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\begin{document}
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\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}
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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 Krylov subspaces which consist in forming a basis of a sequence of successive matrix powers times an initial vector for example the residual. These methods are based on orthogonality of vectors of the Krylov subspace basis to solve linear systems. The most well-known iterative Krylov subspace methods are Conjugate Gradient method and GMRES method (generalized minimal residual).
+
+However, the iterative methods suffer from scalability problems on parallel computing platforms with many processors due to their need for reduction operations and collective communications to perform matrix-vector multiplications. The communications on large clusters with thousands of cores and large sizes of messages can significantly affect the performances of iterative methods. In practice, Krylov subspace iteration methods are often used with preconditioners in order to increase their convergence and accelerate their performances. However, most of the good preconditioners are not scalable on large clusters.
+
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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 $Ax=b$, 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 subspace~\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 subspace 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 $A$ (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}
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\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}
