Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-19T15:19:30.947Z Has data issue: false hasContentIssue false
  • English
  • Français

A higher order approximation technique for restricted linear least-squares problems

Published online by Cambridge University Press:  17 April 2009

Heinz W. Engl  and
C.W. Groetsch
Heinz W. Engl
Affiliation:
Institut fr Mathematik, Universitt, A-9022 KLAGENFURT, Austria. Institut fr Mathematik, Johannes-Kepler-UniversittA-4040 LINZ, Austria
C.W. Groetsch
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH. 45221-0025, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An essential limitation for the method of weighting for equality constrained linear least-squares problems is the sub-optimality of the attainable convergence rate. In this paper, we propose a method, related to the method of iterated Tikhonov regularisation, that (under suitable conditions) gives rise to convergence rates which are arbitrarily near the optimal rate. As a by-product, we develop the theory of iterated Tikhonov regularisation for equations with unbounded linear operators.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 1 , February 1988 , pp. 121 - 130
Copyright
Copyright Australian Mathematical Society 1988

References

1Anderssen, R.S., The linear functional strategy for improperly posed problems, in Inverse Problems, Cannon, J., Hornung, U. (eds.), pp. 1130 (Birkhuser, Basel, 1986).CrossRefGoogle Scholar
2Callon, G.D., Theory and approximation of the restricted pseudosolution, Ph.D thesis, Univ. of Cincinnati.Google Scholar
3Callon, G.D., Finite element approximations for restricted pseudosolutions (to appear).Google Scholar
4Gallon, G. D. and Groetsch, C.W., The method of weighting and approximation of restricted pseudosolutions, J. Approx. Theory 51 (1987), 1118.Google Scholar
5Engl, H.W., Methods for approximating solutions of ill-posed linear operator equations based on numerical functional analysis, in Methods of Functional Analysis in Approximation Theory, Micchelli, C.A., Pai, D.V. and Limaye, B.V. (eds.), pp. 337355 (Birkhuser, Basel, 1986).Google Scholar
6Engl, H.W., On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems, J. Approx. Theory 49 (1987), 5563.Google Scholar
7Engl, H.W. and Neubauer, A., On projection methods for solving linear ill-posed problems (to appear), in Model Optimization in Exploration Geophysics, Edited by Vogel, A. (Vieweg, Wiesbaden, 1987 to appear).Google Scholar
8Engl, H.W. and Neubauer, A., A parameter choice strategy for (iterated) Tikhonov regularization of ill-posed problems leading to superconvergence with optimal rates, Applic. Analysis (to appear).Google Scholar
9Gfrerer, H., An a-posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, Math. Comp. (to appear).Google Scholar
10Groetsch, C.W., The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, 1984).Google Scholar
11Groetsch, C.W., Regularization with equality constraints, in Inverse Problems: Lecture Notes in Math. 1225, Talenti, G. (ed.), pp. 169181 (Springer, New York, 1986).Google Scholar
12Kato, T., Perturbation Theory for Linear Operators, 2nd ed. (Springer, Berlin, 1976).Google Scholar
13King, J.T. and Chillingworth, D., Approximation of generalized inverses by iterated regularization, Numer. Funct. Anal. Optim. 1 (1979), 499513.Google Scholar
14Lee, S.J., Tikhonov's regularization of an arbitrary closed linear operator, Abstracts of papers presented to the Amer. Math. Soc. 8 (1987), p. 82.Google Scholar
15Lee, S.J. and Nashed, M.Z., Constrained minimization problems for linear relations in Hilbert spaces (to appear).Google Scholar
16Locker, J. and Prenter, P., Regularization with differential operators I: General theory, J. Math. Anal. Appl. 74 (1980), 504529.Google Scholar
17Morozov, A., Methods for Solving Incorrectly Posed Problems (Springer, New York, 1984).Google Scholar
18Nashed, M. Z., Aspects of generalized inverses in analysis and regularization, in Generalized Inverses and Applications, Nashed, M. Z. (ed.), pp. 193244 (Academic Press, New York, 1976).Google Scholar
19Schock, E., Approximate solutions to ill-posed problems: arbitrarily slow convergence vs. superconvergence, in Constructive Methods for the Practical Treatment of Integral Equations, Hmmerlin, G. and Hoffmann, K.H. (eds.), pp. 234243 (Birkhuser, Basel, 1985).CrossRefGoogle Scholar
20Trummer, M.R., A method of solving ill-posed linear operator equations, SIAM J. Numer. Anal. 21 (1984), 729737.CrossRefGoogle Scholar