@article{Saad:1993,
- author = {Saad, Youcef},
+ author = {Saad, Y.},
title = {A Flexible Inner-outer Preconditioned GMRES Algorithm},
journal = {SIAM J. Sci. Comput.},
issue_date = {March 1993},
}
@article{Meijerink77,
- author = {Meijerink, J.A. and Vorst, H.A.van der},
+ author = {Meijerink, J.A. and Vorst, H.A. Van der},
title = {An Iterative Solution Method for Linear Systems of Which the Coefficient Matrix is a Symmetric {M}-Matrix},
journal = {Mathematics of Computation},
year = {1977},
}
@techreport{Huang89,
- author = {Huang, Y. and Vorst, H.A. van der},
+ author = {Huang, Y. and Van der Vorst, H.A.},
title = {Some Observations on the Convergence Behavior of {GMRES}},
institution = {Delft Univ. Technology},
type = { Report 89--09},
year = {1989},
}
+@article {Vorst94,
+author = {Van der Vorst, H. A. and Vuik, C.},
+title = {{GMRESR}: a family of nested {GMRES} methods},
+journal = {Numerical Linear Algebra with Applications},
+volume = {1},
+number = {4},
+publisher = {John Wiley & Sons, Ltd},
+issn = {1099-1506},
+url = {http://dx.doi.org/10.1002/nla.1680010404},
+doi = {10.1002/nla.1680010404},
+pages = {369--386},
+year = {1994},
+}
+
\label{sec:02}
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{Saad2003,Meijerink77}. 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{Saad86} 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{Huang89}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.
+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{Saad86} 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, called GMRES($m$), in many cases depends quite critically on the value of $m$~\cite{Huang89}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.
+
+Van der Vorst in~\cite{Vorst94} has proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes, where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process referred to as inner iterations.
%FGMRES , GMRESR, two-stage, communication avoiding
