On the adaptive selection of the parameter in regularization of ill-posed problems

SIAM Journal on Numerical Analysis

Volumn 43, Issue 5, 2005, Pages 2060-2076

  Pereverzev, Sergei ^a   Schock, Eberhard ^b  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^b University of Kaiserslautern ^*  (Germany)

Author keywords

Abel integral equations; Inverse problems in Banach spaces; Lavrentiev regularization for equations with monotone operators; Parameter choice; Profile reconstruction; Scattering

Indexed keywords

INTEGRAL EQUATIONS; INVERSE PROBLEMS; MATHEMATICAL OPERATORS; PARAMETER ESTIMATION; STATISTICAL METHODS;

BANACH SPACES; LAVRENTIEV REGULARIZATION; MONOTONE OPERATORS; PARAMETER CHOICE; PROFILE RECONSTRUCTION;

PROBLEM SOLVING;

EID: 33750173498     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903433819     Document Type: Article

References (31)

1. A. B. BAKUSHINSKII, Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion, Comput. Math. Math. Phys., 24 (1984), pp. 181-182.
2. M. B. BENSON, Errors in Numerical Quadrature for Certain Singular Integrands and the Numerical Solution of Abel Integral Equations, Ph.D. thesis, Dept. of Math., University of Wisconsin, Madison, 1973.
3. G. BRUCKNER, J. ELSCHER, AND M. YAMAMOTO, An Optimization Method for Grating Profile Reconstruction, Preprint 682, WIAS Berlin, 2001.
4. G. BRUCKNER AND S. PEREVERZEV, Self-regularization of projection methods with a posteriori discretization level choice for severally ill-posed problems, Inverse Problems, 19 (2003), pp. 147-156.
5. L. CAVALIER AND A. TSYBAKOV, Sharp adaptation for inverse problems with random noise, Probab. Theory Related Fields, 123 (2002), pp. 323-354.
6. K. DEIMLING, Nonlinear Functional Analysis, Springer, New York, 1985.
7. P. P. B. EGGERMONT, A new analysis of the trapezoidal discretization method for the numerical solution of Abel-type integral equations, J. Integral Equations, 3 (1981), pp. 317-332.
8. H. W. ENGL AND H. GFRERER, A posteriori parameter choice for general regularization methods for solving linear ill-posed problems, Appl. Numer, Math., 4 (1988), pp. 395-417.
9. H. GFRERER, An a posteriori paramter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, Math. Comput., 49 (1987), pp. 507-522.
10. A. GOLDENSHLUGER AND S. PEREVERZEV, Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations, Probab. Theory Related Fields, 118 (2000), pp. 169-186.
11. C. W. GROETSCH, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.
12. H. HARBRECHT, S. PEREVERZEV, AND R. SCHNEIDER, Self-regularization by projection for noisy pseudodifferential equations of negative order, Numer. Math., 95 (2003), pp. 123-143.
13. B. KALTENBACHER, Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems, Inverse Problems, 16 (2000), pp. 137-155.
14. A. KIRSCH AND R. KRESS, An optimization method in inverse acoustic scattering, in Boundary Elements IX, Vol. 3, Springer, Berlin, 1987, pp. 3-18.
15. O. LEPSKII, A problem of adaptive estimation in Gaussian white noise, Theory Probab. Appl., 36 (1990), pp. 454-466.
16. F. LIU AND M. Z. NASHED, Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators, in Partial Differential Equations and Applications, Dekker, New York, 1996, pp. 353-361.
17. M. A. LUKAS, Comparison of parameter choice methods for regularization with discrete noisy data, Inverse Problems, 14 (1998), pp. 161-184.
18. J. T. MARTI, An algorithm for computing minimum norm solutions of Fredholm integral equations of the first kind, SIAM J. Numer. Anal., 15 (1978), pp. 1071-1076.
19. P. MATHÉ AND S. PEREVERZEV, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), pp. 789-803.
20. V. A. MOROZOV, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), pp. 414-417.
21. D. L. PHILLIPS, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach., 9 (1962), pp. 84-97.
22. R. PLATO AND U. HÄMARIK, On pseudo-optimal parameter choices and stopping rules for regularization methods in Banach spaces, Numer. Funct. Anal. Optim., 17 (1996), pp. 181-195.
23. T. RAUS, Residue principle for ill-posed problems, Acta Comment. Univ. Tartu. Math., 672 (1984), pp. 16-26.
24. U. TAUTENHAHN, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems, 18 (2002), pp. 191-207.
25. U. TAUTENHAHN AND U. HÄMARIK, The use of monotonicity for choosing the regularization parameter in ill-posed problems, Inverse Problems, 15 (1999), pp. 1487-1505.
26. A. N. TIKHONOV, Regularization of incorrectly posed problems, Soviet Math. Dokl., 4 (1963), pp. 1624-1627.
27. A. N. TIKHONOV AND V. B. GLASKO, An approximate solution of Fredholm integral equations of the first kind, Zh. Vychisl. Mat. Mat. Fiz., 4 (1964), pp. 564-571.
28. A. N. TIKHONOV AND V. B. GLASKO, Use of the regularization method in non-linear problems, Zh. Vychisl. Mat. Mat. Fiz., 5 (1965), pp. 463-473.
29. A. TSYBAKOV, On the best rate of adaptive estimation in some inverse problems, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), pp. 835-840.
30. R. WEISS, Product integration for the generalized Abel equation, Math. Comp., 26 (1972), pp. 177-190.
31. R. WEISS AND R. S. ANDERSSEN, A product integration for a class of first kind Volterra equations, Numer. Math., 18 (1972), pp. 442-456.