The use of monotonicity for choosing the regularization parameter in ill-posed problems

Inverse Problems

Volumn 15, Issue 6, 1999, Pages 1487-1505

  Tautenhahn, U ^a   Hamarik U ^b  

^a UNIVERSITY OF APPLIED SCIENCES ZITTAU GÖRLITZ   (Germany)

^b UNIVERSITY OF TARTU   (Estonia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0013298158     PISSN: 02665611     EISSN: None     Source Type: Journal    
DOI: 10.1088/0266-5611/15/6/307     Document Type: Article

References (48)

1. Bakushinskii A B 1984 Remarks on choosing a regularization parameter using the quasi-optimahty and ratio criterion USSR Comput. Math Math Phys. 24 181-2
2. Engl H W 1983 On the convergence of regulanzation methods for ill-posed linear operator equations Improperly Posed Problems and Their Numerical Treatment ed G Hammerlin and K H Hoffmann (Basel: Birkhauser) pp 81-95
3. Engl H W and Gfrerer H 1988 A posteriori parameter choice for general regularization methods for solving linear ill-posed problems Appl. Numer. Math. 4 395-417
4. Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Dordrecht: Kluwer)
5. Fridman V 1959 New methods for solving a linear operator equation Dokl. Akad. Nauk SSSR 128 482-4
6. Gellnch C and Hofmann B 1993 A study of regularization by monotonicity Computing 50 105-25
7. Gfrerer H 1987 An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates Math. Comput. 49 507-22
8. Golub G H, Heath M and Wahba G 1979 Generalized cross validation as a method for choosing a good ridge parameter Technometrics 21 215-23
9. Groetsch C W 1984 The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Boston, MA: Pitman)
10. Groetsch C W and Neubauer A 1989 Regularization of ill-posed problems: optimal parameter choice in finite dimensions J. Approx. Theory 58 184-200
11. Hämarik U 1983 Projection methods for the regularization of linear ill-posed problems Proc. Comput. Center Tartu Univ. 50 69-90 (in Russian)
12. Hämarik U 1984 The discrepancy principle for the choice of dimension for solving ill-posed problems by projection methods Acta Comment Univ Tartuesis 672 27-34 (in Russian)
13. Hämarik U 1984 Regularized projection methods for ill-posed problems Proc. Estontan Acad. Sci. Phys. Math 33 266-76 (in Russian)
14. Hämarik U 1992 On the discretization error in regularized projection method with parameter choice by discrepancy principle Proc. Int. Conf. Ill-Posed Problems in Natural Sciences (Moscow. 1991) ed A N Tikhonov et al (Utrecht: VSP) pp 24-8
15. Hämarik U 1993 On the parameter choice in the regularized Ritz-Galerkin method Proc. Estonian Acad. Sd Phys. Math. 42 133-43
16. Hämarik U 1999 Monotonicity of error and choice of the stopping index in iterative regularization methods Differential and integral Equations: Theory and Numerical Analysis ed A Pedas (Tartu: Estonian Mathematical Society) pp 15-30
17. Hamarik U and Raus T 1999 On the a posteriori parameter choice in regulanzation methods Proc. Estonian Acad Sci. Phys. Math. 48 133-45
18. Hanke M and Hansen P C 1993 Regularization methods for large-scale problems Surv. Math. Ind. 3 253-315
19. Hanke M and Raus T 1996 A general heuristic for choosing the regularization parameter in ill-posed problems SIAM J. Sci. Comput 17 956-72
20. Hansen PC 1992 Analysis of discrete ill-posed problems by means of the L-curve SIAM Rev. 34 561-80
21. Hansen P C 1998 Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (Philadelphia. PA: SIAM)
22. Hansen P C and O'Leary D P 1993 The use of the L-curve in the regularization of discrete ill-posed problems SIAM J. Sci. Comput. 14 1487-503
23. Hofmann B 1986 Regularization for Applied Inverse and Ill-Posed Problems (Leipzig: Teubner)
24. Ivanov V V and Kudrinsky V J 1966 Approximate solution of linear operator equations in Hilbert spaces by the method of least squares USSR Comput Math. Math Phys 6 60-75
25. King J T and Neubauer A 1988 A variant of finite dimensional Tikhonov regulanzation with a-posterion parameter choice Computing 40 91-109
26. Leonov AS 1991 On the accuracy of Tikhonov regularizing algorithms and quasioptimal selection of a regularization parameter Sov. Math. Dokl 44 711-6
27. Louis A K 1989 Inverse und schlecht gestellte Probleme (Stuttgart: Teubner)
28. Lukas M A 1998 Asymptotic behaviour of the minimum bound method for choosing the regularization parameter Inverse Problems 14 149-59
29. Morozov V A 1966 On the solution of functional equations by the method of regularization Sov. Math. Dokl. 7 414-7
30. Natterer F 1977 Regularizierung schlecht gestellter Probleme durch Projektionsverfahren Numer. Math 28 329-41
31. Neubauer A 1988 An a-posteriori parameter choice for Tikhonov regularization in presence of modelling error Appl Numer. Math. 4 507-19
32. Neubauer A 1994 On converse and saturation results for regulanzation methods Beiträge zur Angewandten Analysis und Informatik, Helmut Brakhage zu Ehren ed E Schock (Aachen Shaker) pp 262-70
33. Plato R and Hamarik U 1996 On pseudo-optimal parameter choices and stopping rules for regularization methods in Banach spaces Numer Funct. Anal Optimiz. 17 181-95
34. Plato R and Vainikko G M 1990 On the regularization of projection methods for solving ill-posed problems Numer. Math. 57 63-79
35. Raus T 1984 On the discrepancy principle for the solution of ill-posed problems. Acta Comment, Univ Tartuensis 672 16-26 (in Russian)
36. Raus T 1985 On the discrepancy principle for solution of ill-posed problems with non-selfadjoint operators Acta Comment. Univ. Tartuensis 715 12-20 (in Russian)
37. Raus T 1992 On the regularization parameter choice for ill-posed problems with a quasisolution Proc. Int. Conf Ill-Posed Problems in Natural Sciences (Moscow) ed A N Tikhonov el al (Utrecht: VSP) pp 144-8
38. Raus T 1992 About regularization parameter choice in case of approximately given error bounds of data Acta Comment. Univ. Tartuensis 937 77-89
39. Scherzer O, Engl H W and Kunisch K 1993 Optimal a posteriori parameter choice for Tikhonov regularizauon for solving nonlinear ill-posed problems SIAM J. Numer. Anal 30 1796-838
40. Schröter T and Tautenhahn U 1994 On the optimality of regularization methods for solving linear ill-posed problems Z. Anal. Anw. 13 697-710
41. Tautenhahn U 1999 On a new parameter choice rule for ill-posed inverse problems HERCMA'98: Proc. Int. Conf. (Athen, 1998) ed E A Lipitakis (Athens: LEA) to appear
42. Tikhonov A N and Arsenin V Y 1977 Solution of Ill-Posed Problems (New York: Wiley)
43. Tikhonov A N and Glasko V B 1964 On the approximate solution of Fredholm integral equations of the first kind USSR Comput. Math Math. Phys. 4 564-71
44. Vainikko G M 1982 The discrepancy principle for a class of regularization methods USSR Comput. Math. Math. Phys. 22 1-19
45. Vainikko G M 1987 On the optimality of methods for ill-posed problems Z Anal. Anw. 6 351-62
46. Vainikko G M and Hämarik U 1985 Projection methods and self-regularization in ill-posed problems Sov. Math. 29 1-20
47. Vainikko G M and Veretennikov A Y 1986 Iteration Procedures in Ill-Posed Problems (Moscow Nauka) (in Russian)
48. Wahba G 1977 Practical approximate solutions to linear operator equations when the data are noisy SIAM J Numer. Anal. 14 651-67