Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A convergence analysis of regularization by discretization in preimage space


Authors: Barbara Kaltenbacher and Jonas Offtermatt
Journal: Math. Comp. 81 (2012), 2049-2069
MSC (2010): Primary 65J20; Secondary 65M32
DOI: https://doi.org/10.1090/S0025-5718-2012-02596-8
Published electronically: April 2, 2012
MathSciNet review: 2945147
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we investigate the regularizing properties of discretization in preimage space for linear and nonlinear ill-posed operator equations with noisy data. We propose to choose the discretization level, that acts as a regularization parameter in this context, by a discrepancy principle. While general convergence has been shown not to hold, we provide convergence results under appropriate conditions on the exact solution.


References [Enhancements On Off] (What's this?)

  • 1. A.B. Bakushinsky and M. Yu Kokurin, Iterative methods for approximate solution of inverse problems, Springer, Dordrecht, 2004. MR 2133802 (2006e:47025)
  • 2. H. Ben Ameur, G. Chavent, and J. Jaffré, Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities, Inverse Problems 18 (2002), 775-794. MR 1910202 (2003c:76118)
  • 3. Heinz W. Engl, Karl Kunisch, and Andreas Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems, Inverse Problems 5 (1989), no. 4, 523-540. MR 1009037 (91k:65102)
  • 4. H.W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Kluwer, Dordrecht, 1996. MR 1408680 (97k:65145)
  • 5. P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, 1985. MR 775683 (86m:35044)
  • 6. C.W. Groetsch, Inverse problems in mathematical sciences, Vieweg, Braunschweig, 1993. MR 1247696 (94m:00008)
  • 7. C.W. Groetsch and A. Neubauer, Convergence of a general projection method for an operator equation of the first kind, Houston Journal of Mathematics 14 (1988), 201-208. MR 978726 (90b:65109)
  • 8. U. Hämarik, E. Avi, and A. Ganina, On the solution of ill-posed problems by projection methods with a posteriori choice of the discretization level, Mathematical Modelling and Analysis 7 (2002), 241-252. MR 1952913 (2003k:65057)
  • 9. M. Hanke, A. Neubauer, and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math. 72 (1995), 21-37. MR 1359706 (96i:65046)
  • 10. B. Hofmann, P. Mathé, and S.V. Pereverzev, Regularization by projection: Approximation theoretic aspects and distance functions, J. Inverse Ill-Posed Problems 15 (2007), 527-545. MR 2367863 (2008m:65155)
  • 11. J. Kaipio and E. Somersalo, Statistical and computational inverse problems, Applied Mathematical Sciences, vol. 160, Springer, 2005. MR 2102218 (2005g:65001)
  • 12. B. Kaltenbacher, Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems, Inverse Problems 16 (2000), no. 5, 1523-1539. MR 1800607 (2001h:65070)
  • 13. -, On the regularizing properties of a full multigrid method for ill-posed problems, Inverse Problems 17 (2001), 767-788. MR 1861481 (2002h:65094)
  • 14. -, V-cycle convergence of some multigrid methods for ill-posed problems, Mathematics of Computation 72 (2003), 1711-1730. MR 1986801 (2004d:65069)
  • 15. -, Towards global convergence for strongly nonlinear ill-posed problems via a regularizing multilevel approach, Numerical Functional Analysis and Optimization 27 (2006), 637-665. MR 2246581 (2007c:65047)
  • 16. -, Convergence rates of a multilevel method for the regularization of nonlinear ill-posed problems, Journal of Integral Equations and Applications 20 (2008), no. 2, 201-228. MR 2418067 (2009e:47112)
  • 17. B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative regularization methods for nonlinear problems, de Gruyer, Berlin, New York, 2008, Radon Series on Computational and Applied Mathematics.
  • 18. B. Kaltenbacher and J. Offtermatt, A refinement and coarsening indicator algorithm for finding sparse solutions of inverse problems, Inverse Problems and Imaging (2011), 391-406.
  • 19. B. Kaltenbacher and J. Schicho, A multi-grid method with a priori and a posteriori level choice for the regularization of nonlinear ill-posed problems, Numerische Mathematik 93 (2002), 77-107, DOI 10.1007/s002110100375. MR 1938323 (2003h:65076)
  • 20. A. Kirsch, An introduction to the mathematical theory of inverse problems, Springer, New York, 1996. MR 1479408 (99c:34023)
  • 21. R. Kress, Linear integral equations, Springer, Heidelberg, 1989, 2nd ed. 1999. MR 1723850 (2000h:45001)
  • 22. A.K. Louis, Inverse und schlecht gestellte probleme, Teubner, Stuttgart, 1989. MR 1002946 (90g:65075)
  • 23. G.R. Luecke and K.R. Hickey, Convergence of approximate solutions of an operator equation, Houston Journal of Mathematics 11 (1985), no. 3, 345-354. MR 808651 (87a:47019)
  • 24. P. Mathé and N. Schöne, Regularization by projection in variable Hilbert scales, Applicable Analysis 87 (2008), 201-219. MR 2394505 (2009h:47021)
  • 25. V.A. Morozov, Regularization methods for ill-posed problems, CRC Press, Boca Raton, 1993. MR 1244325 (94g:65002)
  • 26. F. Natterer, Regularisierung schecht gestellter Probleme durch Projektionsverfahren, Numerische Mathematik 28 (1977), 329-341. MR 0488721 (58:8238)
  • 27. -, The mathematics of computerized tomography, Teubner, Stuttgart, 1986. MR 856916 (88m:44008)
  • 28. S.V. Pereverzev and S. Prössdorf, On the characterization of self-regularization properties of a fully discrete projection method for Symm's integral equation, Journal of Integral Equations and Applications 12 (2000), no. 2, 113-130. MR 1771514 (2001j:65198)
  • 29. O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, Journal of Mathematical Analysis and Applications 194 (1995), no. 3, 911-933. MR 1350202 (97d:65033)
  • 30. T.I. Seidman, Nonconvergence results for the application of least squares estimation to ill-posed problems, Journal of Optimization Theory and Applications 30 (1980), 535-547. MR 572154 (81g:65073)
  • 31. A. N. Tikhonov and V. A. Arsenin, Methods for solving ill-posed problems, Nauka, Moscow, 1979.
  • 32. G. Vainikko and U. Hämarik, Projection methods and self-regularization in ill-posed problems, Soviet Mathematics 29 (1985), 1-20, in Russian. MR 828379 (87m:65096)
  • 33. G. Vainikko and A. Y. Veterennikov, Iteration procedures in ill-posed problems, Nauka, Moscow, 1986. MR 859375 (88c:47019)
  • 34. V.V. Vasin and A.L. Ageev, Ill-posed problems with a priori information, VSP, Zeist, 1995. MR 1374573 (97j:65100)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65J20, 65M32

Retrieve articles in all journals with MSC (2010): 65J20, 65M32


Additional Information

Barbara Kaltenbacher
Affiliation: Institute for Mathematics, University of Klagenfurt, Universitätsstraße 65-67, A-9020 Klagenfurt, Austria
Email: barbara.kaltenbacher@aau.at

Jonas Offtermatt
Affiliation: Institute for Stochastics and Applications, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email: jonas.offtermatt@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/S0025-5718-2012-02596-8
Received by editor(s): April 10, 2011
Received by editor(s) in revised form: June 26, 2011
Published electronically: April 2, 2012
Additional Notes: Support by the German Science Foundation (DFG) within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart is gratefully acknowledged
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society