Skip to main content
SpringerLink
Log in

An iteratively regularized Gauss–Newton–Halley method for solving nonlinear ill-posed problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper we analyze a second order method for regularizing nonlinear inverse problems, which comes from adapting Halley’s method to the ill-posed situation. The method is related to the iteratively regularized Gauss–Newton method and enhanced by the use of second derivatives of the forward operator. For parameter identification problems in PDEs, these second derivatives often require not much additional computational cost as compared to first order derivatives. We prove convergence and convergence rates under certain conditions on the forward operator in Hilbert spaces. Numerical tests for a problem of identifying a distributed coefficient in an elliptic PDE illustrate the performance of the method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bakushinsky, A.B.: The problem of the convergence of the iteratively regularized Gauss–Newton method. Comput. Math. Math. Phys. 32, 1353–1359 (1992)

    MathSciNet  Google Scholar 

  2. Blaschke (Kaltenbacher), B., Neubauer, A., Scherzer, O.: On convergence rates for the iteratively regularized Gauss–Newton method. IMA J. Numer. Anal. 17, 421–436 (1997)

    Article  MathSciNet  Google Scholar 

  3. Brown Jr, G.H.: On Halley’s variation of Newton’s method. Am. Math. Mon. 84, 726–727 (1977)

    Article  MATH  Google Scholar 

  4. Döring, B.: Einige Sätze ber das Verfahren der tangierenden Hyperbeln in Banach–Räumen. Apl. Mat. 15, 418–464 (1970)

    MATH  MathSciNet  Google Scholar 

  5. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1996)

  6. Engl, H.W., Kunisch, K., Neubauer, A.: Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Probl. 5, 523–540 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hettlich, F., Rundell, W.: A second degree method for nonlinear inverse problems. SIAM J. Numer. Anal. 37, 587–620 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hanke, M., Neubauer, A., Scherzer, O.: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21–37 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hofmann, B., Scherzer, O.: Factors influencing the ill-posedness of nonlinear problems. Inverse Probl. 10, 1277–1297 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kaltenbacher, B.: A convergence rates result for the iteratively regularization Gauss–Newton–Halley method in Banach spaces. arXiv:1409.5655 [math.NA] (submitted)

  11. Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Problems. de Gruyter, Berlin (2008)

    Book  MATH  Google Scholar 

  12. Mathé, P., Hofmann, B.: How general are general source conditions? Inverse Probl. 24, 015009 (5pp) (2008)

  13. Ren, H., Argyros, I.K.: On the semi-local convergence of Halley’s method under a center-Lipschitz condition on the second Fréchet derivative. Appl. Math. Comput. 218, 11488–11495 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author wishes to thank both referees for their detailed reports and fruitful suggestions. Moreover, financial support by the Austrian Science Fund FWF under grant P24970 is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Alpen-Adria Universität Klagenfurt, Universitätsstraße 65-67, 9030, Klagenfurt, Austria

    Barbara Kaltenbacher

Authors
  1. Barbara Kaltenbacher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Barbara Kaltenbacher.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaltenbacher, B. An iteratively regularized Gauss–Newton–Halley method for solving nonlinear ill-posed problems. Numer. Math. 131, 33–57 (2015). https://doi.org/10.1007/s00211-014-0682-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-014-0682-5

Mathematics Subject Classification

Search

Navigation