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Structured Condition Numbers for the Tikhonov Regularization of Discrete Ill-Posed Problems

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||Axb||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].

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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

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