X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/317020501fc8da3f9e443457700821b26f66da55..afeb88ec33744268bc138cf5d9cc5406a9599763:/paper.tex diff --git a/paper.tex b/paper.tex index 436909a..d00cbe1 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 @@ -621,19 +621,21 @@ outer solver periodically applies a least-squares minimization on the residuals At each outer iteration, the sparse linear system $Ax=b$ is partially solved using only $m$ iterations of an iterative method, this latter being initialized with the -best known approximation previously obtained. -GMRES method~\cite{Saad86}, or any of its variants, can be used for instance as an -inner solver. The current approximation of the Krylov method is then stored inside a matrix -$S$ composed by the successive solutions that are computed during inner iterations. - -At each $s$ iterations, the minimization step is applied in order to +last obtained approximation. +GMRES method~\cite{Saad86}, or any of its variants, can potentially be used as +inner solver. The current approximation of the Krylov method is then stored inside a $n \times s$ matrix +$S$, which is composed by the $s$ last solutions that have been computed during +the inner iterations phase. +In the remainder, the $i$-th column vector of $S$ will be denoted by $S_i$. + +At each $s$ iterations, another kind of minimization step is applied in order to 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} \end{equation} -with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$. +with $R=AS$. The new solution $x$ is then computed with $x=S\alpha$. In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$, @@ -663,14 +665,15 @@ appropriate than a single direct method in a parallel context. \label{algo:01} \end{algorithm} -Algorithm~\ref{algo:01} summarizes the principle of our method. The outer -iteration is inside the for loop. Line~\ref{algo:solve}, the Krylov method is +Algorithm~\ref{algo:01} summarizes the principle of the proposed method. The outer +iteration is inside the \emph{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 +equal to the restart number in 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.} +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 $S$, where $S$ is a matrix of size $n\times s$ whose column vector $i$ is denoted by $S_i$. After the +solution $x_k$ into the column $k \mod s$ of $S$. +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 iterations and the threshold to stop the @@ -735,44 +738,55 @@ these operations are easy to implement in PETSc or similar environment. \section{Convergence results} \label{sec:04} -Let us recall the following result, see~\cite{Saad86}. +Let us recall the following result, see~\cite{Saad86} for further readings. \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, 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 @@ -874,7 +888,17 @@ For more technical details on these applications, interested readers are invite 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...}\\ +In the following larger experiments are described on two large scale +architectures: Curie and Juqeen. Both these architectures are supercomputer +composed of 80,640 cores for Curie and 458,752 cores for Juqueen. Those machines +are respectively hosted by GENCI in France and Jülich Supercomputing Centre in +Germany. They belongs with other similar architectures of the PRACE initiative ( +Partnership for Advanced Computing in Europe) which aims at proposing high +performance supercomputing architecture to enhance research in Europe. The Curie +architecture is composed of Intel E5-2680 processors at 2.7 GHz with 2Gb memory +by core. The Juqueen architecture is composed of IBM PowerPC A2 at 1.6 GHz with +1Gb memory per core. + {\bf Description of preconditioners}\\ @@ -1018,13 +1042,22 @@ 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