On Tikhonov regularization in Banach spaces - optimal convergence rates results

Applicable Analysis

Volumn 88, Issue 5, 2009, Pages 653-667

  Hein, Torsten ^a  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Banach space; Bregman distance; Convergence rates; Inverse problem; Regularization

Indexed keywords

EID: 70449637983     PISSN: 00036811     EISSN: 1563504X     Source Type: Journal    
DOI: 10.1080/00036810802555474     Document Type: Article

References (15)

1. M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Probl. 20 (2004), pp. 1411-1421.
2. T. Hein, Convergence rates for regularization of ill-posed problems in Banach spaces by approximate source conditions, Inverse Probl. 24 (2008), p. 045-007, to appear.
3. H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dortrecht, 1996.
4. E. Resmerita, Regularization of ill-posed problems in Banach spaces: Convergence rates, Inverse Probl. 21 (2005), pp. 1303-1314.
5. E. Resmerita and O. Scherzer, Error estimates for non-quadratic regularization and the relation to enhancement, Inverse Probl. 22 (2006), pp. 801-814.
6. B. Hofmann, B. Kaltenbacher, C. Poschl, and O. Scherzer, A convergence rates result in Banach spaces with non-smooth operators, Inverse Probl. 23 (2007), pp. 987-1010.
7. F. Schopfer, A.K. Louis, and T. Schuster, Nonlinear iterative methods for linear ill-posed problems in Banach spaces, Inverse Probl. 22 (2006), pp. 311-329.
8. T. Hein, Regularization of ill-posed problems in Banach spaces-approximative source conditions and convergence rates results. Preprint 2007-28, TU Chemnitz, Faculty of Mathematics (Accepted for publication).
9. H.W. Engl, K. Kunisch, and A. Neubauer, Convergence rates for Tikhonov regularization of non-linear ill-posed problems, Inverse Probl. 5 (1989), pp. 523-540.
10. T.I. Seidman and C.R. Vogel, Well posedness and convergence of some regularisation methods for non-linear ill-posed problems, Inverse Probl. 5 (1989), pp. 227-238.
11. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dortrecht, 1990.
12. Z-B. Xu and G.F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl 157 (1991), pp. 189-210.
13. O.V. Lepskij, A problem of adaptive estimation in Gaussian white noise, Teor. Veroyatn. Primen 35 (1990), pp. 459-470.
14. P. Mathe, The Lepskij principle revisited, Inverse Probl. 22 (2006), pp. L11-L15.
15. F. Bauer and T. Hohage, A Lepskij-type stopping rule for regularized Newton methods, Inverse Probl. 21 (2005), pp. 1979-1991.