非线性问题基于Tikhonov正则化方法的收敛性分析
PDF下载 (9461)袁琼林,王金平 *.非线性问题基于Tikhonov正则化方法的收敛性分析[J].宁波大学学报(理工版),2018,31(4):99-103.DOI:
YUAN Qiong-lin,WANG Jin-ping *.Convergence analysis based on Tikhonov regularization for nonlinear ill-posed problems[J].Journal of Ningbo University(Natural Science & Engineering Edition),2018,31(4):99-103.DOI:
| Title: | Convergence analysis based on Tikhonov regularization for nonlinear ill-posed problems |
| 作者: | 袁琼林, 王金平 * |
| Author(s): | YUAN Qiong-lin, WANG Jin-ping * |
| 关键词: | 变分不等式; 收敛率; Tikhonov正则化; 源条件 |
| Keywords: | variational inequality; convergence rates; Tikhonov regularization; source conditions |
| 分类号: | O177.6; O175.5 |
| 文献标识码: | A |
| 摘要: | 本文主要利用Tikhonov正则化方法研究非线性不适定问题, 与之前的研究所不同的是, 本文主要研究当数据和算子均被噪声破坏时, 利用Tikhonov正则化方法进行收敛性分析. 同时, 充分利用算子的非线性条件及解的光滑性条件, 得到其在Banach空间和Hilbert空间中的收敛率. |
| Abstract: | In this paper, we study the convergence rates of Tikhonov regularization method for solving nonlinear ill-posed problems. Opposed to the previous research, we investigate convergence rates using the Tikhonov regularization method when both the right-hand side and the operator are given approximately with errors. We derive the convergence rates in Banach spaces or Hilbert spaces by taking full advantage of the interplay of the solution smoothness and the nonlinearity structure. |
| 参考文献 /References: | [1] Hein T. Tikhonov regularization in Banach spaces-improved convergence rates results[EB/OL]. [2016-05- 21]. http://iopscience.iop.org/article/10.1088/0266-5611/25/3/035002/meta. [2] Hein T. Convergence rates for regularization of ill-posed problems in Banach spaces by approximate source conditions[EB/OL]. [2016-07-10]. http://iopscience.iop.org/article/10.1088/0266-5611/24/4/045007/meta. [3] Hein T, Hofmann B. Approximate source conditions for nonlinear ill-posed problems—chances and limitations[J]. Inverse Problems, 2009, 25(3):35003-35016. [4] Hofmann B, Kaltenbacher B, P?schl C, et al. A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators[J]. Inverse Problems, 2007, 23(3):987-1010. [5] Hofmann B, Mathé P, Cker H. Regularization in Hilbert space under unbounded operators and general source conditions[J]. Inverse Problems, 2009, 25(11):115013-115015. [6] Lu S, Pereverzev S V, Shao Y, et al. On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales[J]. Journal of Integral Equations & Applications, 2012, 22(2010):483-517. [7] Jin Q, Amato U. A discrete scheme of Landweber iteration for solving nonlinear ill-posed problems[J]. Journal of Mathematical Analysis & Applications, 2001, 253(1):187-203. [8] Neubauer A, Scherzer O. Finite-dimensional approximation of Tikhonov regularized solutions of non-linear ill-posed problems[J]. Numerical Functional Analysis Optimization, 1990, 11(1/2):85-99. [9] Stück R, Burger M, Hohage T. The iteratively regularized Gauss-Newton method with convex constraints and applications in 4Pi microscopy[J]. Inverse Problems, 2011, 28(1):15012-15027. [10] Flemming J. Generalized Tikhonov regularization: Basic theory and comprehensive results on convergence rates[D]. Chemnitz: Technische Universit?t Chemnitz, 2011:109-140. [11] Anzengruber S W, Hofmann B, Mathé P. Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces[J]. Applicable Analysis, 2014, 93(7):1382-1400. |
| 备注/Memo: | 收稿日期: 2018－01－28. 宁波大学学报（理工版）网址: http://journallg.nbu.edu.cn/基金项目: 宁波大学研究生科研创新基金（G17068）.第一作者: 袁琼林（1993－）, 女, 安徽安庆人, 在读硕士研究生, 主要研究方向: 积分变换与图像处理. E-mail: 2296075044@qq.com*通信作者: 王金平（1962－）, 男, 湖北武汉人, 博士/教授, 主要研究方向: 积分变换与图像处理. E-mail: wangjinping@nbu.edu.cn 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ |
