Next Article in Journal / Special Issue
Fixed Point Results for Multi-Valued Contractions in b−Metric Spaces and an Application
Previous Article in Journal
Equilibrium and Stability of Entropy Operator Model for Migratory Interaction of Regional Systems
Previous Article in Special Issue
Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem
Article Menu
Issue 2 (February) cover image

Export Article

Open AccessArticle
Mathematics 2019, 7(2), 131; https://doi.org/10.3390/math7020131

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

Department of Mathematics, NanChang University, Nanchang 330031, China
*
Author to whom correspondence should be addressed.
Received: 31 December 2018 / Revised: 22 January 2019 / Accepted: 28 January 2019 / Published: 1 February 2019
(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)
Full-Text   |   PDF [288 KB, uploaded 1 February 2019]

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. View Full-Text
Keywords: maximally monotone operators; Krasnoselskii–Mann algorithm; Yoshida approximation; resolvent; Douglas–Rachford splitting algorithm maximally monotone operators; Krasnoselskii–Mann algorithm; Yoshida approximation; resolvent; Douglas–Rachford splitting algorithm
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Yang, Y.; Tang, Y.; Zhu, C. Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces. Mathematics 2019, 7, 131.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Mathematics EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top