Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods

SIAM Journal on Numerical Analysis

Volumn 38, Issue 6, 2001, Pages 1999-2021

  Mathé, Peter ^a   Pereverzev, Sergei V ^b  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b INSTITUTE OF MATHEMATICS   (Ukraine)

Author keywords

Gaussian white noise; Ill posed problems; Information complexity; Inverse estimation; Operator equations

Indexed keywords

EID: 0034580744     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S003614299936175X     Document Type: Article

References (42)

1. I. BABUŠKA AND A. K. AZIZ, Survey lectures on the mathematical foundation of the finite element method, in The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations, A. A. K., ed., Academic Press, New York, 1972, pp. 1-359.
2. E. N. BELITSER AND B. Y. LEVIT, On minimax filtering over ellipsoids, Math. Methods Statist., 4 (1995), pp. 259-273.
3. A. BÖTTCHER, A. DIJKSMA, H. LANGER, M. A. DRITSCHEL, J. ROVNYAK, AND M. A. KAASHOEK, Lectures on Operator Theory and Its Applications, AMS, Providence, RI, 1996. Lectures presented at the meeting held at the Fields Institute for Research in Mathematical Sciences, Waterloo, Ontario, September 1994, Edited by Peter Lancaster.
4. G. BRUCKNER, S. PRÖSSDORF, AND G. VAINIKKO, Error bounds of discretization methods for boundary integral equations with noisy data, Appl. Anal., 63 (1996), pp. 25-37.
5. L. CAVALIER AND A. B. TSYBAKOV, Sharp Adaptation for Inverse Problems with Random Noise, Preprint 559, Univ. Paris VI and Paris VII, 1999.
6. P.-L. CHOW, I. A. IBRAGIMOV, AND R. Z. KHASMINSKII, Statistical approach to some ill-posed problems for linear partial differential equations, Probab. Theory Related Fields, 113 (1999), pp. 421-441.
7. W. DAHMEN, A. KUNOTH, AND R. SCHNEIDER, Operator equations, multiscale concepts and complexity, in The Mathematics of Numerical Analysis, AMS, Providence, RI, 1996, pp. 225-261.
8. V. DICKEN AND P. MAASS, Wavelet-Galerkin methods for ill-posed problems, J. Inverse Ill-Posed Prob., 4 (1996), pp. 203-221.
9. D. DONOHO, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition, Appl. Comput. Harmon. Anal., 2 (1995), pp. 101-126.
10. M. R. DOSTANIĆ, Asymptotic behavior of the singular values of fractional integral operators, J. Math. Anal. Appl., 175 (1993), pp. 380-391.
11. H. W. ENGL, M. HANKE, AND A. NEUBAUER, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1996.
12. H. W. ENGL AND A. NEUBAUER, Convergence rates for Tikhonov regularization in finite-dimensional subspaces of Hilbert scales, Proc. Amer. Math. Soc., 102 (1988), pp. 587-592.
13. G. GOLUBEV AND R. Z. KHASMINSKII, Statistical Approach to Inverse Boundary Value Problems for Partial Differential Equations, Preprint 370, WIAS Berlin, 1997.
14. M. HEGLAND, Variable Hilbert scale and their interpolation inequalities with applications to Tikhonov regularization, Appl. Anal., 59 (1995), pp. 207-223.
15. I. M. JOHNSTONE AND B. W. SILVERMAN, Discretization effects in statistical inverse problems, J. Complexity, 7 (1991), pp. 1-34.
16. A. P. KOROSTELEV AND A. B. TSYBAKOV, Minimax Theory of Image Reconstruction, Lecture Notes in Statist. 82, Springer, New York, 1993.
17. K. C. LI, Minimaxity of the method of regularization of stochastic processes, Ann. Statist., 10 (1982), pp. 937-942.
18. A. K. LOUIS, Inverse und schlecht gestellte Probleme, B. G. Teubner, Stuttgart, 1989.
19. M. LUKAS, Comparisons of parameter choice methods for regularization with discrete noisy data, Inverse Problems, 14 (1998), pp. 161-184.
20. B. A. MAIR, Tikhonov regularization for finitely and infinitely smoothing operators, SIAM J. Math. Anal., 25 (1994), pp. 135-147.
21. B. A. MAIR AND F. H. RUYMGAART, Statistical inverse estimation in Hilbert scales, SIAM J. Appl. Math., 56 (1996), pp. 1424-1444.
22. P. MATHÉ, s-numbers in information-based complexity, J. Complexity, 6 (1990), pp. 41-66.
23. V. A. MOROZOV, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984.
24. F. NATTERER, Regularisierung schlecht gestellter Probleme durch Projektionsverfahren, Numer. Math., 28 (1977), pp. 329-341.
25. F. NATTERER, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal., 18 (1984), pp. 29-37.
26. A. NEUBAUER, An a posteriori parameter choice for Tikhonov regularization in Hilbert sacles leading to optimal convergence rates, SIAM J. Numer. Anal., 25 (1988), pp. 1313-1326.
27. M. NUSSBAUM, Degrees of Ill-Posedness in Stochastic and Deterministic Noise Models, Report 101, Dagstuhl, 1994.
28. M. NUSSBAUM AND S. V. PEREVERZEV, The Degree of Ill-Posedness in Stochastic and Deterministic Noise Models, Preprint 509, WIAS Berlin, 1999.
29. D. W. NYCHKA AND D. D. COX, Convergence rates for regularized solutions of integral equations from discrete noisy data, Ann. Statist., 17 (1989), pp. 556-572.
30. A. PIETSCH, Operator Ideals, North-Holland, Amsterdam, 1980.
31. S. G. SAMKO, A generalized Abel equation and fractional integration operators, Differencial'nye Uravnenija, 4 (1968), pp. 298-314.
32. S. G. SAMKO, A. A. KILBAS, AND O. I. MARICHEV, Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, S. M. Nikol'skiǐ, ed., Nauka i Tekhnika, Minsk, 1987.
33. J. SARANEN, A modified discrete spectral collocation method for first kind integral equations with logarithmic kernel, J. Integral Equations Appl., 5 (1993) pp. 547-567.
34. E. SCHOCK, Ritz-regularization versus least-square-regularization. Solution methods for integral equations of the first kind, Z. Anal. Anwendungen, 4 (1985), pp. 277-284.
35. P. SPECKMAN, Spline smoothing and optimal rates of convergence in nonparametric regression models, Ann. Statist., 13 (1985), pp. 970-983.
36. U. TAUTENHAHN, Error estimates for regularization methods in Hilbert scales, SIAM J. Numer. Anal., 33 (1996), pp. 2120-2130.
37. A. N. TIKHONOV AND V. Y. ARSENIN, Solutions of ill-posed problems, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977. Translated from the Russian. Preface by translation editor Fritz John, Scripta Series in Mathematics.
38. J. F. TRAUB, G. W. WASILKOWSKI, AND H. WOŹNIAKOWSKI, Information-Based Complexity, Academic Press, Boston, MA, 1988.
39. G. M. VAǏNIKKO AND U. A. KHYAMARIK, Projection methods and self-regularization in ill-posed problems, Izv. Vyssh. Uchebn. Zaved. Mat., 84 (1985), pp. 3-17.
40. G. M. VAǏNIKKO AND A. Y. VERETENNIKOV, Iteratsionnye protsedury v nekorrektnykh zadachakh, Nauka, Moscow, 1986.
41. G. WAHBA, Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal., 14 (1977), pp. 651-667.
42. A. G. WERSCHULZ, The Computational Complexity of Differential and Integral Equations. An Information-Based Approach, The Clarendon Press, Oxford University Press, New York, 1991.