X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/6fda575b88fae086f925cd4dcf960fc2af0d3b4f..093b40c0c2a74ebfa6e3366e75a49bac1d4f59f2:/paper.tex diff --git a/paper.tex b/paper.tex index 04ad634..012e7f1 100644 --- a/paper.tex +++ b/paper.tex @@ -369,7 +369,7 @@ % % paper title % can use linebreaks \\ within to get better formatting as desired -\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems} +\title{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm to solve large sparse linear systems} @@ -380,7 +380,7 @@ % use a multiple column layout for up to two different % affiliations -\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}} +\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}} \IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\ Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr} \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\ @@ -425,16 +425,17 @@ Email: lilia.ziane@inria.fr} \begin{abstract} -In this article, a two-stage iterative algorithm is proposed to improve the -convergence of Krylov based iterative methods, typically those of GMRES variants. The -principle of the proposed approach is to build an external iteration over the Krylov -method, and to frequently store its current residual (at each -GMRES restart for instance). After a given number of outer iterations, a minimization -step is applied on the matrix composed by the saved residuals, in order to -compute a better solution and to make new iterations if required. It is proven that -the proposal has the same convergence properties than the inner embedded method itself. -Experiments using up to 16,394 cores also show that the proposed algorithm -runs around 5 or 7 times faster than GMRES. +In this article, a two-stage iterative algorithm is proposed to improve the +convergence of Krylov based iterative methods, typically those of GMRES +variants. The principle of the proposed approach is to build an external +iteration over the Krylov method, and to frequently store its current residual +(at each GMRES restart for instance). After a given number of outer iterations, +a least-squares minimization step is applied on the matrix composed by the saved +residuals, in order to compute a better solution and to make new iterations if +required. It is proven that the proposal has the same convergence properties +than the inner embedded method itself. Experiments using up to 16,394 cores +also show that the proposed algorithm runs around 5 or 7 times faster than +GMRES. \end{abstract} \begin{IEEEkeywords} @@ -581,7 +582,7 @@ performances. The present article is organized as follows. Related works are presented in Section~\ref{sec:02}. Section~\ref{sec:03} details the two-stage algorithm using -a least-square residual minimization, while Section~\ref{sec:04} provides +a least-squares residual minimization, while Section~\ref{sec:04} provides convergence results regarding this method. Section~\ref{sec:05} shows some experimental results obtained on large clusters using routines of PETSc toolkit. This research work ends by a conclusion section, in which the proposal @@ -604,7 +605,7 @@ is summarized while intended perspectives are provided. %%%********************************************************* %%%********************************************************* -\section{Two-stage iteration with least-square residuals minimization algorithm} +\section{Two-stage iteration with least-squares residuals minimization algorithm} \label{sec:03} A two-stage algorithm is proposed to solve large sparse linear systems of the form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square @@ -615,7 +616,7 @@ inner-outer iteration solver based on iterative Krylov methods. The main key-points of the proposed solver are given in Algorithm~\ref{algo:01}. It can be summarized as follows: the inner solver is a Krylov based one. In order to accelerate its convergence, the -outer solver periodically applies a least-square minimization on the residuals computed by the inner one. %Tsolver which does not required to be changed. +outer solver periodically applies a least-squares minimization on the residuals computed by the inner one. %Tsolver which does not required to be changed. At each outer iteration, the sparse linear system $Ax=b$ is partially solved using only $m$ @@ -636,7 +637,7 @@ with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$. In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$, -with $s\ll n$. In order to minimize~\eqref{eq:01}, a least-square method such as +with $s\ll n$. In order to minimize~\eqref{eq:01}, a least-squares method such as CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Remark that these methods are more appropriate than a single direct method in a parallel context. @@ -647,15 +648,15 @@ appropriate than a single direct method in a parallel context. \begin{algorithmic}[1] \Input $A$ (sparse matrix), $b$ (right-hand side) \Output $x$ (solution vector)\vspace{0.2cm} - \State Set the initial guess $x^0$ + \State Set the initial guess $x_0$ \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 $x_k=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve} \State retrieve error - \State $S_{k \mod s}=x^k$ \label{algo:store} + \State $S_{k \mod s}=x_k$ \label{algo:store} \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$} \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul} - \State $\alpha=Solve\_Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:} - \State $x^k=S\alpha$ \Comment{compute new solution} + \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:} + \State $x_k=S\alpha$ \Comment{compute new solution} \EndIf \EndFor \end{algorithmic} @@ -668,8 +669,8 @@ 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 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 +$\epsilon_{tsirm}$). 