Skip to main content
SpringerLink
Log in

On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We consider a regularized Levenberg–Marquardt method for solving nonlinear ill-posed inverse problems. We use the discrepancy principle to terminate the iteration. Under certain conditions, we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source-wise representations.

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

Similar content being viewed by others

References

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

    MathSciNet  Google Scholar 

  2. Groetsch C.W.: Stable Approximate Evaluation of Unbounded Operators. Lecture Notes in Mathematics, vol. 1894. Springer, Berlin (2007)

    Google Scholar 

  3. Hanke M.: A regularizing Levenberg–Marquardt scheme with applications to inverse groundwater filtration problems. Inverse Probl. 13, 79–95 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hanke, M.: The regularizing Levenberg–Marquardt scheme is of optimal order (2009, preprint)

  5. Hanke M., Groetsch C.W.: Nonstationary iterated Tikhonov regularization. J. Optim. Theory Appl. 97(1), 37–53 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Jin Q.N.: On the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed problems. Math. Comput. 69(232), 1603–1623 (2000)

    Article  MATH  Google Scholar 

  8. Jin Q.N., Hou Z.Y.: On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems. Numer. Math. 83(1), 139–159 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jin Q.N., Tautenhahn U.: On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems. Numer. Math. 111, 509–558 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kaltenbacher B., Neubauer A., Scherzer O.: Iterative regularization methods for nonlinear ill-posed problems. de Gruyter, Berlin (2008)

    Book  MATH  Google Scholar 

  11. Rieder A.: On the regularization of nonlinear ill-posed problems via inexact Newton iterations. Inverse Probl. 15, 309–327 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rieder A.: On convergence rates of inexact Newton regularizations. Numer. Math. 88, 347–365 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Qinian Jin

    Present address: Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Qinian Jin

Authors
  1. Qinian Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qinian Jin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jin, Q. On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems. Numer. Math. 115, 229–259 (2010). https://doi.org/10.1007/s00211-009-0275-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-009-0275-x

Mathematics Subject Classification (2000)

Search

Navigation