Structured Condition Numbers for the Tikhonov Regularization of Discrete Ill-Posed Problems
Year: 2017
Author: Lingsheng Meng, Bing Zheng
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 2 : pp. 169–186
Abstract
The possibly most popular regularization method for solving the least squares problem minx||Ax−b||2 with a highly ill-conditioned or rank deficient coefficient matrix A is the Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when A has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al. [Numer. Linear Algebra Appl., 18 (2011), 87-103].
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1608-m2015-0279
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 2 : pp. 169–186
Published online: 2017-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18
Keywords: Tikhonov regularization Discrete ill-posed problem Structured least squares problem Structured condition number.
Author Details
Lingsheng Meng Email
Bing Zheng Email
-
Condition numbers of the generalized ridge regression and its statistical estimation
Kong, Jing | Wang, ShaoxinAIMS Mathematics, Vol. 9 (2024), Iss. 2 P.4178
https://doi.org/10.3934/math.2024205 [Citations: 0] -
Perturbation analysis and condition numbers for the Tikhonov regularization of total least squares problem and their statistical estimation
Samar, Mahvish | Lin, Fu-RongJournal of Computational and Applied Mathematics, Vol. 411 (2022), Iss. P.114230
https://doi.org/10.1016/j.cam.2022.114230 [Citations: 7]