Abstract
In this note, we study the convergence of the Levenberg–Marquardt regularization scheme for nonlinear ill-posed problems. We consider the case that the initial error satisfies a source condition. Our main result shows that if the regularization parameter does not grow too fast (not faster than a geometric sequence), then the scheme converges with optimal convergence rates. Our analysis is based on our recent work on the convergence of the exponential Euler regularization scheme (Hochbruck et al. in Inverse Probl 25(7):075009, 2009).
Similar content being viewed by others
References
Hanke M.: A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Probl. 13(1), 79–95 (1997)
Hanke M., Neubauer A., Scherzer O.: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72(1), 21–37 (1995)
Hochbruck M., Hönig M., Ostermann A.: A convergence analysis of the exponential Euler iteration for nonlinear ill-posed problems. Inverse Probl. 25(7), 075009 (2009)
Jin, Q.: On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems. Tech. rep., University of Texas at Austin (2009)
Jin Q., Tautenhahn U.: On the discrepancy principles for some Newton type methods for solving nonlinear inverse problems. Numer. Math. 111(4), 509–558 (2009)
Kaltenbacher B., Neubauer A., Scherzer O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. De Gruyter, Berlin, New York (2008)
Rieder A.: On the regularization of nonlinear ill-posed problems via inexact Newton iterations. Inverse Probl. 15(1), 309–327 (1999)
Rieder A.: On convergence rates of inexact Newton regularizations. Numer. Math. 88(2), 347–365 (2001)
Tautenhahn U.: On the asymptotical regularization of nonlinear ill-posed problems. Inverse Probl. 10(6), 1405–1418 (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hochbruck, M., Hönig, M. On the convergence of a regularizing Levenberg–Marquardt scheme for nonlinear ill-posed problems. Numer. Math. 115, 71–79 (2010). https://doi.org/10.1007/s00211-009-0268-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-009-0268-9