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

Yixuan Yang, Yuchao Tang and Chuanxi Zhu
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Yixuan Yang: Department of Mathematics, NanChang University, Nanchang 330031, China
Yuchao Tang: Department of Mathematics, NanChang University, Nanchang 330031, China
Chuanxi Zhu: Department of Mathematics, NanChang University, Nanchang 330031, China

Mathematics, 2019, vol. 7, issue 2, 1-16

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.

Keywords: maximally monotone operators; Krasnoselskii–Mann algorithm; Yoshida approximation; resolvent; Douglas–Rachford splitting algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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