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2015, 2015(special): 686-695. doi: 10.3934/proc.2015.0686

Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space

Barbara Kaltenbacher 1, and  Ivan Tomba 2,

1. 

Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt
2. 

Università di Bologna, Piazza Porta S. Donato, 5, 40127 - Bologna, Italy

Received  August 2014 Revised  June 2015 Published  November 2015

This paper is a close follow-up of [9] and [11], where Newton-Landweber iterations have been shown to converge either (unconditionally) without rates or (under an additional regularity assumption) with rates. The choice of the parameters in the method were different in each of these two cases. We now found a unified and more general strategy for choosing these parameters that enables both convergence and convergence rates. Moreover, as opposed to the previous one, this choice yields strong convergence as the noise level tends to zero, also in the case of no additional regularity. Additionally, the resulting method appears to be more efficient than the one from [9], as our numerical tests show.
Keywords: Newton's method, Regularization, Banach space, Landweber iteration., nonlinear inverse problems.
Mathematics Subject Classification: Primary: 65J20; Secondary: 65M3.
Citation: Barbara Kaltenbacher, Ivan Tomba. Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space. Conference Publications, 2015, 2015 (special) : 686-695. doi: 10.3934/proc.2015.0686
References:
[1]

A. B. Bakushinsky and M. Yu. Kokurin, Iterative methods for approximate solution of inverse problems, Springer, Dordrecht, 2004.

[2]

M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Problems, 20(5) (2004), 1411-1421.

[3]

C. Clason and B. Jin, A semi-smooth Newton method for nonlinear parameter identification problems with impulsive noise, SIAM J. Imaging Sci, 5 (2012), 505-538.

[4]

M. Hanke, A. Neubauer, and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.

[5]

T. Hein and B. Hofmann, Approximate source conditions for nonlinear ill-posed problems - chances and limitations, Inverse Problems, 25:035003 (16pp), 2009.

[6]

T. Hein and K. S. Kazimierski, Accelerated Landweber iteration in Banach spaces, Inverse Problems, 26:055002 (17pp), 2010.

[7]

T. Hein and K. S. Kazimierski, Modified Landweber iteration in Banach spaces - convergence and convergence rates, Numerical Functional Analysis and Optimization, 31(10) (2010), 1158-1189.

[8]

B. Hofmann, B. Kaltenbacher, C. Pöschl, and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23(3) (2007), 987-1010.

[9]

B. Kaltenbacher and I. Tomba, Convergence rates for an iteratively regularized Newton-Landweber iteration in Banach space, Inverse Problems, 29:025010 (18pp), 2013.

[10]

B. Kaltenbacher and I. Tomba, Enhanced choice of the parameters in an iteratively regularized Newton- Landweber iteration in Banach space, arXiv:1408.5026 [math.NA], 2014.

[11]

Q. Jin, Inexact Newton-Landweber iteration for solving nonlinear inverse problems in Banach spaces, Inverse Problems, 28:065002 (14pp), 2012.

[12]

Q. Jin and L.Stals, Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces, Inverse Problems, 28:104011 (15pp), 2012.

[13]

B. Kaltenbacher, Convergence rates for the iteratively regularized Landweber iteration in Banach space, Proceedings of the 25th IFIP TC7 Conference on System Modeling and Optimization, (eds. D. Hömberg and F.Tröltzsch), Springer, (2013), 38-48.

[14]

B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized gauss-newton method in Banach spaces, Inverse Problems, 26:035007 (21pp), 2010.

[15]

B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-posed Problems, de Gruyter, 2007.

[16]

B. Kaltenbacher, F. Schöpfer, and T. Schuster, Convergence of some iterative methods for the regularization of nonlinear ill-posed problems in Banach spaces, Inverse Problems, 25:065003, 2009.

[17]

A. Neubauer, T. Hein, B. Hofmann, S. Kindermann, and U. Tautenhahn, Improved and extended results for enhanced convergence rates of Tikhonov regularization in Banach spaces, Appl. Anal., 89(11) (2010), 1729-1743.

[18]

A. Rieder, On convergence rates of inexact Newton regularizations, Numer. Math. 88 (2001), 347-365.

[19]

A. Rieder, Inexact Newton regularization using conjugate gradients as inner iteration, SIAM J. Numer. Anal. 43 (2005), 604-622.

