A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions

Inverse Problems

Volumn 14, Issue 5, 1998, Pages 1081-1106

  Deuflhard, Peter ^a,b   Engl, Heinz W ^c   Scherzer, Otmar ^c  

^a ZUSE INSTITUTE BERLIN   (Germany)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

^c JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0345875999     PISSN: 02665611     EISSN: None     Source Type: Journal    
DOI: 10.1088/0266-5611/14/5/002     Document Type: Article

References (25)

1. [Bak92] Bakushinskii A B 1992 The problem of the convergence of the iteratively regularized Gauss-Newton method Comput. Math. Math. Phys. 32 1353-9
2. [BD96] Bornemann F A and Deuflhard P 1996 The cascadic multigrid method for elliptic problems Numer. Math. 75 135-52
3. [BG94] Bakushinsky A B and Goncharsky A V 1994 Ill-Posed Problems: Theory and Applications (Dordrecht: Kluwer Academic)
4. [BHS96] Binder A, Hanke M and Scherzer O 1996 On the Landweber iteration for nonlinear ill-posed problems J. Inverse Ill-Posed Problems 4 381-9
5. [Bla96] Blaschke B 1996 Some Newton Type Methods for the Regularization of Nonlinear Ill-Posed Problems (Linz: Universitätsverlag Rudolf Trauner) (PhD Thesis Schriften der Johannes-Kepler-Universität)
6. [BNS97] Blaschke B, Neubauer A and Scherzer O 1997 On convergence rates for the iteratively regularized Gauss-Newton method IMA J. Numer. Anal. 17 421-36
7. [DH79] Deuflhard P and Heindl G 1979 Affine invariant convergence theorems for Newton's method and extensions to related methods SIAM J. Numer. Anal. 16 1-10
8. [EHN96] Engl H W and Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Dordrecht: Kluwer Academic)
9. [EKN89] Engl H W, Kunisch K and Neubauer A 1989 Convergence rates for Tikhonov regularization of nonlinear ill-posed problems Inverse Problems 5 523-40
10. [Han97] Hanke M 1997 A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems Inverse Problems 13 79-95
11. [HNS95] Hanke M, Neubauer A and Scherzer O 1995 A convergence analysis of Landweber iteration for nonlinear ill-posed problems Numer. Math. 72 21-37
12. [Hof93] Hofmann B 1993 Regularization of nonlinear problems and the degree of ill-posedness Inverse Problems: Principles and Applications in Geophysics, Technology, and Medicine (Proc. Int. Conf. (Potsdam)) ed G Anger et al (Berlin: Akademie) pp 174-88
13. [Hoh97] Hohage T 1997 Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem Inverse Problems 13 1279-300
14. [HR96] Hettlich F and Rundell W 1996 Iterative methods for the reconstruction of an inverse potential problem Inverse Problems 12 251-66
15. [Kal98] Kaltenbacher B 1998 On Broyden's method for nonlinear ill-posed problems Numer. Funct. Anal. Opt. to appear
16. [LN96a] Liu F and Nashed M Z 1996 Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators Lecture Notes in Pure and Applied Mathematics 177 ed P Marcellini et al (New York: Dekker) pp 353-61 (Partial differential equations and applications, collected papers in honor of Carlo Pucci on the occasion of his 70th birthday)
17. [LN96b] Liu F and Nashed M Z 1996 On nonlinear ill-posed problems II: monotone operator equations and monotone variational inequalities Lecture Notes in Pure and Applied Mathematics 178 ed A G Kartsatos (New York: Dekker) pp 223-40 (Theory and applications of nonlinear operators of accretive and monotone type)
18. [Lou89] Louis A K 1989 Inverse und Schlecht Gestellte Probleme (Stuttgart: Teubner)
19. [Nas76] Nashed M Z 1976 Generalized Inverses and Applications (New York: Academic)
20. [NS90] Neubauer A and Scherzer O 1990 Finite-dimensional approximation of Tikhonov regularized solutions of non-linear ill-posed problems Numer. Funct. Anal. Opt. 11 85-99
21. [PR86] Potra F A and Rheinboldt W C 1986 On the monotone convergence of Newton's method Computing 36 81-90
22. [Ram97] Ramlau R 1997 A modified Landweber-method for inverse problems Report Universität Potsdam,
23. Germany [Sch85] Schock E 1985 Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence Constructive Methods for the Practical Treatment of Integral Equations (Proc. Conf. Oberwolfach, 1984) (ISNM 73) pp 234-43
24. [Sch97] Scherzer O 1997 A multi-level algorithm for the solution of nonlinear ill-posed problems Numer. Math. to appear
25. [VV86] Vainikko G M and Veretennikov A Y 1986 Iteration Procedures in Ill-Posed Problems (Moscow: Nauka) (in Russian)