X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/e73422d5f48b17486f47f19b98698ec34d487873..b3eead38d6fdb71b38c446e5c30b295b35dd599e:/paper.tex

diff --git a/paper.tex b/paper.tex
index 8232b7e..89487b6 100644
--- a/paper.tex
+++ b/paper.tex
@@ -348,6 +348,20 @@
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 
+
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{amsmath}
+\usepackage{amssymb}
+
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+
+
+
 \begin{document}
 %
 % paper title
@@ -417,7 +431,7 @@ Email: lilia.ziane@inria.fr}
 \end{abstract}
 
 \begin{IEEEkeywords}
-Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à voir... 
+Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à voir... 
 \end{IEEEkeywords}
 
 
@@ -541,6 +555,42 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à
 %%%*********************************************************
 %%%*********************************************************
 \section{A Krylov two-stage algorithm}
+We propose a two-stage algorithm to solve large sparse linear systems of the form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector and $b\in\mathbb{R}^n$ is the right-hand side. The algorithm is implemented as an inner-outer iteration solver based on iterative Krylov methods. The main key points of our solver are given in Algorithm~\ref{algo:01}. 
+
+The outer iteration is implemented as an iterative Krylov method which minimizes some error function over a Krylov sub-space~\cite{saad96}. At every iteration, the sparse linear system $Ax=b$ is solved iteratively with an iterative method as GMRES method~\cite{saad86} and the Krylov sub-space that we used is spanned by a basis $S$ composed of successive solutions issued from the inner iteration
+\begin{equation}
+  S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
+\end{equation} 
+The advantage of such a Krylov sub-space is that we neither need an orthogonal basis nor any synchronization between processors to generate this basis. The algorithm is periodically restarted every $s$ iterations with a new initial guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov sub-space spanned by vectors of $S$, where $\alpha$ is a solution of the normal equations
+\begin{equation}
+  R^TR\alpha = R^Tb,
+\end{equation}
+which is associated with the least-squares problem
+\begin{equation}
+   \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
+\label{eq:01}
+\end{equation}
+such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$, $s\ll n$, and $R^T$ denotes the transpose of matrix $R$. We use an iterative method to solve the least-squares problem~(\ref{eq:01}) as CGLS~\cite{hestenes52} or LSQR~\cite{paige82} which is more appropriate that a direct method in the parallel context.
+
+\begin{algorithm}[t]
+\caption{A Krylov two-stage algorithm}
+\begin{algorithmic}[1]
+  \Input $A$ (sparse matrix), $b$ (right-hand side)
+  \Output $x$ (solution vector)\vspace{0.2cm}
+  \State Set the initial guess $x^0$
+  \For {$k=1,2,3,\ldots$ until convergence}
+    \State Solve iteratively $Ax^k=b$
+    \State Add vector $x^k$ to Krylov basis $S$
+    \If {$k$ mod $s=0$ {\bf and} not convergence}
+      \State Compute dense matrix $R=AS$
+      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
+      \State Compute minimizer $x^k=S\alpha$
+      \State Reinitialize Krylov basis $S$
+    \EndIf
+  \EndFor
+\end{algorithmic}
+\label{algo:01}
+\end{algorithm}
 %%%*********************************************************
 %%%*********************************************************
 
@@ -601,10 +651,13 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à
 % (used to reserve space for the reference number labels box)
 \begin{thebibliography}{1}
 
-\bibitem{IEEEhowto:kopka}
-%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
-%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
+\bibitem{saad86} Y.~Saad and M.~H.~Schultz, \emph{GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems}, SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, 1986.
+
+\bibitem{saad96} Y.~Saad and M.~H.~Schultz, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.
+
+\bibitem{hestenes52} M.~R.~Hestenes and E.~Stiefel, \emph{Methods of conjugate gradients for solving linear system}, Journal of Research of National Bureau of Standards, B49:409--436, 1952.
 
+\bibitem{paige82} C.~C.~Paige and A.~M.~Saunders, \emph{LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, ACM Trans. Math. Softw. 8(1):43--71, 1982.
 \end{thebibliography}