Strengthened Splitting Methods for Computing Resolvents

TL;DR

This paper introduces a systematic framework for computing resolvents of sums of monotone operators using strengthening, enabling efficient algorithms for problems like approximation, denoising, and PDEs.

Contribution

It proposes a novel strengthening approach for set-valued operators, facilitating new iterative schemes for resolvent computation in various applications.

Findings

Developed iterative schemes for resolvent computation

Applied methods to image denoising and PDEs

Demonstrated computational efficiency and versatility

Abstract

In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the "strengthening" of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs.