\usepackage{algorithm}
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+\usepackage{amsmath}
+\usepackage{amssymb}
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+Iterative methods are become more attractive than direct ones to solve large sparse linear systems. They are more effective in a parallel context and require less memory and arithmetic operations than direct methods. A number of iterative methods are proposed and adapted by many researchers and the increased need for solving very large sparse linear systems triggered the development of efficient iterative techniques suitable for the parallel processing.
+
+The most successful iterative methods currently available are those based on the Krylov subspace which consists in forming a basis of a sequence of successive matrix powers times the initial residual. These methods are based on orthogonality of vectors of the Krylov subspace basis to solve generalized linear systems. The most well-known iterative Krylov subspace methods are Conjugate Gradient method and GMRES method (generalized minimal residual).
+
+%les chercheurs ont développer différentes méthodes exemple de méthode iteratives stationnaires et non stationnaires (krylov)
+%problème de convergence et difficulté dans le passage à l'échelle
+
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\section{A Krylov two-stage algorithm}
-
-
-\begin{algorithm}[!t]
+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}.
+
+In order to accelerate the convergence, 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 subspace 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 subspace 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} methods which is more appropriate than a direct method in the parallel context.
+
+\begin{algorithm}[t]
\caption{A Krylov two-stage algorithm}
\begin{algorithmic}[1]
-\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
-\Output $X_\ell$ (solution sub-vector)\vspace{0.2cm}
-\State Load $A_\ell$, $B_\ell$
-\State Set the initial guess $x^0$
-\State Set the minimizer $\tilde{x}^0=x^0$
-\For {$k=1,2,3,\ldots$ until the global convergence}
-\State Restart with $x^0=\tilde{x}^{k-1}$:
-\For {$j=1,2,\ldots,s$}
-\State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
-\State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$
-\State Exchange local values of $X_\ell^j$ with the neighboring clusters
-\State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$
-\EndFor
-\State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
-\State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$
-\State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring clusters
-\EndFor
-
-\Statex
-
-\Function {InnerSolver}{$x^0$, $j$}
-\State Compute local right-hand side $Y_\ell = B_\ell - \sum^L_{\substack{m=1\\m\neq \ell}}A_{\ell m}X_m^0$
-\State Solving local splitting $A_{\ell \ell}X_\ell^j=Y_\ell$ using parallel GMRES method, such that $X_\ell^0$ is the initial guess
-\State \Return $X_\ell^j$
-\EndFunction
-
-\Statex
-
-\Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
-\State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using parallel CGNR method
-\State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
-\State \Return $\tilde{X}_\ell^k$
-\EndFunction
+ \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 subspace 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 subspace basis $S$
+ \EndIf
+ \EndFor
\end{algorithmic}
\label{algo:01}
\end{algorithm}
% (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, \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}