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 $S$, where $S$ is a matrix of size $n\times s$ whose column vector $i$ is denoted by $S_i$. 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 iterations and the threshold to stop the @@ -680,19 +681,19 @@ Let us summarize the most important parameters of TSIRM: \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; -\item $\epsilon_{ls}$: the threshold used to stop the least-square method. +\item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method; +\item $\epsilon_{ls}$: the threshold used to stop the least-squares method. \end{itemize} -The parallelisation of TSIRM relies on the parallelization of all its -parts. More precisely, except the least-square step, all the other parts are +The parallelization of TSIRM relies on the parallelization of all its +parts. More precisely, except the least-squares 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 line~\ref{algo:matrix_mul} the matrix-matrix multiplication is implemented and efficient since the matrix $A$ is sparse and since the matrix $S$ contains few -colums in practice. As explained previously, at least two methods seem to be -interesting to solve the least-square minimization, CGLS and LSQR. +columns in practice. As explained previously, at least two methods seem to be +interesting to solve the least-squares minimization, CGLS and LSQR. In the following we remind the CGLS algorithm. The LSQR method follows more or less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR. @@ -702,19 +703,21 @@ less the same principle but it takes more place, so we briefly explain the paral \begin{algorithmic}[1] \Input $A$ (matrix), $b$ (right-hand side) \Output $x$ (solution vector)\vspace{0.2cm} - \State $r=b-Ax$ - \State $p=A'r$ - \State $s=p$ - \State $g=||s||^2_2$ - \For {$k=1,2,3,\ldots$ until convergence (g$<\epsilon_{ls}$)} \label{algo2:conv} - \State $q=Ap$ - \State $\alpha=g/||q||^2_2$ - \State $x=x+alpha*p$ - \State $r=r-alpha*q$ - \State $s=A'*r$ - \State $g_{old}=g$ - \State $g=||s||^2_2$ - \State $\beta=g/g_{old}$ + \State Let $x_0$ be an initial approximation + \State $r_0=b-Ax_0$ + \State $p_1=A^Tr_0$ + \State $s_0=p_1$ + \State $\gamma=||s_0||^2_2$ + \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv} + \State $q_k=Ap_k$ + \State $\alpha_k=\gamma/||q_k||^2_2$ + \State $x_k=x_{k-1}+\alpha_kp_k$ + \State $r_k=r_{k-1}-\alpha_kq_k$ + \State $s_k=A^Tr_k$ + \State $\gamma_{old}=\gamma$ + \State $\gamma=||s_k||^2_2$ + \State $\beta_k=\gamma/\gamma_{old}$ + \State $p_{k+1}=s_k+\beta_kp_k$ \EndFor \end{algorithmic} \label{algo:02} @@ -734,15 +737,54 @@ these operations are easy to implement in PETSc or similar environment. \label{sec:04} Let us recall the following result, see~\cite{Saad86}. \begin{proposition} +\label{prop:saad} Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies: \begin{equation} ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| , \end{equation} -where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves +where $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$, which proves the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$. \end{proposition} +We can now claim that, +\begin{proposition} +If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, +let $r_k$ be the +$k$-th residue of TSIRM, then +we still have: +\begin{equation} +||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| , +\end{equation} +where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}. +\end{proposition} + +\begin{proof} +We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, +$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$ + +The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}. + +Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$. +We will show that the statement holds too for $r_k$. Two situations can occur: +\begin{itemize} +\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ by the inductive hypothesis. +\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$, and a least squares resolution. +Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\ +$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$ + +$\begin{array}{ll} +& = \min_{x \in span\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\ +& = \min_{x \in span\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\ +& \leqslant \min_{x \in span\left( x_{k-1} \right)} ||b-Ax ||_2\\ +& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\ +& \leqslant ||b-Ax_{k-1}||_2 . +\end{array}$ +\end{itemize} +\end{proof} + +We can remark that, at each iterate, the residue of the TSIRM algorithm is lower +than the one of the GMRES method. %%%********************************************************* %%%********************************************************* @@ -814,7 +856,7 @@ torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\ \hline \end{tabular} -\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.} +\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.} \label{tab:02} \end{center} \end{table} @@ -829,22 +871,21 @@ 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 equal to 4 and 4 extra-diagonals - representing the neighbors in each directions is equal to -1. This example is + representing the neighbors in each directions are equal to -1. This example is used in many physical phenomena, for example, heat and fluid flow, wave - propagation... + propagation, etc. \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 $\alpha$. + coefficient in embedded circle 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. +For more technical details on these applications, interested readers are invited +to read the codes available in the PETSc sources. Those problems have been +chosen because they are scalable with many cores which is not the case of other problems that we have tested. In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\ -{\bf Description of preconditioners} +{\bf Description of preconditioners}\\ \begin{table*}[htbp] \begin{center} @@ -865,27 +906,27 @@ In the following larger experiments are described on two large scale architectur \hline \end{tabular} -\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.