Banach空间中迭代正则化方法的收敛性分析

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英文篇名：Convergence analysis of iterative regularization methods in Banach spaces 作者：王佳敏 ; 王金平 英文作者：WANG Jia-min;WANG Jin-ping;Faculty of Science, Ningbo University; 关键词：Landweber-Kaczmarz迭代算法 ; 收敛性 ; Bregman距离 ; Fréchet可微 英文关键词：Landweber-Kaczmarz iterative algorithm;;convergence;;Bregman distance;;Fréchet differentiable 中文刊名：NBDZ 英文刊名：Journal of Ningbo University(Natural Science & Engineering Edition) 机构：宁波大学理学院; 出版日期：2019-01-10 出版单位：宁波大学学报(理工版) 年：2019 期：v.32;No.115 基金：国家自然科学基金(61271398);; 宁波大学研究生科研创新基金(G17068) 语种：中文; 页：NBDZ201901015 页数：5 CN：01 ISSN：33-1134/N 分类号：90-94

摘要

利用Landweber-Kaczmarz迭代算法研究非线性不适定问题.首先,在Banach空间引入Bregman距离,构造合适的步长,说明Bregman距离序列在迭代算法中是单调递减的.然后,由凸分析、对偶映射和Fréchet可微的性质得到迭代算法具有收敛性.
        In this paper, we study the Landweber-Kaczmarz iterative algorithm for nonlinear ill-posed problems. We first introduce the iterative step, and then derive the sequence of Bregman distance which is monotonously decreasing in Banach space. With the convex analysis, duality mapping and Fréchet differentiable, we succeed in obtaining the convergence of the iterative algorithm.

引文

[1]Hanke M,Neabuaer A,Scherzer O.A convergence analysis of the Landweber iteration for nonlinear ill-posed problems[J].Numerische Mathematik,1995,72(1):21-37.
    [2]Haltmeier M,Kowar R,Leit?o A,et al.Kaczmarz methods for regularizing nonlinear ill-posed equations II:Applications[J].Inverse Problems Imaging,2017,1(3):507-523.
    [3]Hofmann B,Kaltenbacher B,P?schl C,et al.Aconvergence rates result for Tikhonov regularization in Banach spaces with non-smooth operator[J].Inverse Problems,2007,23(3):987-1010.
    [4]Sch?pfer F,Louis A K,Schuster T.Nonlinear iterative methods for linear ill-posed problems in Banach spaces[J].Inverse Problems,2006,22(1):311-329.
    [5]Kaltenbacher B,Sch?pfer F,Schuster T.Iterative methods for nonlinear ill-posed problems in Banach spaces:Convergence and applications to parameter identification problems[J].Inverse Problems,2009,26(6):65003-65021.
    [6]Jin Q.Inexact Newton-Landweber iteration for solving nonlinear inverse problems in Banach spaces[J].Inverse Problems,2012,28(28):1012-1020.
    [7]Jin Q,Wang W.Landweber iteration of Kaczmarz type with general non-smooth convex penalty functionals[J].Inverse Problems,2013,29(8):1400-1416.
    [8]Leit?o A,Alves M A.On Landweber-Kaczmarz method for regularizing systems of ill-posed equations in Banach spaces[EB/OL].[2017-12-03].http://iopscience.iop.org/article/10.1088/0266-5611/28/10/104008/pdf.
    [9]管金林,胡长松.Banach空间中几种迭代算法的收敛性[D].黄石:湖北师范学院,2014.
    [10]杜玲玲,王金平.巴拿赫空间中Landweber迭代算法的收敛性[J].宁波大学学报(理工版),2016,29(2):85-88.