From: Christophe Guyeux Date: Fri, 10 Oct 2014 14:32:47 +0000 (+0200) Subject: Relecture X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/commitdiff_plain/7c0145001ac26bf858bc9cd4e4abb507a3e63ea2?ds=inline Relecture --- diff --git a/paper.tex b/paper.tex index a4545fd..896ac71 100644 --- a/paper.tex +++ b/paper.tex @@ -626,6 +626,7 @@ 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 @@ -667,11 +668,12 @@ appropriate than a single direct method in a parallel context. 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 @@ -736,7 +738,7 @@ 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: