X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/7d86b81c520197567b7aecfcfbbdaadd251ecfe9..1ecf754b036f2415b4209f6512c1156d9e0d0de9:/paper.tex

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 \section{Related works}
 \label{sec:02} 
-Krylov subspace iteration methods have increasingly become very useful and popular for solving linear equations. 
+Krylov subspace iteration methods have increasingly become useful and successful techniques for solving linear and nonlinear systems and eigenvalue problems, especially since the increase development of the preconditioners~\cite{}. One reason of the popularity of these methods is their generality, simplicity and efficiency to solve systems of equations arising from very large and complex problems. A Krylov method is based on a projection process onto a Krylov subspace spanned by vectors and it forms a sequence of approximations by minimizing the residual over the subspace formed~\cite{}.
 
+GMRES is one of the most widely used Krylov iterative method for solving sparse and large linear systems. It is developed by Saad and al.~\cite{} as a generalized method to deal with unsymmetric and non-Hermitian problems, and indefinite symmetric problems too. In its original version called full GMRES, it minimizes the residual
+over the current Krylov subspace until convergence in at most $n$ iterations, where $n$ is the size of the sparse matrix. It should be noted that full GMRES is too expensive in the case of large matrices since the required orthogonalization process per iteration grows quadratically with the number of iterations. For that reason, in practice GMRES is restarted after each $m\ll n$ iterations to avoid the storage of a large orthonormal basis. However, the convergence behavior of the restarted GMRES in many cases depends quite critically on the value of $m$~\cite{}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.
 
-%GMRES method is one of the most widely used iterative solvers chosen to deal with the sparsity and the large order of linear systems. It was initially developed by Saad \& al.~\cite{Saad86} to deal with non-symmetric and non-Hermitian problems, and indefinite symmetric problems too. The convergence of the restarted GMRES with preconditioning is faster and more stable than those of some other iterative solvers. 
+%FGMRES , GMRESR, two-stage, communication avoiding 
 
-%The next two chapters explore a few methods which are considered currently to be among the most important iterative techniques available for solving large linear systems. These techniques are based on projection processes, both orthogonal and oblique, onto Krylov subspaces, which are subspaces spanned by vectors of the form p(A)v where p is a polynomial. In short, these techniques approximate A −1 b by p(A)b, where p is a “good” polynomial. This chapter covers methods derived from, or related to, the Arnoldi orthogonalization. The next chapter covers methods based on Lanczos biorthogonalization.
-
-%Krylov subspace techniques have inceasingly been viewed as general purpose iterative methods, especially since the popularization of the preconditioning techniqes.
-
-%Preconditioned Krylov-subspace iterations are a key ingredient in many modern linear solvers, including in solvers that employ support preconditioners. 
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