An analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption

Inverse Problems

Volumn 23, Issue 1, 2007, Pages 217-230

  Lu, Shuai ^a   Pereverzev, Sergei V ^a   Ramlau, Ronny ^a  

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

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; HILBERT SPACES; OPTIMIZATION; PARAMETER ESTIMATION; PROBLEM SOLVING;

NONLINEAR ILL-POSED PROBLEMS; NONLINEAR OPERATORS; REGULARIZED APPROXIMATION; TIKHONOV FUNCTIONALS;

INVERSE PROBLEMS;

EID: 33947620567     PISSN: 02665611     EISSN: 13616420     Source Type: Journal    
DOI: 10.1088/0266-5611/23/1/011     Document Type: Article

References (24)

1. Bakushinskii A and Smirnova A 2005 On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems Numer. Funct. Anal. Optim. 26 35-48
2. Bauer F and Hohage T 2005 A Lepskij-type stopping rule for regularized Newton methods Inverse Problems 21 1975-91
3. Deuflhard P, Engl H W and Scherzer O 1998 A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions Inverse Problems 14 1081-106
4. Engl H W, Hanke M and Neubauer A 1996 Regularization in Inverse Problems (Dordrecht: Kluwer)
5. Engl H W, Kunish K and Neubauer A 1989 Convergence rates for Tikhonov regularization of non-linear ill-posed problems Inverse Problems 5 523-40
6. Hanke M, Neubauer A and Scherzer O 1995 A convergence analysis of the Landweber iteration for nonlinear ill-posed problems Numer. Math. 72 21-37
7. Hofmann B 1994 On the degree of ill-posedness for nonlinear problems J. Inv. Ill-Posed Probl. 2 61-76
8. Hofmann B and Scherzer O 1994 Factors influencing the ill-posedness of nonlinear problems Inverse Problems 10 1277-97
9. Mathé P 2006 The Lepskii principle revisited Inverse Problems 22 L11-5
10. Mathé P and Pereverzev S V 2003 Geometry of linear ill-posed problems in variable Hilbert scales Inverse Problems 19 789-803
11. Natterer F 1984 Error bounds for Tikhonov regularization in Hilbert scales Appl. Anal. 18 29-37
12. Neubauer A 1989 Tikhonov regularization for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation Inverse Problems 5 541-7
13. Neubauer A 2000 On Landweber iteration for nonlinear ill-posed problems in Hilbert scales Numer. Math. 85 309-28
14. Pereverzev S and Schock E 2005 On the adaptive selection of the parameter in regularization of ill-posed problems SIAM J. Numer. Anal. 43 2060-76
15. Pereverzyev S S, Pinnau R and Siedow N 2006 Regularized fixed-point iterations for nonlinear inverse problems Inverse Problems 22 1-22
16. Ramlau R 2002 A steepest descent algorithm for the global minimization of Tikhonov Phillips functional Inverse Problems 18 381-405
17. Ramlau R 2003 TIGRA-an iterative algorithm for regularizing nonlinear ill-posed problems Inverse Problems 19 433-65
18. Ramlau R 2005 On the use of fixed point iterations for the regularization of nonlinear ill-posed problems J. Inv. Ill-posed Probl. 13 175-200
19. Ramlau R and Teschke G 2005 Tikhonov replacement functionals for iteratively solving nonlinear operator equations Inverse Problems 21 1571-92
20. Scherzer O 1993 The use of Morozov's discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems Computing 51 45-60
21. Scherzer O, Engl H W and Kunish K 1993 Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems SIAM J. Numer. Anal. 30 1796-838
22. Seidman T I and Vogel C R 1989 Well posedness and convergence of some regularisation methods for non-linear ill posed problems Inverse Problems 5 227-38
23. Tautenhahn U and Jin Q N 2003 Tikhonov regularization and a posteriori rules for solving nonlinear ill-posed problems Inverse Problems 19 1-21
24. Tikhonov A N, Leonov A S and Yagola A G 1998 Nonlinear ill-posed problems: vols 1, 2 Applied Mathematics and Mathematical Computation p 14 (Translated from the Russian)