A multi-grid method with a priori and a posteriori level choice for the regularization of nonlinear ill-posed problems

Numerische Mathematik

Volumn 93, Issue 1, 2002, Pages 77-107

  Kaltenbacher, Barbara ^a   Schicho, Josef ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036432049     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02679438     Document Type: Article

References (29)

1. D. Braess, W. Hackbusch: A new convergence proof for the multi-grid method including the V-cycle. SIAM J. Numer. Anal. 20, 967-975 (1983)
2. J. Bramble: Multigrid methods. Pitman Research Notes in Mathematics Series 294, Longman Scientific & Technical, Harlow, UK, 1993
3. A. Brandt: Multilevel adaptive solutions to boundary value problems Math. Comp. 31, 333-390 (1977)
4. G.G. Ciarlet: The Finite Element Method for Elliptic Problems, North Holland, Amsterdam - New York - Oxford, 1978
5. P. Deuflhard, H.W. Engl, O. Scherzer: A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Problems 14, 1081-1106 (1998)
6. P. Deufihard, G. Heindl: Affine invariant convergence theorems for Newton's method and extensions to related methods. SIAM J. Numer. Anal. 16, 1-10 (1979)
7. H.W. Engl: Integralgleichungen. Springer, Wien - New York, 1997
8. H.W. Engl, A. Neubauer: On projection methods for solving linear ill-posed problems. In: A. Vogel, ed., Model Optimization in Exploration Geophysics, Vieweg, Braun-schweig, 1987, 73-92
9. H.W. Engl, M. Hanke, A. Neubauer: Regularization of Inverse Problems. Kluwer Dordrecht, 1996
10. C.W. Groetsch: Inverse Problems in Mathematical Sciences. Vieweg, Braunschweig, 1993
11. C.W. Groetsch, A. Neubauer: Convergence of a general projection method for an operator equation of the first kind. Houston J. Mathem. 14, 201-207 (1988)
12. W. Hackbusch: Multi-Grid Methods and Applications. Springer, Berlin, 1985
13. M. Hanke, A. Neubauer, O. Scherzer: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21-37 (1995)
14. M. Hanke, C.R. Vogel: Two-level preconditioners for regularized inverse problems I: Theory. Numer. Math. 83, 385-402 (1999)
15. B. Blaschke-Kaltenbacher, H.W. Engl: Regularization methods for nonlinear ill-posed problems with applications to phase reconstruction. H.W. Engl, A.K. Louis, W. Rundell, eds., Inverse Problems in Medical Imaging and Nondestructive Testing, Springer, Vienna-New York, 1997
16. B. Kaltenbacher: On Broyden's method for nonlinear ill-posed problems. Numerical Functional Analysis and Optimization 19, 807-833 (1998)
17. B. Kaltenbacher: A projection-regularized Newton method for nonlinear ill-posed problems with application to parameter identification problems with finite element discretization. SIAM J. Numer. Anal. 37 (2000) 1885-1908
18. B. Kaltenbacher: On the regularizing properties of a full multigrid method for ill-posed problems. Inverse Problems 17 (2001), special issue to celebrate Pierre Sabatier's 65th birthday, 767-788
19. B. Kaltenbacher: Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems. Inverse Problems 16, 1523-1539 (2000)
20. J.T. King: Multilevel algorithms for ill-posed problems. Numer. Math. 61, 311-334 (1992)
21. A.K. Louis: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart, 1989
22. V. A. Morozov: Regularization Methods for Ill-Posed Problems. CRC Press, Boca Raton, 1993
23. F. Natterer: Regularisierung schlecht gestellter Probleme durch Projektionsverfahren. Numer. Math. 28, 329-341 (1977)
24. F. Natterer: The Mathematics of Computerized Tomography. Teubner, Stuttgart, 1986
25. A. Rieder: A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization. Numer. Math. 75, 501-522 (1997)
26. O. Scherzer: Convergence criteria of iterative methods based on Landweber iteration for nonlinear problems. J. Math. Anal. Appl. 194, 911-933 (1995)
27. O. Scherzer, H.W. Engl, K. Kunisch: Optimal a posteriori parameter choice for Tikhonov regularization leading to optimal convergence rates. SIAM J. Numer. Anal. 30, 1796-1838 (1993)
28. A.N. Tikhonov, V.A. Arsenin: Methods for Solving Ill-Posed Problems. Nauka, Moskau, 1979
29. G.M. Vainikko, A. Y. Veterennikov: Iteration Procedures in III-Posed Problems. Nauka, Moscow, 1986. In Russian