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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.
+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.
-%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
+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 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
+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 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
+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}
\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 sub-space basis $S$
+ \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 sub-space basis $S$
+ \State Reinitialize Krylov subspace basis $S$
\EndIf
\EndFor
\end{algorithmic}
