135
Views
4
CrossRef citations to date
0
Altmetric
Original Articles

Approximate source conditions in Tikhonov regularization‐new analytical results and some numerical studies

, &
Pages 41-56 | Received 03 Nov 2005, Published online: 14 Oct 2010

References

  • Bakushinksy , A.B. and Kokurin , M.Yu. 2004 . Iterative Methods for Approximate Solutions of Inverse Problems Springer, Dordrecht
  • Baumeister , J. 1987 . Stable Solution of Inverse Problems , Braunschweig : Vieweg .
  • Boettcher , A. , Hofmann , B. , Tautenhahn , U. and Yamamoto , M. 2006 . Convergence rates for Tikhonov regularization from different kinds of smoothness conditions . Applicable Analysis , 85 In press. Available electronically as DOI: 10.1080/00036810500474838
  • Engl , H.W. , Hanke , M. and Neubauer , A. 1996 . Regularization of Inverse Problems , Dordrecht : Kluwer .
  • Preitag , M. and Hofmann , B. 2005 . Analytical and numerical studies on the influence of multiplication operators for the ill‐posedness of inverse problems . Journal of Inverse and Ill‐Posed Problems , 13 : 123 – 148 .
  • Groetsch , C.W. 1984 . The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind , Boston : Pitman .
  • Hofmann , B. Approximate source conditions in regularization and an application to multiplication operators . Mathematical Modelling and Analysis 2005. Proc. 10th Int. Conf. MMA2005 & CMAM2, Trakai, Lithuania, 2005 . Edited by: Čiegis , R. pp. 29 – 34 . Vilnius : Technika .
  • Hofmann , B. 2006 . Approximate source conditions in Tikhonov‐Phillips regularization and consequences for inverse problems with multiplication operators . Mathematical Methods in the Applied Sciences , 29 : 351 – 371 .
  • Hofmann , B. and Wolfersdorf , L. von . 2005 . Some results and a conjecture on the degree of ill‐posedness for integration operators with weights . Inverse Problems , 21 : 427 – 433 .
  • Hofmann , B. and Yamamoto , M. 2005 . Convergence rates for Tikhonov regularization based on range inclusions . Inverse Problems , 21 : 805 – 820 .
  • Hohage , T. 2000 . Regularization of exponentially ill‐posed problems . Numerical Functional Analysis and Optimization , 21 : 439 – 464 .
  • Krumbiegel , K. 2005 . On the influence of approximate source conditions on accuracy and convergence rates of regularized solutions , Germany : Faculty of Mathematics, Chemnitz University of Technology, Chemnitz . Diploma thesis (in German)
  • Locker , J. and Prenter , P.M. 1980 . Regularization with differential operators‐first part: General Theory . Journal of Mathematical Analysis and Applications , 74 : 504 – 529 .
  • Mathé , P. and Pereverzev , S.V. 2002 . Moduli of continuity for operator valued functions . Numer. Fund. Anal. Optim. , 23 : 623 – 631 .
  • Mathé , P. and Pereverzev , S.V. 2003 . Geometry of linear ill‐posed problems in variable Hilbert scales . Inverse Problems , 19 : 789 – 803 .
  • Mathé , P. and Pereverzev , S.V. 2006 . Regularization of some linear ill‐posed problems with discretized random noisy data . Mathematics of Computation , 75 In press
  • Nair , M.T. , Pereverzev , S.V. and Tautenhahn , U. 2005 . Regularization in Hilbert scales under general smoothing conditions . Inverse Problems , 21 : 1851 – 1869 .
  • Nair , M.T. , Schock , E. and Tautenhahn , U. 2003 . Morozov's discrepancy principle under general source conditions . Zeitschrift fuer Analysis u. Anwendungen , 22 : 199 – 214 .
  • Neubauer , A. 1997 . On converse and saturation results for Tikhonov regularization of linear ill‐posed problems . SIAM J. Numer Anal. , 34 : 517 – 527 .
  • Tautenhahn , U. 1998 . Optimality for ill‐posed problems under general source conditions . Numerical Functional Analysis and Optimization , 19 : 377 – 398 .
  • Tikhonov , A.N. and Arsenin , V.Y. 1977 . Solutions of Ill‐Posed Problems , New York : Wiley .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.