Abstract
In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method.
Acknowledgements
The authors are thankful to both the referees for their valuable suggestions and comments on the paper which indeed helped a lot in bringing the paper in the present form.
References
[1] A. Bakushinsky and A. Goncharsky, Iterative Methods for the Solution of Incorrect Problems (in Russian), Nauka, Moscow, 1989. Search in Google Scholar
[2] A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Math. Appl. (New York) 577, Springer, Dordrecht, 2004. 10.1007/978-1-4020-3122-9Search in Google Scholar
[3] B. Blaschke, A. Neubauer and O. Scherzer, On convergence rates for the iteratively regularized Gauss–Newton method, IMA J. Numer. Anal. 17 (1997), no. 3, 421–436. 10.1093/imanum/17.3.421Search in Google Scholar
[4] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Math. Appl. 62, Kluwer Academic, Dordrecht, 1990. 10.1007/978-94-009-2121-4Search in Google Scholar
[5] H. Egger, Accelerated Newton–Landweber iterations for regularizing nonlinear inverse problems, RICAM-Report 2005-01, Johann Radon Institute for Computational and Applied Mathematics, Linz, 2005. Search in Google Scholar
[6] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Search in Google Scholar
[7] M. Hanke, Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math. 60 (1991), no. 1, 341–373. 10.1007/BF01385727Search in Google Scholar
[8] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math. 72 (1995), no. 1, 21–37. 10.1007/s002110050158Search in Google Scholar
[9] T. Hein and K. S. Kazimierski, Modified Landweber iteration in Banach spaces—convergence and convergence rates, Numer. Funct. Anal. Optim. 31 (2010), no. 10, 1158–1184. 10.1080/01630563.2010.510977Search in Google Scholar
[10] B. Hofmann and P. Mathé, Analysis of profile functions for general linear regularization methods, SIAM J. Numer. Anal. 45 (2007), no. 3, 1122–1141. 10.1137/060654530Search in Google Scholar
[11] T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss–Newton method for an inverse potential and an inverse scattering problem, Inverse Problems 13 (1997), no. 5, 1279–1299. 10.1088/0266-5611/13/5/012Search in Google Scholar
[12] Q. Jin, A general convergence analysis of some Newton-type methods for nonlinear inverse problems, SIAM J. Numer. Anal. 49 (2011), no. 2, 549–573. 10.1137/100804231Search in Google Scholar
[13] Q. Jin, Inexact Newton–Landweber iteration for solving nonlinear inverse problems in Banach spaces, Inverse Problems 28 (2012), no. 6, Article ID 065002. 10.1088/0266-5611/28/6/065002Search in Google Scholar
[14] Q. Jin and U. Tautenhahn, Inexact Newton regularization methods in Hilbert scales, Numer. Math. 117 (2011), no. 3, 555–579. 10.1007/s00211-010-0342-3Search in Google Scholar
[15] B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized Gauss–Newton method in Banach spaces, Inverse Problems 26 (2010), no. 3, Article ID 035007. 10.1088/0266-5611/26/3/035007Search in Google Scholar
[16] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-posed Problems, Radon Ser. Comput. Appl. Math. 6, Walter de Gruyter, Berlin, 2008. 10.1515/9783110208276Search in Google Scholar
[17] B. Kaltenbacher, F. Schöpfer and T. Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces: Convergence and applications to parameter identification problems, Inverse Problems 25 (2009), no. 6, Article ID 065003. 10.1088/0266-5611/25/6/065003Search in Google Scholar
[18] B. Kaltenbacher and I. Tomba, Convergence rates for an iteratively regularized Newton-Landweber iteration in Banach space, Inverse Problems 29 (2013), no. 2, Article ID 025010. 10.1088/0266-5611/29/2/025010Search in Google Scholar
[19] F. Margotti, Mixed gradient-Tikhonov methods for solving nonlinear ill-posed problems in Banach spaces, Inverse Problems 32 (2016), no. 12, Article ID 125012. 10.1088/0266-5611/32/12/125012Search in Google Scholar
[20] A. Rieder, On the regularization of nonlinear ill-posed problems via inexact Newton iterations, Inverse Problems 15 (1999), no. 1, 309–327. 10.1088/0266-5611/15/1/028Search in Google Scholar
[21] O. Scherzer, A modified Landweber iteration for solving parameter estimation problems, Appl. Math. Optim. 38 (1998), no. 1, 45–68. 10.1007/s002459900081Search in Google Scholar
[22] Z. B. Xu and G. F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl. 157 (1991), no. 1, 189–210. 10.1016/0022-247X(91)90144-OSearch in Google Scholar
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