On the equivalence between SOR-type methods for linear systems and discrete gradient methods for gradient systems
For researchers in numerical linear algebra and dynamical systems, this work offers a novel theoretical connection that may lead to improved understanding and development of iterative methods, though the practical impact is not demonstrated.
The paper establishes a new equivalence between SOR-type methods for linear systems and discrete gradient methods for gradient systems, providing new interpretations and derivations for SOR-type methods.
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 methods. The focus of the discussion is the equivalence between SOR-type methods and discrete gradient methods applied to gradient systems. The discussion leads to new interpretations for SOR-type methods. For example, we found a new way to derive these methods; these methods monotonically decrease a certain quadratic function and obtain a new interpretation of the relaxation parameter. We also obtained a new discrete gradient while studying the new connection.