Iterative algorithm for solving the multiple-sets split equality problem with split self-adaptive step size in Hilbert spaces

Dianlu Tian,Luoyi Shi,Rudong Chen

doi:10.1186/s13660-016-0982-7

Dianlu Tian, Luoyi Shi + Show 1 more

Open Access

https://doi.org/10.1186/s13660-016-0982-7

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Abstract

The split equality problem is a generalization of the split feasibility problem, meanwhile it is a special case of multiple-sets split equality problems. In this paper, we propose an iterative algorithm for solving the multiple-sets split equality problem whose iterative step size is split self-adaptive. The advantage of the split self-adaptive step size is that it could be obtained directly from the iterative procedure without needing to have any information of the spectral norm of the related operators. Under suitable conditions, we establish the theoretical convergence of the algorithm proposed in Hilbert spaces, and several numerical results confirm the effectiveness of the algorithm proposed.

Highlights

There arise various linear inverse problems in phase retrieval, radiation therapy treatment, signal processing, and medical image reconstruction etc
We propose an iterative algorithm with split self-adaptive step size where we need not calculate or estimate the spectral norms of related operators
In Section, we present an iterative algorithm with split self-adaptive step size and provide the proof of its convergence

Summary

Introduction

There arise various linear inverse problems in phase retrieval, radiation therapy treatment, signal processing, and medical image reconstruction etc. Censor and Elfving [ ] summarized one of these classes of problems and proposed a new concept which is called the split feasibility problem (SFP), and the SFP can be characterized mathematically as finding x ∈ C which satisfies Ax ∈ Q, where C, Q are closed, convex, and nonempty subsets of the Hilbert spaces H and H , respectively, and A : H → H is a bounded and linear operator. Where C, Q are closed, convex, and nonempty subsets of Hilbert spaces H and H , respectively, and H is a Hilbert space, A : H → H , B : H → H are two bounded and linear operators.

Results

Conclusion