A proportional-derivative control strategy for restarting the GMRES( m ) algorithm

A proportional-derivative control strategy for restarting the GMRES( m )... Restarted GMRES (or GMRES( m )) is normally used for solving large linear systems A x = b with a general, possibly nonsymmetric, matrix A . Although, the restarted GMRES consumes less computational time than its counterpart full GMRES, if the restarting parameter is not correctly chosen its convergence cannot be guaranteed and the method may converge slowly. Unfortunately, it is difficult to know how to choose this parameter a priori. In this article, we regard the GMRES( m ) method as a control problem, in which the parameter m is the controlled variable and propose a new control-inspired strategy for choosing the parameter m adaptively at each iteration. The advantage of this control strategy method is that only a few additional vectors need to be stored and the controller has the capacity to modify the dimension of the Krylov subspace whenever any convergence problem is detected. Numerical experiments, based on benchmark problems, show that the proposed control strategy accelerates the convergence of GMRES ( m ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Computational and Applied Mathematics Elsevier

A proportional-derivative control strategy for restarting the GMRES( m ) algorithm

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Publisher
Elsevier
Copyright
Copyright © 2018 Elsevier B.V.
ISSN
0377-0427
eISSN
1879-1778
D.O.I.
10.1016/j.cam.2018.01.009
Publisher site
See Article on Publisher Site

Abstract

Restarted GMRES (or GMRES( m )) is normally used for solving large linear systems A x = b with a general, possibly nonsymmetric, matrix A . Although, the restarted GMRES consumes less computational time than its counterpart full GMRES, if the restarting parameter is not correctly chosen its convergence cannot be guaranteed and the method may converge slowly. Unfortunately, it is difficult to know how to choose this parameter a priori. In this article, we regard the GMRES( m ) method as a control problem, in which the parameter m is the controlled variable and propose a new control-inspired strategy for choosing the parameter m adaptively at each iteration. The advantage of this control strategy method is that only a few additional vectors need to be stored and the controller has the capacity to modify the dimension of the Krylov subspace whenever any convergence problem is detected. Numerical experiments, based on benchmark problems, show that the proposed control strategy accelerates the convergence of GMRES ( m ) .

Journal

Journal of Computational and Applied MathematicsElsevier

Published: Aug 1, 2018

References

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