A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators

Inverse Problems

Volumn 23, Issue 3, 2007, Pages 987-1010

  Hofmann, B ^a   Kaltenbacher, B ^b   Poschl C ^c   Scherzer, O ^c  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF STUTTGART   (Germany)

^c UNIVERSITY OF INNSBRUCK   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; HILBERT SPACES; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; OPTIMIZATION; PROBLEM SOLVING; VARIATIONAL TECHNIQUES;

CONVERGENCE RATE; PHASE RETRIEVAL PROBLEMS; VARIATIONAL INEQUALITY;

BANACH SPACES;

EID: 34249721452     PISSN: 02665611     EISSN: 13616420     Source Type: Journal    
DOI: 10.1088/0266-5611/23/3/009     Document Type: Article

References (40)

1. Adams R A 1975 Sobolev Spaces (New York: Academic)
2. Appell J and Zabrejko P P 1990 Nonlinear Superposition Operators (Cambridge: Cambridge University Press)
3. Avellaneda M, Friedman C, Holmes R and Samperi D 1997 Calibrating volatility surfaces via relative entropy minimization Appl. Math. Finance 4 3-64
4. Blaschke B, Engl H W, Grever W and Klibanov M V 1996 An application of Tikhonov regularization to phase retrieval Nonlinear World 3 771-86
5. Blaschke-Kaltenbacher B and Engl H W 1997 Regularization methods for nonlinear ill-posed problems with applications to phase reconstruction Inverse Problems in Medical Imaging and Nondestructive Testing ed H W Engl, A K Louis and W Rundell (New York: Springer) pp 1-18
6. Borwein J M and Lewis A S 1991 Convergence of best entropy estimates SIAM J. Optim. 1 191-205
7. Bouchouev I and Isakov V 1997 The inverse problem of option pricing Inverse Problems 13 L1-7
8. Bouchouev I and Isakov V 1999 Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets Invere Problems 15 R95-116
9. Burger M and Osher S 2004 Convergence rates of convex variational regularization Inverse Problems 20 1411-21
10. Butnariu D and Iusem A N 2000 Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization (Dordrecht: Kluwer)
11. Butnariu D and Resmerita E 2001 The outer Bregman projection method for solving stochastic convex feasibility problems in Banach spaces Inherently Parallel Algorithms in Feasibility and Optimization and their Applications ed D Butnariu, Y Censor and S Reich (Amsterdam: Elsevier) pp 69-86
12. Butnariu D and Resmerita E 2006 Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces Abstract and Applied Analysis 2006 84919 (39 pages)
13. Chadan K and Sabatier P C 1989 Inverse Problems in Quantum Scattering Theory (New York: Springer)
14. Champeney D C 1987 A Handbook of Fourier Theorems (Cambridge: Cambridge University Press)
15. Chavent G and Kunisch K 1997 Regularization of linear least squares problems by total bounded variation ESAIM Control Optim. Calc. Var. 2 359-76 (electronic)
16. Cont R and Tankov P 2006 Retrieving Lévy processes from option prices: regularization of an il-posed inverse problem SIAM J. Control Optim. 45 1-25
17. Crépey S 2003 Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization SIAM J. Math. Anal. 34 1183-206
18. Dobson D 1992 Phase reconstruction via nonlinear least squares Inverse Problems 8 541-57
19. Egger H and Engl H W 2005 Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates Inverse Problems 21 1027-45
20. Egger H, Hein T and Hofmann B 2006 On decoupling of volatility smile and term structure in inverse option pricing Inverse Problems 22 1247-59
21. Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Dordrecht: Kluwer)
22. Engl H W, Kunisch K and Neubauer A 1989 Convergence rates for Tikhonov regularization of nonlinear ill-posed problems Inverse Problems 5 523-40
23. Engl H W and Zou J 2000 A new approach to convergence rate analysis of Tikhonov regularization Inverse Problems 16 1907-23
24. Felcher G and Russell T 1991 Proc. Workshop on Methods of Analysis and Interpretation of Neutron Reflectivity Data Physica B vol 173
25. Hanke M and Rösler E 2005 Computation of local volatilities from regularized Dupire equations Int. J. Theor. Appl. Finance 8 207-22
26. Hein T and Hofmann B 2003 On the nature of ill-posedness of an inverse problem arising in option pricing Inverse Problems 19 1319-38
27. Hofmann B and Krämer R 2005 On maximum entropy regularization for a specific inverse problem of option pricing J. Inv. Ill-Posed Problems 13 41-63
28. Hofmann B and Scherzer O 1994 Factors influencing the ill-posedness of nonlinear problems Inverse Problems 10 1277-97
29. Hurt N E 1989 Phase Retrieval and Zero Crossings (Dordrecht: Kluwer)
30. Jackson N, Süli E and Howison S 1998 Computation of deterministic volatility surfaces J. Math. Finance 2 5-32
31. Klibanov M V and Sacks P E 1992 Phaseless inverse scattering and the phase problem in optics J. Math. Phys. 33 3913-821
32. Klibanov M V, Sacks P E and Tikhonravov A V 1995 The phase retrieval problem Inverse Problems 11 1-28
33. Kwok Y K 1998 Mathematical Models of Financial Derivatives (Singapore: Springer)
34. Lagnado R and Osher S 1997 A technique for calibrating derivative security pricing models: numerical solution of the inverse problem J. Comput. Finance 1 13-25
35. Lions J L and Magenes E 1972 Non-Homogeneous Boundary Value Problems and Applications I-III (New York: Springer)
36. Lishang J and Youshan T 2001 Identifying the volatility of underlying assets from option prices Inverse Problems 17 137-55
37. Morozov V A 1984 Methods for Solving Incorrectly Posed Problems (Berlin: Springer)
38. Resmerita E and Scherzer O 2006 Error estimates for non-quadratic regularization and the relation to enhancing Inverse Problems 22 801-14
39. Schöpfer F, Louis A K and Schuster T 2006 Nonlinear iterative methods for linear ill-posed problems in Banach space Inverse Problems 22 311-29
40. Zubelli J P 2005 Inverse problems in finance: a short survey of calibration techniques Proc. 2nd Brazilian Conf. on Statistical Modelling in Insurance and Finance (Maresias, Brazil) ed N Kolev and P Morettin (So Paulo: Institute of Mathematics and Statistics, University of So Paulo) pp 64-75