Abstract
We consider the perturbation analysis of two important problems for solving ill-conditioned or rank-deficient linear least squares problems. The Tikhonov regularized problem is a linear least squares problem with a regularization term balancing the size of the residual against the size of the weighted solution. The weight matrix can be a non-square matrix (usually with fewer rows than columns). The minimum-norm problem is the minimization of the size of the weighted solutions given by the set of solutions to the, possibly rank-deficient, linear least squares problem.
It is well known that the solution of the Tikhonov problem tends to the minimum-norm solution as the regularization parameter of the Tikhonov problem tends to zero. Using this fact and the generalized singular value decomposition enable us to make a perturbation analysis of the minimum-norm problem with perturbation results for the Tikhonov problem. From the analysis we attain perturbation identities for Tikhonov inverses and weighted pseudoinverses.
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REFERENCES
Å. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996.
L. Eldèn, A weighted pseudoinverse, generalized singular values and constrained least squares problems, BIT, 22(1982), pp. 487–502.
G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
M. E. Gulliksson and P.-Å Wedin, The use of filter matrices in perturbation theory for regularized Tikhonov inverses. Tech. Report, Department of Computer Science, Ume?a University, Umeå, Sweden, 1998.
A. Neubauer H. Engl, and M. Hanke, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996.
P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, PA, 1997.
P.-Å. Wedin, Perburbation theory for pseudo-inverses, BIT, 13(1973), pp. 217–232.
M. Wei, Perturbation theory for the rank-deficient equality constrained and weighted least squares problem, SIAM J. Numer. Anal., 29(1992), pp. 1462–1481.
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Gulliksson, M.E., Wedin, PÅ. & Wei, Y. Perturbation Identities for Regularized Tikhonov Inverses and Weighted Pseudoinverses. BIT Numerical Mathematics 40, 513–523 (2000). https://doi.org/10.1023/A:1022319830134
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DOI: https://doi.org/10.1023/A:1022319830134