Volume 35, Issue 2
Structured Condition Numbers for the Tikhonov Regularization of Discrete Ill-Posed Problems

J. Comp. Math., 35 (2017), pp. 169-186.

Published online: 2017-04

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• Abstract

The possibly most popular regularization method for solving the least squares problem $\mathop{\rm min}\limits_x$$||Ax-b||_2 with a highly ill-conditioned or rank deficient coefficient matrix A is the Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when A has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al. [Numer. Linear Algebra Appl., 18 (2011), 87-103]. • AMS Subject Headings 65F35, 65F20. • Copyright COPYRIGHT: © Global Science Press • Email address menglsh@nwnu.edu.cn (Lingsheng Meng) bzheng@lzu.edu.cn (Bing Zheng) • BibTex • RIS • TXT @Article{JCM-35-169, author = {Meng , Lingsheng and Zheng , Bing}, title = {Structured Condition Numbers for the Tikhonov Regularization of Discrete Ill-Posed Problems}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {2}, pages = {169--186}, abstract = { The possibly most popular regularization method for solving the least squares problem \mathop{\rm min}\limits_x$$||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].

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1608-m2015-0279}, url = {http://global-sci.org/intro/article_detail/jcm/9768.html} }
TY - JOUR T1 - Structured Condition Numbers for the Tikhonov Regularization of Discrete Ill-Posed Problems AU - Meng , Lingsheng AU - Zheng , Bing JO - Journal of Computational Mathematics VL - 2 SP - 169 EP - 186 PY - 2017 DA - 2017/04 SN - 35 DO - http://doi.org/10.4208/jcm.1608-m2015-0279 UR - https://global-sci.org/intro/article_detail/jcm/9768.html KW - Tikhonov regularization, Discrete ill-posed problem, Structured least squares problem, Structured condition number. AB -

The possibly most popular regularization method for solving the least squares problem $\mathop{\rm min}\limits_x$$||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].

Lingsheng Meng & Bing Zheng. (2020). Structured Condition Numbers for the Tikhonov Regularization of Discrete Ill-Posed Problems. Journal of Computational Mathematics. 35 (2). 169-186. doi:10.4208/jcm.1608-m2015-0279
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