On the equivalence between SOR-type methods for linear systems and the discrete gradient methods for gradient systems

摘要

Inspired by the iterative nature of many discretization methods for continuous dynamical systems, connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems have been studied since 1970s. For stationary iterative methods solving linear systems, Chu (1988, 2008) discussed a connection to continuous dynamical systems by using the explicit Euler method, however, further understanding of stationary iterative methods might be limited due to the use of the explicit Euler method. This paper presents a new connection, based on the so-called discrete gradient methods, between SOR-type methods and gradient systems. There, the key of the discussion is the equivalence between SOR-type methods and the discrete gradient methods applied to gradient systems. The discussion leads to new interpretations for SOR-type methods. For example, a new derivation of SOR-type methods is found, these methods monotonically decrease a certain quadratic function, and a new interpretation of the relaxation parameter is obtained. Besides, while studying the new connection, a new discrete gradient is also obtained.