Optimal regularization for ill-posed problems in metric spaces

Journal of Inverse and Ill-Posed Problems

Volumn 15, Issue 2, 2007, Pages 137-148

  Bauer, F ^a   Munk, A ^a  

^a UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

Lepskij type balancing principle; Regularization in normed spaces

Indexed keywords

EID: 34250161643     PISSN: 09280219     EISSN: None     Source Type: Journal    
DOI: 10.1515/JIIP.2007.007     Document Type: Article

References (18)

1. F. Bauer and S. Pereverzev, Regularization without preliminary knowledge of smoothness and error behavior. European J. Appl. Math. 16 (2005), 303-317. (Pubitemid 41742826)
2. N. Bissantz, T. Hohage, and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise. Inverse Problems 20 (2004), 1773-1791.
3. L. Devroye and G. Lugosi, Combinatorial Methods in Density Estimation. Springer Series in Statistics. Springer, New York, 2001.
4. H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems. Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, Boston, London, 1996.
5. P. Hansen, Rank-deficient and discrete ill-posed problems. SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998, Numerical aspects of linear inversion.
6. B. Hofmann, Regularization of Applied Inverse and Ill-Posed Problems. Teubner, Leipzig, 1986.
7. O. V. Lepskiǐ, On a problem of adaptive estimation in Gaussian white noise. Theory of Probability and its Applications 35 (1990), 454-466.
8. B. A. Mair and F. H. Ruymgaart, Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 (1996), 1424-1444.
9. P. Mathé and S. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert spaces. Inverse Problems 19 (2003), 789-803.
10. V. A. Morozov, Methods for Solving Incorrectly Posed Problems. Springer-Verlag, New York, 1984.
11. M. Nikolova, Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares. Multiscale Model. Simul. 4 (2005), 960-991 (electronic). (Pubitemid 44340295)
12. W. Rudin, Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Company, New York, 1973.
13. O. Scherzer, Taut-string algorithm and regularization programs with G-norm data fit. J. Math. Imaging Vision 23 (2005), 135-143. (Pubitemid 41201585)
14. F. Schöpfer, A. K. Louis, and T. Schuster, Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Problems 22 (2006), 311-329. (Pubitemid 43185103)
15. A. V. Skorokhod, Integration in Hilbert space. Springer, Berlin, 1974.
16. A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems. Wiley, New York, 1977.
17. A. van der Vaart and J. Wellner, Weak Convergence and Empirical Processes. With Applications to Statistics. Springer Series in Statistics. Springer, New York, 1996.
18. G. Wahba, Practical approximate solutions to linear operator equations when the data are noisy. SIAM Journal on Numerical Analysis 14 (1977), 651-667.