X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/9300a8195f1fd0aba96819dbd29244ca8bc90a33..eacf4c2eeca7315f0a4a7bc9dec99dc6778843d1:/paper.tex diff --git a/paper.tex b/paper.tex index 812326f..4d59b93 100644 --- a/paper.tex +++ b/paper.tex @@ -748,8 +748,9 @@ these operations are easy to implement in PETSc or similar environment. We can now claim that, \begin{proposition} \label{prop:saad} -If $A$ is either a definite positive or a positive matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, -let $r_k$ be the +If $A$ is either a definite positive or a positive 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 have the following boundaries: \begin{itemize} @@ -770,26 +771,31 @@ where $M$ is the symmetric part of $A$, $\alpha = \lambda_{min}(M)^2$ and $\beta \begin{proof} Let us first recall that the residue is under control when considering the GMRES algorithm on a positive definite matrix, and it is bounded as follows: \begin{equation*} -\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, . +\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{k/2} \|r_0\| . \end{equation*} Additionally, when $A$ 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 $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}, which proves -the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$. -Such well-known results can be found, \emph{e.g.}, in~\cite{Saad86}. +the convergence of GMRES($m$) for all $m$ under such assumptions regarding $A$. +These well-known results can be found, \emph{e.g.}, in~\cite{Saad86}. We will now prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||$ when $A$ is positive, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ when $A$ is positive definite. -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}. +The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ that follows the inductive hypothesis due, to the results recalled above. -Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$. +Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ in the definite positive one. 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. +\item If $k \not\equiv 0 ~(\textrm{mod}\ m)$, then the TSIRM algorithm consists in executing GMRES once. In that case and by using the inductive hypothesis, we obtain either $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ if $A$ is positive, or $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite case. +\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies: +\begin{itemize} +\item $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case, +\item $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite one, +\end{itemize} +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,\\ $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$ @@ -800,14 +806,16 @@ $\begin{array}{ll} & \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||, +& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||, \textrm{ if $A$ is positive,}\\ +& \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|, \textrm{ if $A$ is}\\ +& \textrm{positive definite,} \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 -than the one of the GMRES method. +%We can remark that, at each iterate, the residue of the TSIRM algorithm is lower +%than the one of the GMRES method. %%%********************************************************* %%%********************************************************* @@ -815,13 +823,13 @@ than the one of the GMRES method. \label{sec:05} -In order to see the influence of our algorithm with only one processor, we first -show a comparison with GMRES or FGMRES and our algorithm. In Table~\ref{tab:01}, -we show the matrices we have used and some of them characteristics. Those -matrices are chosen from the Davis collection of the University of -Florida~\cite{Dav97}. They are matrices arising in real-world applications. For -all the matrices, the name, the field, the number of rows and the number of -nonzero elements are given. +In order to see the behavior of the proposal when considering only one processor, a first +comparison with GMRES or FGMRES and the new algorithm detailed previously has been experimented. +Matrices that have been used with their characteristics (names, fields, rows, and nonzero coefficients) are detailed in +Table~\ref{tab:01}. These latter, which are real-world applications matrices, +have been extracted + from the Davis collection, University of +Florida~\cite{Dav97}. \begin{table}[htbp] \begin{center} @@ -841,8 +849,9 @@ torso3 & 2D/3D problem & 259,156 & 4,429,042 \\ \label{tab:01} \end{center} \end{table} - -The following parameters have been chosen for our experiments. As by default +Chosen parameters are detailed below. +%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 (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is chosen to minimize the least-squares problem with the following parameters: @@ -922,8 +931,16 @@ by core. The Juqueen architecture is composed of IBM PowerPC A2 at 1.6 GHz with speed network. +In many situations, using preconditioners is essential in order to find the +solution of a linear system. There are many preconditioners available in PETSc. +For parallel applications all the preconditioners based on matrix factorization +are not available. In our experiments, we have tested different kinds of +preconditioners, however as it is not the subject of this paper, we will not +present results with many preconditioners. In practise, we have chosen to use a +multigrid (mg) and successive over-relaxation (sor). For more details on the +preconditioner in PETSc please consult~\cite{petsc-web-page}. + -{\bf Description of preconditioners}\\ \begin{table*}[htbp] \begin{center} @@ -951,8 +968,7 @@ speed network. Table~\ref{tab:03} shows the execution times and the number of iterations of 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 +are studied ranging from 2,048 up-to 16,383 with the two preconditioners {\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