X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/317020501fc8da3f9e443457700821b26f66da55..c315caa973c79514a908f77115d9f95687851199:/paper.tex diff --git a/paper.tex b/paper.tex index 436909a..6a83f52 100644 --- a/paper.tex +++ b/paper.tex @@ -381,7 +381,7 @@ % affiliations \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\\ +\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche-Comt\'e, France\\ Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr} \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\ Email: lilia.ziane@inria.fr} @@ -564,7 +564,7 @@ gradient and GMRES ones (Generalized Minimal RESidual). However, iterative methods suffer from scalability problems on parallel computing platforms with many processors, due to their need of reduction -operations, and to collective communications to achive matrix-vector +operations, and to collective communications to achieve matrix-vector multiplications. The communications on large clusters with thousands of cores and large sizes of messages can significantly affect the performances of these iterative methods. As a consequence, Krylov subspace iteration methods are often used @@ -742,37 +742,48 @@ Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the res \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, we still have +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_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| , +||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} -Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the -$k$-th iterate of TSIRM. -We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$. +We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, +$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$ -Each step of the TSIRM algorithm \\ +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}. -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 vectors $S$. So,\\ +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,\\ $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$ $\begin{array}{ll} -& = \min_{x \in span\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\ -& = \min_{x \in span\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\ -& \leqslant \min_{x \in span\left( x_{k-1} \right)} ||b-Ax ||_2\\ -& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\ -& \leqslant ||b-Ax_{k-1}||_2 . +& = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\ +& = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\ +& \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_2\\ +& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\ +& \leqslant ||b-Ax_{k}||_2\\ +& = ||r_k||_2\\ +& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||, \end{array}$ +\end{itemize} +which concludes the induction and the proof. \end{proof} We can remark that, at each iterate, the residue of the TSIRM algorithm is lower @@ -1018,13 +1029,22 @@ 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 parameters\\ -- adaptative number of outer iterations to minimize\\ -- other methods to minimize the residuals?\\ -- implement our solver inside PETSc +A novel two-stage iterative algorithm has been proposed in this article, +in order to accelerate the convergence Krylov iterative methods. +Our TSIRM proposal acts as a merger between Krylov based solvers and +a least-squares minimization step. +The convergence of the method has been proven in some situations, while +experiments up to 16,394 cores have been led to verify that TSIRM runs +5 or 7 times faster than GMRES. + + +For future work, the authors' intention is to investigate +other kinds of matrices, problems, and inner solvers. The +influence of all parameters must be tested too, while +other methods to minimize the residuals must be regarded. +The number of outer iterations to minimize should become +adaptative to improve the overall performances of the proposal. +Finally, this solver will be implemented inside PETSc. % conference papers do not normally have an appendix