STRUCTURED CONDITION NUMBERS FOR THE TIKHONOV REGULARIZATION OF DISCRETE ILL-POSED PROBLEMS  

作　　者：LingshengMeng Bing Zheng 

机构地区：[1]School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

出　　处：《Journal of Computational Mathematics》2017年第2期169-186,共18页计算数学（英文）

基　　金：The authors would like to thank the anonymous referees for their valu- able suggestions and comments. This work was supported by the National Natural Science Foundation of China （No. 11571004 and No. 11171371）.

摘　　要：The possibly most popular regularization method for solving the least squares problem rain ‖Ax - b‖2 with a highly ill-conditioned or rank deficient coefficient matrix A is the x 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].The possibly most popular regularization method for solving the least squares problem rain ‖Ax - b‖2 with a highly ill-conditioned or rank deficient coefficient matrix A is the x 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].

关 键 词：Tikhonov regularization Discrete ill-posed problem Structured least squaresproblem Structured condition number. 

分 类 号：O[理学]