Home Mathematics Simplified Iteratively Regularized Gauss–Newton Method in Banach Spaces Under a General Source Condition
Article
Licensed
Unlicensed Requires Authentication

Simplified Iteratively Regularized Gauss–Newton Method in Banach Spaces Under a General Source Condition

  • EMAIL logo and
Published/Copyright: March 8, 2019

Abstract

In this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method.

References

[1] M. Burger and B. Kaltenbacher, Regularizing Newton–Kaczmarz methods for nonlinear ill-posed problems, SIAM J. Numer. Anal. 44 (2006), no. 1, 153–182. 10.1137/040613779Search in Google Scholar

[2] 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

[3] T. Hein and B. Hofmann, Approximate source conditions for nonlinear ill-posed problems—chances and limitations, Inverse Problems 25 (2009), no. 3, Article ID 035003. 10.1088/0266-5611/25/3/035003Search in Google Scholar

[4] B. Hofmann, Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators, Math. Methods Appl. Sci. 29 (2006), no. 3, 351–371. 10.1002/mma.686Search in Google Scholar

[5] 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

[6] T. Hohage, Regularization of exponentially ill-posed problems, Numer. Funct. Anal. Optim. 21 (2000), no. 3–4, 439–464. 10.1080/01630560008816965Search in Google Scholar

[7] Q. Jin and L. Stals, Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces, Inverse Problems 28 (2012), no. 10, Article ID 104011. 10.1088/0266-5611/28/10/104011Search in Google Scholar

[8] Q. Jin and M. Zhong, On the iteratively regularized Gauss–Newton method in Banach spaces with applications to parameter identification problems, Numer. Math. 124 (2013), no. 4, 647–683. 10.1007/s00211-013-0529-5Search in Google Scholar

[9] Q. Jin and M. Zhong, Nonstationary iterated Tikhonov regularization in Banach spaces with uniformly convex penalty terms, Numer. Math. 127 (2014), no. 3, 485–513. 10.1007/s00211-013-0594-9Search in Google Scholar

[10] B. Kaltenbacher, A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems, Numer. Math. 79 (1998), no. 4, 501–528. 10.1007/s002110050349Search in Google Scholar

[11] B. Kaltenbacher, A note on logarithmic convergence rates for nonlinear Tikhonov regularization, J. Inverse Ill-Posed Probl. 16 (2008), no. 1, 79–88. 10.1515/jiip.2008.006Search in Google Scholar

[12] 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

[13] 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

[14] 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

[15] S. Langer and T. Hohage, Convergence analysis of an inexact iteratively regularized Gauss–Newton method under general source conditions, J. Inverse Ill-Posed Probl. 15 (2007), no. 3, 311–327. 10.1515/jiip.2007.017Search in Google Scholar

[16] P. Mahale, Simplified generalized Gauss–Newton iterative method under Morozove type stopping rule, Numer. Funct. Anal. Optim. 36 (2015), no. 11, 1448–1470. 10.1080/01630563.2015.1067822Search in Google Scholar

[17] P. Mahale and S. K. Dixit, Convergence analysis of simplified iteratively regularized Gauss–Newton method in a Banach space setting, Appl. Anal. 97 (2018), no. 15, 2686–2719. 10.1080/00036811.2017.1386785Search in Google Scholar

[18] P. Mahale and S. K. Dixit, Error estimates for the simplified iteratively regularized Gauss–Newton method in Banach spaces under a Morozov-type stopping rule, J. Inverse Ill-Posed Probl. 26 (2018), no. 3, 311–333. 10.1515/jiip-2017-0059Search in Google Scholar

[19] P. Mahale and M. T. Nair, A simplified generalized Gauss–Newton method for nonlinear ill-posed problems, Math. Comp. 78 (2009), no. 265, 171–184. 10.1090/S0025-5718-08-02149-2Search in Google Scholar

[20] 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

[21] S. Pereverzev and E. Schock, Morozov’s discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces, Numer. Funct. Anal. Optim. 21 (2000), no. 7–8, 901–916. 10.1080/01630560008816993Search in Google Scholar

[22] W. Sun and Y.-X. Yuan, Optimization Theory and Methods. Nonlinear Programming, Springer Optim. Appl. 1, Springer, New York, 2006. Search in Google Scholar

[23] U. Tautenhahn, On a general regularization scheme for nonlinear ill-posed problems, Inverse Problems 13 (1997), no. 5, 1427–1437. 10.1088/0266-5611/13/5/020Search in Google Scholar

[24] C. Zálinscu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, 2002. 10.1142/5021Search in Google Scholar

Received: 2018-07-02
Revised: 2018-11-27
Accepted: 2019-02-12
Published Online: 2019-03-08
Published in Print: 2020-04-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 20.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/cmam-2018-0165/html
Scroll to top button