Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces

Abstract

The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator. Second, we propose a fixed point approach for computing this resolvent operator in a general case. Based on the Krasnoselskii–Mann algorithm for finding fixed points of non-expansive operators, we prove the strong convergence of the sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator, which has wide applications in image restoration and image reconstruction problems. Furthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper, lower semi-continuous convex functions.

An inertial three-operator splitting algorithm with applications to image inpainting

“…The traditional operator splitting algorithms include the forward-backward splitting algorithm [31], the Douglas-Rachford splitting algorithm [27], and the forward-backward-forward splitting algorithm [44], which are originally designed for solving the monotone inclusion of the sum of two maximally monotone operators, where one of which is assumed to be cocoercive or just Lipschitz continuous. In recent years, the monotone inclusion problems with the sum of more than two operators have been received much attention; see, for example, [15,16,17,20,41,45,46,48] and the references therein.…”