelization of all its +The parallelisation of TSIRM relies on the parallelization of all its parts. More precisely, except the least-square step, all the other parts are obvious to achieve out in parallel. In order to develop a parallel version of our code, we have chosen to use PETSc~\cite{petsc-web-page}. For @@ -759,7 +759,7 @@ Table~\ref{tab:01}, we show the matrices we have used and some of them characteristics. For all the matrices, the name, the field, the number of rows and the number of nonzero elements is given. -\begin{table*} +\begin{table*}[htbp] \begin{center} \begin{tabular}{|c|c|r|r|r|} \hline @@ -783,7 +783,7 @@ the restart of GMRES is performed every 30 iterations, we have chosen to stop the GMRES every 30 iterations, $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is chosen to minimize the least-squares problem with the following parameters: $\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to -$\epsilon_{tsarm}=1e-10$. Those experiments have been performed on a Intel(R) +$\epsilon_{tsirm}=1e-10$. Those experiments have been performed on a Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc. @@ -791,20 +791,20 @@ In Table~\ref{tab:02}, some experiments comparing the solving of the linear systems obtained with the previous matrices with a GMRES variant and with out 2 stage algorithm are given. In the second column, it can be noticed that either gmres or fgmres is used to solve the linear system. According to the matrices, -different preconditioner is used. With TSARM, the same solver and the same -preconditionner is used. This Table shows that TSARM can drastically reduce the +different preconditioner is used. With TSIRM, the same solver and the same +preconditionner is used. This Table shows that TSIRM can drastically reduce the number of iterations to reach the convergence when the number of iterations for the normal GMRES is more or less greater than 500. In fact this also depends on tow parameters: the number of iterations to stop GMRES and the number of iterations to perform the minimization. -\begin{table} +\begin{table}[htbp] \begin{center} \begin{tabular}{|c|c|r|r|r|r|} \hline - \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} \\ + \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} \\ \cline{3-6} & precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline @@ -849,12 +849,12 @@ In the following larger experiments are described on two large scale architectur {\bf Description of preconditioners} -\begin{table*} +\begin{table*}[htbp] \begin{center} \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & precond & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ + nb. cores & precond & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ \cline{3-8} & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline 2,048 & mg & 403.49 & 18,210 & 73.89 & 3,060 & 77.84 & 3,270 & 5.46 \\ @@ -868,7 +868,7 @@ In the following larger experiments are described on two large scale architectur \hline \end{tabular} -\caption{Comparison of FGMRES and TSARM with FGMRES for example ex15 of PETSc with two preconditioner (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.} +\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioner (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.} \label{tab:03} \end{center} \end{table*} @@ -881,12 +881,16 @@ problems) per processor is fixed to 25,000. This number can seem relatively small. In fact, for some applications that need a lot of memory, the number of components per processor requires sometimes to be small. -In this Table, we can notice that TSARM is always faster than FGMRES. The last -column shows the ratio between FGMRES and the best version of TSARM according to -the minimization procedure: CGLS or LSQR. +In this Table, we can notice that TSIRM is always faster than FGMRES. The last +column shows the ratio between FGMRES and the best version of TSIRM according to +the minimization procedure: CGLS or LSQR. Even if we have computed the worst +case between CGLS and LSQR, it is clear that TSIRM is alsways faster than +FGMRES. For this example, the multigrid preconditionner is faster than SOR. The +gain between TSIRM and FGMRES is more or less similar for the two +preconditioners -\begin{figure} +\begin{figure}[htbp] \centering \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen} \caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03}} @@ -897,12 +901,12 @@ the minimization procedure: CGLS or LSQR. -\begin{table*} +\begin{table*}[htbp] \begin{center} \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ + nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ \cline{3-8} & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline 2,048 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\ @@ -924,12 +928,12 @@ the minimization procedure: CGLS or LSQR. -\begin{table*} +\begin{table*}[htbp] \begin{center} \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ + nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ \cline{2-7} \cline{9-11} & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & GMRES & TS CGLS & TS LSQR\\\hline \hline 512 & 3,969.69 & 33,120 & 709.57 & 5,790 & 622.76 & 5,070 & 6.37 & 1 & 1 & 1 \\ -- 2.39.5