X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/99819ede8e00abfefaff7e709c42952c94e96f4c..176107a654cef5a1e3e376218e5da656e515453b:/paper.tex diff --git a/paper.tex b/paper.tex index deff6f3..18340f6 100644 --- a/paper.tex +++ b/paper.tex @@ -354,6 +354,7 @@ \usepackage{amsmath} \usepackage{amssymb} \usepackage{multirow} +\usepackage{graphicx} \algnewcommand\algorithmicinput{\textbf{Input:}} \algnewcommand\Input{\item[\algorithmicinput]} @@ -583,8 +584,7 @@ performances. The present paper is organized as follows. In Section~\ref{sec:02} some related works are presented. Section~\ref{sec:03} presents our two-stage algorithm using a least-square residual minimization. Section~\ref{sec:04} describes some -convergence results on this method. In Section~\ref{sec:05}, parallization -details of TSARM are given. Section~\ref{sec:06} shows some experimental +convergence results on this method. Section~\ref{sec:05} shows some experimental results obtained on large clusters of our algorithm using routines of PETSc toolkit. Finally Section~\ref{sec:06} concludes and gives some perspectives. %%%********************************************************* @@ -615,7 +615,7 @@ points of our solver are given in Algorithm~\ref{algo:01}. In order to accelerate the convergence, the outer iteration periodically applies a least-square minimization on the residuals computed by the inner solver. The -inner solver is a Krylov based solver which does not required to be changed. +inner solver is based on a Krylov method which does not require to be changed. At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$ iterations, using an iterative method restarting with the previous solution. For @@ -644,11 +644,11 @@ appropriate than a direct method in a parallel context. \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$<\epsilon_{kryl}$)} \label{algo:conv} + \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsarm}$)} \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} - \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{kryl}$} + \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{tsarm}$} \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul} \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:} \State $x^k=S\alpha$ \Comment{compute new solution} @@ -664,7 +664,7 @@ called for a maximum of $max\_iter_{kryl}$ iterations. In practice, we sugges 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 (i.e. -$\epsilon$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the +$\epsilon_{tsarm}$). 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 @@ -673,25 +673,13 @@ method. To summarize, the important parameters of TSARM are: \begin{itemize} -\item $\epsilon_{kryl}$ the threshold to stop the method of the krylov method +\item $\epsilon_{tsarm}$ the threshold to stop the TSARM 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 \item $\epsilon_{ls}$ the threshold to stop the least-square method \end{itemize} -%%%********************************************************* -%%%********************************************************* - -\section{Convergence results} -\label{sec:04} - - - -%%%********************************************************* -%%%********************************************************* -\section{Parallelization} -\label{sec:05} The parallelisation of TSARM relies on the parallelization of all its parts. More precisely, except the least-square step, all the other parts are @@ -733,10 +721,21 @@ In each iteration of CGLS, there is two matrix-vector multiplications and some classical operations: dots, norm, multiplication and addition on vectors. All these operations are easy to implement in PETSc or similar environment. + + +%%%********************************************************* +%%%********************************************************* + +\section{Convergence results} +\label{sec:04} + + + + %%%********************************************************* %%%********************************************************* \section{Experiments using petsc} -\label{sec:06} +\label{sec:05} In order to see the influence of our algorithm with only one processor, we first @@ -766,12 +765,10 @@ torso3 & 2D/3D problem & 259,156 & 4,429,042 \\ The following parameters have been chosen for our experiments. As by default the restart of GMRES is performed every 30 iterations, we have chosen to stop -the GMRES every 30 iterations (line \ref{algo:solve} in -Algorithm~\ref{algo:01}). $s$ is set to 8. CGLS is chosen to minimize the -least-squares problem. Two conditions are used to stop CGLS, either the -precision is under $1e-40$ or the number of iterations is greater to $20$. The -external precision is set to $1e-10$ (line \ref{algo:conv} in -Algorithm~\ref{algo:01}). Those experiments have been performed on a Intel(R) +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) Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc. @@ -814,9 +811,27 @@ torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\ -Larger experiments ....\\ -Describe the problems ex15 and ex54 +In the following we describe the applications of PETSc we have +experimented. Those applications are available in the ksp part which is suited +for scalable linear equations solvers: +\begin{itemize} +\item ex15 is an example which solves in parallel an operator using a finite + difference scheme. The diagonal is equals to 4 and 4 extra-diagonals + representing the neighbors in each directions is equal to -1. This example is + used in many physical phenomena , for exemple, heat and fluid flow, wave + propagation... +\item ex54 is another example based on 2D problem discretized with quadrilateral + finite elements. For this example, the user can define the scaling of material + coefficient in embedded circle, it is called $\alpha$. +\end{itemize} +For more technical details on these applications, interested reader are invited +to read the codes available in the PETSc sources. Those problem have been +chosen because they are scalable with many cores. We have tested other problem +but they are not scalable with many cores. + + + \begin{table*} \begin{center} @@ -843,6 +858,17 @@ Describe the problems ex15 and ex54 \end{table*} +\begin{figure} +\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}} +\label{fig:01} +\end{figure} + + + + + \begin{table*} \begin{center} \begin{tabular}{|r|r|r|r|r|r|r|r|r|} @@ -900,7 +926,7 @@ Describe the problems ex15 and ex54 %%%********************************************************* %%%********************************************************* \section{Conclusion} -\label{sec:07} +\label{sec:06} %The conclusion goes here. this is more of the conclusion %%%********************************************************* %%%*********************************************************