X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/3aa4a4a5934c50b8ee7ac223ae1ebc31796c820c..860c8942588be4c114424b511f6a525063e42087:/paper.tex diff --git a/paper.tex b/paper.tex index 2372b61..b0ec851 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} @@ -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 $m$ @@ -626,8 +627,8 @@ inner solver. The current approximation of the Krylov method is then stored insi $S$ composed by the successive solutions that are computed during inner iterations. At each $s$ iterations, the minimization step is applied in order to -compute a new solution $x$. For that, the previous residuals are computed with -$(b-AS)$. The minimization of the residuals is obtained by +compute a new solution $x$. For that, the previous residuals of $Ax=b$ are computed by +the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by \begin{equation} \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2 \label{eq:01} @@ -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. @@ -654,7 +655,7 @@ appropriate than a single direct method in a parallel context. \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 Solve least-square problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:} + \State $\alpha=Solve\_Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:} \State $x^k=S\alpha$ \Comment{compute new solution} \EndIf \EndFor @@ -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 +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. +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} @@ -897,7 +900,7 @@ 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} @@ -965,6 +968,13 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect \end{center} \end{table*} +\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} (strong scaling)} +\label{fig:02} +\end{figure} + %%%********************************************************* %%%*********************************************************