[20]

O. Scherzer, A modified Landweber iteration for solving parameter estimation problems, Appl. Math. Optim., 38 (1998), 45-68.

[21]

F. Schöpfer, A. K. Louis, and T. Schuster, Nonlinear iterative methods for linear ill-posed problems in Banach spaces, Inverse Problems, 22(1) (2006), 311-329.

[22]

T. Schuster, B. Kaltenbacher, B. Hofmann, and K. Kazimierski, Regularization Methods in Banach Spaces, de Gruyter, Berlin, New York, 2012.

[23]

Z.-B. Xu and G. F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, Journal of Mathematical Analysis and Applications, 157 (1991), 189-210.

show all references

References:
[1]

A. B. Bakushinsky and M. Yu. Kokurin, Iterative methods for approximate solution of inverse problems, Springer, Dordrecht, 2004.

[2]

M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Problems, 20(5) (2004), 1411-1421.

[3]

C. Clason and B. Jin, A semi-smooth Newton method for nonlinear parameter identification problems with impulsive noise, SIAM J. Imaging Sci, 5 (2012), 505-538.

[4]

M. Hanke, A. Neubauer, and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.

[5]

T. Hein and B. Hofmann, Approximate source conditions for nonlinear ill-posed problems - chances and limitations, Inverse Problems, 25:035003 (16pp), 2009.

[6]

T. Hein and K. S. Kazimierski, Accelerated Landweber iteration in Banach spaces, Inverse Problems, 26:055002 (17pp), 2010.

[7]

T. Hein and K. S. Kazimierski, Modified Landweber iteration in Banach spaces - convergence and convergence rates, Numerical Functional Analysis and Optimization, 31(10) (2010), 1158-1189.

[8]

B. Hofmann, B. Kaltenbacher, C. Pöschl, and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23(3) (2007), 987-1010.

[9]

B. Kaltenbacher and I. Tomba, Convergence rates for an iteratively regularized Newton-Landweber iteration in Banach space, Inverse Problems, 29:025010 (18pp), 2013.

[10]

B. Kaltenbacher and I. Tomba, Enhanced choice of the parameters in an iteratively regularized Newton- Landweber iteration in Banach space, arXiv:1408.5026 [math.NA], 2014.

[11]

Q. Jin, Inexact Newton-Landweber iteration for solving nonlinear inverse problems in Banach spaces, Inverse Problems, 28:065002 (14pp), 2012.

[12]

Q. Jin and L.Stals, Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces, Inverse Problems, 28:104011 (15pp), 2012.

[13]

B. Kaltenbacher, Convergence rates for the iteratively regularized Landweber iteration in Banach space, Proceedings of the 25th IFIP TC7 Conference on System Modeling and Optimization, (eds. D. Hömberg and F.Tröltzsch), Springer, (2013), 38-48.

[14]

B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized gauss-newton method in Banach spaces, Inverse Problems, 26:035007 (21pp), 2010.

[15]

B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-posed Problems, de Gruyter, 2007.

[16]

B. Kaltenbacher, F. Schöpfer, and T. Schuster, Convergence of some iterative methods for the regularization of nonlinear ill-posed problems in Banach spaces, Inverse Problems, 25:065003, 2009.

[17]

A. Neubauer, T. Hein, B. Hofmann, S. Kindermann, and U. Tautenhahn, Improved and extended results for enhanced convergence rates of Tikhonov regularization in Banach spaces, Appl. Anal., 89(11) (2010), 1729-1743.

[18]

A. Rieder, On convergence rates of inexact Newton regularizations, Numer. Math. 88 (2001), 347-365.

[19]

A. Rieder, Inexact Newton regularization using conjugate gradients as inner iteration, SIAM J. Numer. Anal. 43 (2005), 604-622.

[20]

O. Scherzer, A modified Landweber iteration for solving parameter estimation problems, Appl. Math. Optim., 38 (1998), 45-68.

[21]

F. Schöpfer, A. K. Louis, and T. Schuster, Nonlinear iterative methods for linear ill-posed problems in Banach spaces, Inverse Problems, 22(1) (2006), 311-329.

[22]

T. Schuster, B. Kaltenbacher, B. Hofmann, and K. Kazimierski, Regularization Methods in Banach Spaces, de Gruyter, Berlin, New York, 2012.

[23]

Z.-B. Xu and G. F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, Journal of Mathematical Analysis and Applications, 157 (1991), 189-210.

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