Abstract
Inexact Newton regularization methods have been proposed by Hanke and Rieder for solving nonlinear ill-posed inverse problems. Every such a method consists of two components: an outer Newton iteration and an inner scheme providing increments by regularizing local linearized equations. The method is terminated by a discrepancy principle. In this paper we consider the inexact Newton regularization methods with the inner scheme defined by Landweber iteration, the implicit iteration, the asymptotic regularization and Tikhonov regularization. Under certain conditions we obtain the order optimal convergence rate result which improves the suboptimal one of Rieder. We in fact obtain a more general order optimality result by considering these inexact Newton methods in Hilbert scales.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dembo R.S., Eisenstat S.C., Steihaug T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)
Deuflhard P., Engl H.W., Scherzer O.: A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Probl. 14, 1081–1106 (1998)
Engl H.W., Hanke M., Neubauer A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)
Hanke M.: A regularizing Levenberg-Marquardt scheme with applications to inverse groundwater filtration problems. Inverse Probl. 13, 79–95 (1997)
Hanke M.: The regularizing Levenberg-Marquardt scheme is of optimal order. J. Integral Equ. Appl. 22(2), 259–283 (2010)
Hanke M., Neubauer A., Scherzer O.: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21–37 (1995)
Hohage T., Pricop M.: Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Probl. Imaging 2, 271–290 (2008)
Jin Q.: A general convergence analysis of some Newton-type methods for nonlinear inverse problems. SIAM J. Numer. Anal. 49, 549–573 (2011)
Jin Q., Tautenhahn U.: Inexact Newton regularization methods in Hilbert scales. Numer. Math. 117, 555–579 (2011)
Lechleiter A., Rieder A.: Towards a general convergence theory for inexact Newton regularizations. Numer. Math. 114(3), 521–548 (2010)
Neubauer A.: On Landweber iteration for nonlinear ill-posed problems in Hilbert scales. Numer. Math. 85, 309–328 (2000)
Rieder A.: On the regularization of nonlinear ill-posed problems via inexact Newton iterations. Inverse Probl. 15, 309–327 (1999)
Rieder A.: On convergence rates of inexact Newton regularizations. Numer. Math. 88, 347–365 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jin, Q. On the order optimality of the regularization via inexact Newton iterations. Numer. Math. 121, 237–260 (2012). https://doi.org/10.1007/s00211-011-0435-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-011-0435-7