Adaptive SOR methods based on the Wolfe conditions

TL;DR

This paper introduces new adaptive SOR methods for symmetric positive definite systems that utilize Wolfe conditions for step size control, avoiding extra matrix-vector products and improving efficiency.

Contribution

The paper presents adaptive SOR algorithms that interpret the relaxation parameter as a step size and adapt it using line search techniques like Wolfe conditions, without additional matrix-vector computations.

Findings

Favorable numerical performance demonstrated

Effective step size adaptation using Wolfe conditions

Applicable to a broad class of symmetric positive definite systems

Abstract

Because the expense of estimating the optimal value of the relaxation parameter in the successive over-relaxation (SOR) method is usually prohibitive, the parameter is often adaptively controlled. In this paper, new adaptive SOR methods are presented that are applicable to a variety of symmetric positive definite linear systems and do not require additional matrix-vector products when updating the parameter. To this end, we regard the SOR method as an algorithm for minimising a certain objective function, which yields an interpretation of the relaxation parameter as the step size following a certain change of variables. This interpretation enables us to adaptively control the step size based on some line search techniques, such as the Wolfe conditions. Numerical examples demonstrate the favourable behaviour of the proposed methods.