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ON MOROZOV’S DISCREPANCY PRINCIPLE FOR NONLINEAR ILL-POSED EQUATIONS

Part of: Numerical analysis in abstract spaces Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  13 March 2009

M. T. NAIR
M. T. NAIR*
Affiliation:
Department of Mathematics, I.I.T. Madras, Chennai 600 036, India (email: [email protected])
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Abstract

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Morozov’s discrepancy principle is one of the simplest and most widely used parameter choice strategies in the context of regularization of ill-posed operator equations. Although many authors have considered this principle under general source conditions for linear ill-posed problems, such study for nonlinear problems is restricted to only a few papers. The aim of this paper is to apply Morozov’s discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems under general source conditions.

Keywords

inverse problemsill-posed problemsTikhonov regularizationdiscrepancy principle

MSC classification

Secondary: 65J20: Improperly posed problems; regularization 65M30: Improperly posed problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 337 - 342
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Engl, H. W., Hanke, M. and Neubauer, A., Regularization of Inverse Problems (Kluwer, Dordrecht, 1993).Google Scholar
[2] Engl, H. W., Kunisch, K. and Neubauer, A., ‘Convergence rates for Tikhonov regularization of nonlinear ill-posed problems’, Inverse Problems 5 (1989), 523–540.CrossRefGoogle Scholar
[3] Hegland, M., ‘An optimal order regularization method which does not use additional smoothness assumptions’, SIAM J. Numer. Anal. 29 (1992), 1446–1461.Google Scholar
[4] Hegland, M., ‘Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization’, Appl. Anal. 59 (1995), 207–223.CrossRefGoogle Scholar
[5] Hohage, T., ‘Logarithmic convergence rates of the iteratively regularized Gauss–Newton method for an inverse potential and an inverse scattering problem’, Inverse Problems 13 (1997), 1279–1299.Google Scholar
[6] Hohage, T., ‘Regularization of exponentially ill-posed problems’, Numer. Funct. Anal. Optim. 21 (2000), 439–464.CrossRefGoogle Scholar
[7] Lu, S., Pereverzev, S. V. and Ramlau, R., ‘An analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption’, Inverse Problems 23 (2007), 217–230.Google Scholar
[8] Mahale, P. and Nair, M. T., ‘Tikhonov regularization of nonlinear ill-posed equations under general source conditon’, J. Inverse Ill-Posed Probl. 15 (2007), 813–829.CrossRefGoogle Scholar
[9] Nair, M. T., Functional Analysis: A First Course (Prentice-Hall of India, New Delhi, 2002).Google Scholar
[10] Nair, M. T., ‘On improving error estimates for Tikhonov regularization using an unbounded operator’, J. Anal. 14 (2006), 143–157.Google Scholar
[11] Nair, M. T. and Pereverzev, S., ‘Regularized collocation method for Fredholm integral equations of the first kind’, J. Complexity 23 (2007), 454–467.CrossRefGoogle Scholar
[12] Nair, M. T., Pereverzev, S. and Tautenhahn, U., ‘Regularization in Hilbert scales under general smoothing conditions’, Inverse Problems 21 (2005), 1851–1869.CrossRefGoogle Scholar
[13] Nair, M. T., Schock, E. and Tautenhahn, U., ‘Morozov’s discrepancy principle under general source conditions’, J. Anal. Anw. 22 (2003), 199–214.CrossRefGoogle Scholar
[14] Pereverzev, S. and Schock, E., ‘Morozov’s discrepancy principle for Tikhonov regularization of severely ill-posed problems’, Numer. Funct. Anal. Optim. 21 (2000), 901–920.CrossRefGoogle Scholar
[15] Ramlau, R., ‘Morozov’s discrepancy principle for Tikhonov regularization of nonlinear operators’, Numer. Funct. Anal. Optim. 23 (2002), 147–172.CrossRefGoogle Scholar
[16] Rudin, W., Real and Complex Analysis, 3rd edn (McGraw-Hill, New York, 1987).Google Scholar
[17] Tautenhahn, U., ‘Optimality for ill-posed problems under general source conditions’, Numer. Funct. Anal. Optim. 19 (1998), 377–398.CrossRefGoogle Scholar