From 2fc464daa5a04774f5fa10bd96d36c53cc908a4a Mon Sep 17 00:00:00 2001 From: lilia Date: Fri, 10 Oct 2014 10:00:17 +0200 Subject: [PATCH] 10-10-2014 01 --- paper.tex | 15 ++++++--------- 1 file changed, 6 insertions(+), 9 deletions(-) diff --git a/paper.tex b/paper.tex index 463fe2c..54e35d3 100644 --- a/paper.tex +++ b/paper.tex @@ -370,10 +370,7 @@ % 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} -%où -%\title{A two-stage algorithm with error minimization to solve large sparse linear systems} -%où -%\title{???} + @@ -607,7 +604,7 @@ is summarized while intended perspectives are provided. %%%********************************************************* %%%********************************************************* -\section{Two-stage algorithm with least-square residuals minimization} +\section{Two-stage iteration with least-square 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 @@ -639,7 +636,7 @@ 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-square 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. @@ -675,7 +672,7 @@ $\epsilon_{tsirm}$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in cop solution $x_k$ into the column $k~ mod~ s$ of the matrix $S$. 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 iteration and the threshold to stop the +required for that: the maximum number of iterations and the threshold to stop the method. Let us summarize the most important parameters of TSIRM: @@ -698,7 +695,7 @@ colums in practice. As explained previously, at least two methods seem to be interesting to solve the least-square minimization, CGLS and LSQR. In the following we remind the CGLS algorithm. The LSQR method follows more or -less the same principle but it take more place, so we briefly explain the parallelization of CGLS which is similar to LSQR. +less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR. \begin{algorithm}[t] \caption{CGLS} @@ -725,7 +722,7 @@ less the same principle but it take more place, so we briefly explain the parall In each iteration of CGLS, there is two matrix-vector multiplications and some -classical operations: dots, norm, multiplication and addition on vectors. All +classical operations: dot product, norm, multiplication and addition on vectors. All these operations are easy to implement in PETSc or similar environment. -- 2.39.5