Accelerated Additive Schwarz Methods for Convex Optimization with Adaptive Restart

TL;DR

This paper introduces an acceleration scheme for additive Schwarz methods in convex optimization, leveraging gradient method techniques like momentum and adaptive restarting to improve convergence without prior problem smoothness information.

Contribution

It presents a novel acceleration approach that enhances additive Schwarz methods, applicable to diverse convex problems without needing problem-specific smoothness parameters.

Findings

Significant convergence rate improvements demonstrated

Applicable to linear, nonlinear, nonsmooth, and nonsharp problems

Numerical results confirm broad effectiveness

Abstract

Based on an observation that additive Schwarz methods for general convex optimization can be interpreted as gradient methods, we propose an acceleration scheme for additive Schwarz methods. Adopting acceleration techniques developed for gradient methods such as momentum and adaptive restarting, the convergence rate of additive Schwarz methods is greatly improved. The proposed acceleration scheme does not require any a priori information on the levels of smoothness and sharpness of a target energy functional, so that it can be applied to various convex optimization problems. Numerical results for linear elliptic problems, nonlinear elliptic problems, nonsmooth problems, and nonsharp problems are provided to highlight the superiority and the broad applicability of the proposed scheme.