Abstract
In this paper, we propose a new Newton-Landweber iteration for nonlinear inverse problems. We show the convergence of this method without any convergence rate under a weak nonlinearity condition. Also, this iteration will converge and inherit a certain monotonicity of the iteration error like Landweber iteration, if we restrict our inner iteration steps. Furthermore, we obtain optimal convergence rate of the new Newton-Landweber iteration under stronger nonlinearity conditions and parameter choice rules. Numerical experiments have shown some attractive results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blaschke, B., Neubauer, A., Scherzer, O.: On convergence rates for the iteratively regularized Gauss-Newton method. IMA J. Numer. Anal. 17, 421–436 (1997)
Egger, H.: Accelerated Newton-Landweber iterations for regularizing nonlinear inverse problems. SFB-Report 2005-3 (2005)
Egger, H.: Fast fully iterative Newton-type methods for inverse problems. J. Inverse Ill-Posed Probl. 15, 257–275 (2007)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996)
Hanke, M.: Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. Numer. Funct. Anal. Optim. 18, 971–993 (1997)
Hanke, M.: A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Probl. 13, 79–95 (1997)
Hanke, M., Neubauer, A., Scherzer, O.: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21–37 (1995)
Kaltenbacher, B.: Some Newton-type methods for the regularization of nonlinear ill-posed problems. Inverse Probl. 13, 729–753 (1997)
Kaltenbacher, B.: A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems. Numer. Math. 10, 261–280 (1998)
Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer, Berlin (1996)
Kransnoselskii, M.A., Zabreiko, P.P., Pustylnik, E.I., Sbolevskii, P.E.: Integral Operators in Spaces of Summable Functions. Nordhoff International Publishing, Leyden (1976)
Landweber, L.: An iteration formula for Fredholm integral equations of the first kind. Am. J. Math. 73, 615–624 (1951)
Louis, A.K.: Inverse und Schlecht Gestellte Probleme. Teubner, Stuttgart (1989)
Morozov, V.A.: On the solution of functional equations by the method of regularization. Sov. Math. Dokl. 7, 414–417 (1966)
Morozov, V.A.: Regularization Methods for Ill-Posed Problems. CRC Press, Boca Raton (1993)
Neubauer, A.: On Landweber iteration for nonlinear ill-posed problems in Hilbert scales. Numer. Math. 85, 309–328 (2000)
Rodrigue, G.H.: A uniform approach to gradient methods for linear operator equations. J. Math. Anal. Appl. 49, 275–285 (1975)
Tikhonov, A.N., Arsenin, V.A.: Methods for Solving Ill-Posed Problems. Nauka, Moscow (1979)
Vainikko, G.M., Veretennikov, A.Y.: Iteration Procedures in Ill-Posed Problems. Nauka, Moscow (1986), (in Russian) McCormick, S.F
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xiao, C., Deng, Y. A new Newton-Landweber iteration for nonlinear inverse problems. J. Appl. Math. Comput. 36, 489–505 (2011). https://doi.org/10.1007/s12190-010-0415-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-010-0415-6