Three-Operator Splitting Method with Two-Step Inertial Extrapolation

TL;DR

This paper analyzes a three-operator splitting method with two-step inertial extrapolation, demonstrating improved convergence and efficiency in solving monotone inclusion problems, with applications in image restoration and statistical penalty problems.

Contribution

It introduces a novel two-step inertial extrapolation technique for the three-operator splitting method, improving convergence analysis and practical performance.

Findings

Enhanced convergence without summability conditions.

Numerical results show improved efficiency in image restoration.

Two-step inertial extrapolation outperforms one-step methods.

Abstract

The aim of this paper is to study the weak convergence analysis of sequence of iterates generated by a three-operator splitting method of Davis and Yin incorporated with two-step inertial extrapolation for solving monotone inclusion problem involving the sum of two maximal monotone operators and a co-coercive operator in Hilbert spaces. Our results improve on the setbacks observed recently in the literature that one-step inertial Douglas-Rachford splitting method may fail to provide acceleration. Our convergence results also dispense with the summability conditions imposed on inertial parameters and the sequence of iterates assumed in recent results on multi-step inertial methods in the literature. Numerical illustrations from image restoration problem and Smoothly Clipped Absolute Deviation (SCAD) penalty problem are given to show the efficiency and advantage gained by incorporating two-step inertial extrapolation over one-step inertial extrapolation for three-operator splitting method.