} +\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (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*} Table~\ref{tab:03} shows the execution times and the number of iterations of -example ex15 of PETSc on the Juqueen architecture. Differents number of cores -are studied rangin from 2,048 upto 16,383. Two preconditioners have been -tested. For those experiments, the number of components (or unknown of the -problems) per processor is fixed to 25,000, also called weak scaling. This +example ex15 of PETSc on the Juqueen architecture. Different numbers of cores +are studied ranging from 2,048 up-to 16,383. Two preconditioners have been +tested: {\it mg} and {\it sor}. For those experiments, the number of components (or unknowns of the +problems) per core is fixed to 25,000, also called weak scaling. 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 TSIRM is always faster than FGMRES. The last +In Table~\ref{tab:03}, 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 +case between CGLS and LSQR, it is clear that TSIRM is always faster than +FGMRES. For this example, the multigrid preconditioner is faster than SOR. The gain between TSIRM and FGMRES is more or less similar for the two preconditioners. Looking at the number of iterations to reach the convergence, it is obvious that TSIRM allows the reduction of the number of iterations. It @@ -897,16 +938,16 @@ corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15. \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}} +\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03} (weak scaling)} \label{fig:01} \end{figure} In Figure~\ref{fig:01}, the number of iterations per second corresponding to -Table~\ref{tab:01} is displayed. It can be noticed that the number of -iterations per second of FMGRES is constant whereas it decrease with TSIRM with -both preconditioner. This can be explained by the fact that when the number of -core increases the time for the minimization step also increases but, generally, +Table~\ref{tab:03} is displayed. It can be noticed that the number of +iterations per second of FMGRES is constant whereas it decreases with TSIRM with +both preconditioners. This can be explained by the fact that when the number of +cores increases the time for the least-squares minimization step also increases but, generally, when the number of cores increases, the number of iterations to reach the threshold also increases, and, in that case, TSIRM is more efficient to reduce the number of iterations. So, the overall benefit of using TSIRM is interesting. @@ -921,7 +962,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting. \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ + nb. cores & threshold & \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 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\ @@ -934,7 +975,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting. \hline \end{tabular} -\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12), time is expressed in seconds.} +\caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie (restart=30, s=12), time is expressed in seconds.} \label{tab:04} \end{center} \end{table*} @@ -948,9 +989,9 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ + nb. cores & \multicolumn{2}{c|}{FGMRES} & \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 + & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & FGMRES & 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 \\ 1024 & 1,530.06 & 25,860 & 290.95 & 4,830 & 307.71 & 5,070 & 5.25 & 1.30 & 1.21 & 1.01 \\ 2048 & 919.62 & 31,470 & 237.52 & 8,040 & 194.22 & 6,510 & 4.73 & 1.08 & .75 & .80\\ @@ -960,7 +1001,7 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect \hline \end{tabular} -\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5), time is expressed in seconds.} +\caption{Comparison of FGMRES and TSIRM with FGMRES for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshold 5e-5), time is expressed in seconds.} \label{tab:05} \end{center} \end{table*} @@ -968,7 +1009,7 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect \begin{figure}[htbp] \centering \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex54_curie} -\caption{Number of iterations per second with ex54 and the same parameters than in Table~\ref{tab:05}} +\caption{Number of iterations per second with ex54 and the same parameters than in Table~\ref{tab:05} (strong scaling)} \label{fig:02} \end{figure} @@ -988,7 +1029,7 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect future plan : \\ - study other kinds of matrices, problems, inner solvers\\ -- test the influence of all the parameters\\ +- test the influence of all parameters\\ - adaptative number of outer iterations to minimize\\ - other methods to minimize the residuals?\\ - implement our solver inside PETSc @@ -1003,7 +1044,7 @@ future plan : \\ %%%********************************************************* \section*{Acknowledgment} This paper is partially funded by the Labex ACTION program (contract -ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resource +ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resources Curie and Juqueen respectively based in France and Germany. @@ -1046,5 +1087,3 @@ Curie and Juqueen respectively based in France and Germany. % that's all folks \end{document} - -