10-10-2014 01

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 463fe2c1bc0b383aadacfd6a31924d31e54cfd1e..54e35d38f3a810d26fbe5585e02b1c3a8d48f89b 100644 (file)
--- 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}
 % 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
 \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  $\mathbb{R}^{n\times s}$,
 
 
 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.
 
 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
 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:
 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
 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}
 
 \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
 
 
 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.
 
 
 these operations are easy to implement in PETSc or similar environment.