Choice of the regularization parameter in ill-posed problems with rough estimate of the noise level of data

WSEAS Transactions on Mathematics

Volumn 4, Issue 2, 2005, Pages 76-81

  Hämarik, Uno ^a   Raus, Toomas ^a  

^a UNIVERSITY OF TARTU   (Estonia)

Author keywords

Convergence; Discrepancy principle; Ill posed problem; L curve; Noise level; Parameter choice; Regularization

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; ESTIMATION; HEURISTIC METHODS; ITERATIVE METHODS; MATHEMATICAL OPERATORS;

DISCREPANCY PRINCIPLE; ILL POSED PROBLEM; L-CURVE; NOISE LEVEL; PARAMETER CHOICE; REGULARIZATION APPROXIMATION;

APPROXIMATION THEORY;

EID: 21944449591     PISSN: 11092769     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (19)

1. A. Bakushinskii, Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion. U.S.S.R. Comput. Math. Math. Physics, Vol.24, No. 4, 1984, pp. 181-182.
2. H. Gfrerer, An a-posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, Math. Comp., Vol.49, No.180, 1987, pp. 507-522.
3. G. H. Golub, Urs. von Matt, Generalized cross-validation for large-scale problems. J. Comput. Graph. Statist., Vol.6, No.1, 1997, pp. 1-34.
4. U. Hämarik, T. Raus, On regularization parameter choice for solving self-adjoint ill-posed problems with approximately given noise level of data, Proc. Estonian Acad. Sci. Phys. Math., Vol.53, No.2, 2004, pp. 77-83.
5. U. Hämarik, T. Raus, On the regularization parameter choice in case of approximately given error bounds of data, ENUMATH2003 - Proceedings of the 5th European Conference on Numerical Mathematics and Advanced Applications, (eds. M. Feistauer, V. Dolejši, P. Knobloch, K. Najzar), Springer, 2004, 400-409.
6. U. Hämarik, U. Tautenhahn, On the monotone error rule for parameter choice in iterative and continuous regularization methods, BIT, Vol.41, No.5, 2001, pp. 1029-1038.
7. M. Hanke, T. Raus, A general heuristic for choosing the regularization parameter in ill-posed problems, SIAM J. Sci. Comput., Vol.17, No.4, 1996, pp. 954-972..
8. P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., Vol.34, No.4, 1992, pp. 561-580.
9. V. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., Vol.7, 1966, pp. 414-417.
10. T. Raus, Residual principle for ill-posed problems, Acta Comment. Univ. Tartuensis, Vol.672, 1984, pp. 16-26 (in Russian).
11. T. Raus, On the discrepancy principle for solution of ill-posed problems with non-selfadjoint operators, Acta Comment. Univ. Tartuensis, Vol.715, 1985, pp. 12-20 (in Russian).
12. T. Raus, An a posteriori choice of the regularization parameter in case of approximately given error bound of data, Acta Comment. Univ. Tartuensis, Vol.913, 1990, pp. 73-87.
13. T. Raus, About regularization parameter choice in case of approximately given error bounds of data, Acta Comment. Univ. Tartuensis, Vol.937, 1992, pp. 77-89.
14. U. Tautenhahn, U. Hämarik, The use of monotonicity for choosing the regularization parameter in ill-posed problems, Inverse Problems, Vol.15, No.6, 1999, pp. 1487-1505.
15. A. N. Tikhonov, V. B. Glasko, Use of the regularization method in non-linear problems, USSR Comput. Math. Math. Phys., Vol.5, No.3, 1965, pp. 93-107.
16. A. Tikhonov, V. Arsenin, Solution of Ill-Posed Problems. Wiley, New York, 1977.
17. G. Vainikko, The discrepancy principle for a class of regularization methods, U.S.S.R. Comput. Math. Math. Phys., Vol.22, No.3, 1982, pp. 1-19.
18. G. Vainikko, A. Veretennikov, Iteration Procedures in Ill-Posed Problems, Nauka, Moscow, 1986 (in Russian).
19. G. Wahba, Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal., Vol.14, No.4, 1977, pp. 651-657.