Gradient Methods with Selection Technique for the Multiple-Sets Split Equality Problem

Dianlu Tian,Lining Jiang,Luoyi Shi

doi:10.3390/math7100928

Dianlu Tian, Lining Jiang + Show 1 more

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https://doi.org/10.3390/math7100928

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Abstract

The inverse problem is one of the four major problems in computational mathematics. There is an inverse problem in medical image reconstruction and radiotherapy that is called the multiple-sets split equality problem. The multiple-sets split equality problem is a unified form of the split feasibility problem, split equality problem, and split common fixed point problem. In this paper, we present two iterative algorithms for solving it. The suggested algorithms are based on the gradient method with a selection technique. Based on this technique, we only need to calculate one projection in each iteration.

Highlights

The inverse problem is one of the four major problems in computational mathematics
The rapid development of the inverse problem has been a feature of recent decades; it can be found in computer vision, machine learning, statistics, geography, medical imaging, remote sensing, ocean acoustics, tomography, aviation, physics, and other fields
There is an inverse problem in medical image reconstruction and radiotherapy that can be expressed as a split feasibility problem [1,2,3,4,5,6,7,8,9], split equality problem [10,11,12,13], and split common fixed point problem [14,15,16,17,18,19]

Summary

Introduction

The inverse problem is one of the four major problems in computational mathematics. The rapid development of the inverse problem has been a feature of recent decades; it can be found in computer vision, machine learning, statistics, geography, medical imaging, remote sensing, ocean acoustics, tomography, aviation, physics, and other fields. The multiple-sets split equality problem (MSSEP for short) can be formulated as f inding x∈. H2 = H3 and B is the identity operator on H2 It is the split common fixed point problem if we take x ∈ Ci to x = PCi x, y ∈ Q j to y ∈ PQ j where PCi , PQ j are the metric projections on Ci , Q j. To solve the minimization problem (4), a classical method is the gradient algorithm, which takes the iterative issue w n + 1 = w n − γn ∇ f ( w n ) ,. In iteration (6), we only need to implement a projection once in each step Motivated by this point, we present Algorithms 1 and 2 in Section 3 to solve problem (3).

Preliminaries

Main Results

Numerical Experiments