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A class of discrepancy principles for the simplified regularization of ill-posed problems

Published online by Cambridge University Press:  17 February 2009

Santhosh George  and
M. Thamban Nair
M. Thamban Nair
Affiliation:
Department of Mathematics, Goa University, Goa — 403 203, India
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Abstract

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A class of discrepancy principles for the choice of parameters for the simplified regularization of ill-posed problems is proposed. This procedure does not require knowledge of the unknown solution, and if the smoothness of the unknown solution is known then the convergence rate obtained is optimal. The results of this paper include the Arcangeli's method considered by Groetsch and Guacaneme (1987) for which the convergence rate was not known and also of a result of Guacaneme (1988) for which there is a gap in the proof.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 2 , October 1994 , pp. 242 - 248
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Engl, H. W., “Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates”, J. Optim, Theory Appl. 52 (1987) 209–215.CrossRefGoogle Scholar
[2]Engl, H. W. and Gfrerer, H., “A posteriori parameter choice for the general regularization methods for solving linear ill-posed problems”, Appl. Numer. Math. 4 (1988) 395–417.Google Scholar
[3]Engl, H. W. and Neubauer, A., “Optimal discrepancy principle for the Tikhonov regularization of integral equations of the first kind”, in Constructive methods for the Practical Treatment of Integral Equations (eds. Hammerlin, G. and Hoffman, K. H.), ISNM 73, (Birkhauser Verlag, Basel, 1985) 120–141.CrossRefGoogle Scholar
[4]Groetsch, C. W., The theory of Tikhonov regularization method for Fredholm integral equations of the first kind (Pitman, Boston, 1984).Google Scholar
[5]Groetsch, C. W. and Guacaneme, J., “Arcangeli's method for Fredholm equations of the first kind”, Proc. Amer. Math. Soc. 99 (1987) 256–260.Google Scholar
[6]Guacaneme, J. E., “On simplified regularization”, J. Optim. Theory Appl. 58 (1988) 133–138.CrossRefGoogle Scholar
[7]Guacaneme, J. E., “An optimal parameter choice for regularized ill-posed problems”, Integral Equations Operator Theory 11 (1988) 610–613.Google Scholar
[8]Nair, M. Thamban, “A generalization of Arcangeli's method for ill-posed problems leading to optimal rates”, Integral Equations Operator Theory 15 (1992) 1042–1046.Google Scholar
[9]Schock, E., “On the asymptotic order of accuracy of Tikhonov regularization”, J. Optim. Theory Appl. 44 (1984) 95–104.CrossRefGoogle Scholar
[10]Schock, E., “Parameter choice by discrepancy principles for the approximate solution of ill-posed problems”, Integral Equations Operator Theory 7 (1984) 895–898.CrossRefGoogle Scholar
[11]Schock, E., “Ritz regularization versus least square regularization: Solution methods for integral equations of the first kind”, Zeitschrift fur Analysis und ihre Anwedungen 2 (1985).Google Scholar