2018
DOI: 10.1016/j.cam.2018.04.013
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On the equivalence between SOR-type methods for linear systems and the discrete gradient methods for gradient systems

Abstract: The iterative nature of many discretisation methods for continuous dynamical systems has led to the study of the connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems. Certain researchers have used the explicit Euler method to understand this connection, but this method has its limitation. In this study, we present a new connection between successive over-relaxation (SOR)-type methods and gradient systems; this connection is based on discrete gradient meth… Show more

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Cited by 19 publications
(27 citation statements)
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“…Our previous work [13] considers an adaptive SOR method that employs the locally optimal step size of the steepest descent method. We review this method before presenting new approaches.…”
Section: Approach Based On the Locally Optimal Step Size Of The Steepmentioning
confidence: 99%
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“…Our previous work [13] considers an adaptive SOR method that employs the locally optimal step size of the steepest descent method. We review this method before presenting new approaches.…”
Section: Approach Based On the Locally Optimal Step Size Of The Steepmentioning
confidence: 99%
“…By taking this background into account, the aim of this paper is to develop a new type of adaptive SOR method that is applicable to a variety of symmetric positive definite linear systems, without additional assumptions or a requirement for additional matrix-vector products when updating the relaxation parameter. To achieve this goal, we develop new adaptive SOR methods based on our previous work [13], which shows that for any symmetric positive definite linear system the SOR method can be regarded as an algorithm for solving a certain minimisation problem. The consequence of this discussion is that the relaxation parameter can be interpreted as the step size following the change of variables h = 2ω/(2 − ω).…”
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confidence: 99%
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“…The book by Stuart and Humphries [38] and the article by Chu [8] provide a broad overview of the relation between algorithms and dynamical systems. A lot of recent research focuses on the discrete steps in the solution to linear systems and the relation to continuous dynamics on manifolds of matrices [8,15,22,29]. In particular, algorithms for eigenvalue problems have been studied in this context, with the relation to integrable systems (Lax-pair formulation) [40].…”
mentioning
confidence: 99%