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The trade-off between regularity and stability in Tikhonov regularization


Authors: M. Thamban Nair, Markus Hegland and Robert S. Anderssen
Journal: Math. Comp. 66 (1997), 193-206
MSC (1991): Primary 65R30; Secondary 65J20, 45B05
DOI: https://doi.org/10.1090/S0025-5718-97-00811-9
MathSciNet review: 1388891
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Abstract: When deriving rates of convergence for the approximations generated by the application of Tikhonov regularization to ill-posed operator equations, assumptions must be made about the nature of the stabilization (i.e., the choice of the seminorm in the Tikhonov regularization) and the regularity of the least squares solutions which one looks for. In fact, it is clear from works of Hegland, Engl and Neubauer and Natterer that, in terms of the rate of convergence, there is a trade-off between stabilization and regularity. It is this matter which is examined in this paper by means of the best-possible worst-error estimates. The results of this paper provide better estimates than those of Engl and Neubauer, and also include and extend the best possible rate derived by Natterer. The paper concludes with an application of these results to first-kind integral equations with smooth kernels.


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Additional Information

M. Thamban Nair
Affiliation: Centre for Mathematics and Its Applications, Australian National University, Canberra ACT 0200, Australia
Address at time of publication: Department of Mathematics, Indian Institute of Technology, Madras - 600 036, India
Email: mtnair@acer.iitm.ernet.in

Markus Hegland
Affiliation: Centre for Mathematics and Its Applications, Australian National University, Canberra ACT 0200, Australia
Address at time of publication: Computer Sciences Laboratory, RSISE, Australian National University, Canberra ACT 0200, Australia
Email: Markus.Hegland@anu.edu.au

Robert S. Anderssen
Affiliation: Centre for Mathematics and Its Applications, Australian National University, Canberra ACT 0200, Australia
Address at time of publication: CSIRO Division of Mathematics and Statistics, GPO Box 1965, Canberra ACT 2601, Australia
Email: boba@cbr.dms.csiro.au

DOI: https://doi.org/10.1090/S0025-5718-97-00811-9
Received by editor(s): April 13, 1994
Received by editor(s) in revised form: November 13, 1995
Article copyright: © Copyright 1997 American Mathematical Society