\usepackage{algorithm}
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+\usepackage{amsmath}
+\usepackage{amssymb}
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\section{A Krylov two-stage algorithm}
-We propose a two-stage algorithm to solve large sparse linear systems of the form $Ax=b$ based on iterative Krylov sub-space methods.
-
-
-\begin{algorithm}[!h]
+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)
\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 $\|b-R\alpha\|_2$
+ \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
% (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}
