Error estimates for regularization methods in Hilbert scales

SIAM Journal on Numerical Analysis

Volumn 33, Issue 6, 1996, Pages 2120-2130

  Tautenhahn, Ulrich ^a  

^a UNIVERSITY OF APPLIED SCIENCES ZITTAU GÖRLITZ   (Germany)

Author keywords

A priori parameter choice; Discrepancy principle; General regularization scheme; Hilbert scales; Linear ill posed problems; Order optimal error bounds

Indexed keywords

EID: 0000414254     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142994269411     Document Type: Article

References (23)

1. J. BAUMEISTER, Stable Solution of Inverse Problems, Vieweg, Braunschweig, 1987.
2. H. W. ENGL, Regularization methods for the stable solution of inverse problems, Surv. Math. Ind., 3 (1993), pp. 71-143.
3. H. W. ENGL AND H. GFRERER, A posteriori parameter choice for general regularization methods for solving linear ill-posed problems. Appl. Numer. Math., 4 (1988), pp. 395-417.
4. R. GORENFLO AND S. VESSELLA, Abel Integral Equations - Analysis and Applications, Lecture Notes in Mathematics, Vol. 1461, Springer, Berlin, Heidelberg, 1991.
5. C. W. GROETSCH, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.
6. M. HANKE, Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math., 60 (1991), pp. 341-373.
7. M. HEGLAND, An optimal order regularization method which does not use additional smoothness assumptions, SIAM J. Numer. Anal., 29 (1992), pp. 1447-1461.
8. B. HOFMANN, Regularization for Applied Inverse and Ill-Posed Problems, Teubner, Leipzig, 1986.
9. S. KREIN AND Y. I. PETUNIN, Scales of Banach spaces, Russian Math. Surveys, 21 (1966), pp. 85-159.
10. J. LOCKER AND P. M. PRENTER, Regularization with differential operators. I. General theory. J. Math. Anal. Appl., 74 (1980), pp. 504-529.
11. A. K. LOUIS, Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989.
12. V. A. MOROZOV, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), pp. 414-417.
13. F. NATTERER, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal., 18 (1984), pp. 29-37.
14. A. NEUBAUER, Ana posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates, SIAM J. Numer. Anal., 25 (1988), pp. 1313-1326.
15. R. PLATO AND G. M. VAINIKKO, On the regularization of projection methods for solving ill-posed problems, Numer. Math., 57 (1990), pp. 63-79.
16. T. SCHRÖTER AND U. TAUTENHAHN, Error estimates for Tikhonov regularization in Hilbert scales, Numer. Funct. Anal. Optim., 15 (1994), pp. 155-168.
17. _, On the optimality of regularization methods for solving linear ill-posed problems, Z. Anal. Anwend., 13 (1994), pp. 697-710.
18. U. TAUTENHAHN, Regularisierungsverfahren für lineare und nichtlineare nichtkorrekte inverse Aufgaben, Habil., Techn. Univ. Chemnitz-Zwickau, 1994.
19. A. N. TIKHONOV AND V. Y. ARSENIN, Solution of Ill-Posed Problems, Wiley, New York, 1977.
20. G. M. VAINIKKO, The critical level of discrepancy in regularization methods, U.S.S.R. Comput. Math. and Math. Phys., 23 (1983), pp. 1-9.
21. _, On the optimality of methods for ill-posed problems, Z. Anal. Anwend., 6 (1987), pp. 351-362.
22. G. M. VAINIKKO AND A. Y. VERETENNIKOV, Iteration Procedures in Ill-Posed Problems, Nauka, Moscow, 1986. (In Russian.)
23. G. WAHBA, Ill-posed problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill-posed Problems with Noisy Data, Technical report 595, Univ. Wisconsin, Madison, WI, 1980.