Approximate source conditions in Tikhonov-Phillips regularization and consequences for inverse problems with multiplication operators

Mathematical Methods in the Applied Sciences

Volumn 29, Issue 3, 2006, Pages 351-371

  Hofmann, Bernd ^a  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Convergence rates; Distance function; Inverse and ill posed problems; Multiplication operator; Source conditions; Superposition operator; Tikhonov Phillips regularization

Indexed keywords

FUNCTIONS; LINEAR EQUATIONS; MATHEMATICAL OPERATORS; NATURAL SCIENCES;

LINEAR OPERATORS; REGULARIZATION; SOURCE CONDITIONS;

INVERSE PROBLEMS;

EID: 30844472201     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.686     Document Type: Article

References (38)

1. Hofmann B, Scherzer O. Factors influencing the ill-posedness of nonlinear problems. Inverse Problems 1994; 10:1277-1297.
2. Bakushinsky A, Goncharsky A. Ill-Posed Problems: Theory and Applications. Kluwer Academic Publishers: Dordrecht, 1994.
3. Baumeister J. Stable Solution of Inverse Problems. Vieweg: Braunschweig, 1987.
4. Engl HW, Hanke M, Neubauer A. Regularization of Inverse Problems. Kluwer Academic Publishers: Dordrecht, 1996.
5. Groetsch CW. The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind. Pitman: Boston, 1984.
6. Kirsch A. An Introduction to the Mathematical Theory of Inverse Problems. Springer: New York, 1996.
7. Kress R. Linear Integral Equations. Springer: Berlin, 1989.
8. Louis AK. Inverse und schlecht gestellte Probleme. Teubner: Stuttgart, 1989.
9. Tikhonov AN, Arsenin VY. Solutions of Ill-Posed Problems. Wiley: New York, 1997.
10. Vainikko GM, Vertennikov AY. Iteration Procedures in Ill-Posed Problems. Nauka: Moscow, 1986 (in Russian).
11. Schock E. Approximate solution of ill-posed equations: arbitrarily slow convergence vs. superconvergence. In Constructive Methods for the Practical Treatment of Integral Equations, Hämmerlin G, Hoffmann KH (eds). Birkhäuser: Basel, 1985; 234-243.
12. Böttcher A, Hoftmnn B, Tautenhahn U, Yamamoto M. Convergence rates for Tikhonov regularization from different kinds of smoothness conditions. Preprint 2005-9, Preprintreihe der Fakultät für Mathematik der TU Chemnitz, 2005. http://www.mathematik.tu-chemnitz.de/preprint/2005/PREPRINT.09.html.
13. Hegland M. Variable Hubert scales and their interpolation inequalities with applications to Tikhonov regularization. Applicable Analysis 1995; 59:207-223.
14. Hofmann B, Yamamoto M. Convergence rates for Tikhonov regularization based on range inclusions. Inverse Problems 2005; 21:805-820.
15. Hohage T. Regularization of exponentially ill-posed problems. Numerical Functional Analysis and Optimization 2000; 21:439-464.
16. Mair BA. Tikhonov regularization for finitely and infinitely smoothing operators. SIAM Journal on Mathematical Analysis 1994; 25:135-147.
17. Mathé P, Pereverzev SV. Discretization strategy for linear ill-posed problems in variable Hubert scales. Inverse Problems 2003; 19:1263-1277.
18. Nair MT, Schock E, Tautenhahn U. Morozov's discrepancy principle under general source conditions. Zeitschrift für Analysis und ihre Anwendungen 2003; 22:199-214.
19. Tautenhahn U. Optimality for ill-posed problems under general source conditions. Numerical Functional Analysis and Optimization 1998; 19:377-398.
20. Mathé P, Pereverzev SV. Geometry of linear ill-posed problems in variable Hubert scales. Inverse Problems 2003; 19:789-803.
21. Math́ P, Pereverzev SV. Regularization of some linear ill-posed problems with discretized random noisy data. Mathematics of Computation, in press.
22. Neubauer A. On converse and saturation results for Tikhonov regularization of linear ill-posed problems. SIAM Journal on Numerical Analysis 1997; 34:517-527.
23. Mathé P, Pereverzev SV. Moduli of continuity for operator valued functions. Numerical Functional Analysis and Optimization 2002; 23:623-631.
24. Bakushinsky AB, Kokurin MYu. Iterative Methods for Approximate Solutions of Inverse Problems. Springer: Dordrecht, 2004.
25. Hofmann B, Fleischer G. Stability rates for linear ill-posed problems with compact and non-compact operators. Zeitschrift für -Analysis und ihre Anwendungen 1999; 18:267-286.
26. Zeidler E (ed.). Teubner-Taschenbuch der Mathematik. Teubner: Leipzig, 1996.
27. Hofmann B. A local stability analysis of nonlinear inverse problems. In Inverse Problems in Engineering-Theory and Practice, Delaunay D et al. (eds). The American Society of Mechanical Engineers: New York, 1998; 313-320.
28. Hofmann B. Mathematik inverser Probleme. Teubner: Stuttgart, 1999.
29. Hein T, Hofmann B. On the nature of ill-posedness of an inverse problem arising in option pricing. Inverse Problems 2003; 19:1319-1338.
30. Hofmann B, Krämer R. On maximum entropy regularization for a specific inverse problem of option pricing. Journal of Inverse and Ill-Posed Problems 2005; 13:41-63.
31. Groetsch CW. Inverse Problems in the Mathematical Sciences. Vieweg: Braunschweig, 1993.
32. Anger G. Inverse Problems in Differential Equations. Akademie: Berlin, 1990.
33. Appell J, Zabrejko PP. Nonlinear Superposition Operators. Cambridge University Press: Cambridge, 1990.
34. Kwok YK. Mathematical Models of Financial Derivatives. Springer: Singapore, 1998.
35. Hofmann B. The potential for ill-posedness of multiplication operators occurring in inverse problems. Preprint 2004-17, Preprintreihe der Fakultät für Mathematik der TU Chemnitz, 2004. http://www.mathematik. tu-chemnitz.de/preprint/2004/PREPRINT.17.html.
36. Hofmann B. Regularization for Applied Inverse and Ill-Posed Problems. Teubner: Leipzig, 1986.
37. Hofmann B, von Wolfersdorf L. Some results and a conjecture on the degree of ill-posedness for integration operators with weights. Inverse Problems 2005; 21:427-433.
38. Freitag M, Hofmann B. Analytical and numerical studies on the influence of multiplication operators for the ill-posedness of inverse problems. Journal of Inverse and Ill-Posed Problems 2005; 13:123-148.