American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models
  • PROC Home
  • This Issue
  • Next Article
    Stability of neutral delay differential equations modeling wave propagation in cracked media
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, (2004).   Google Scholar

[2]

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

[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.   Google Scholar

[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.   Google Scholar

[5]

T. Hein and B. Hofmann, Approximate source conditions for nonlinear ill-posed problems - chances and limitations,, Inverse Problems, (2009).   Google Scholar

[6]

T. Hein and K. S. Kazimierski, Accelerated Landweber iteration in Banach spaces,, Inverse Problems, (2010).   Google Scholar

[7]

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

[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 (2007), 987.   Google Scholar

[9]

B. Kaltenbacher and I. Tomba, Convergence rates for an iteratively regularized Newton-Landweber iteration in Banach space,, Inverse Problems, (2013).   Google Scholar

[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).   Google Scholar

[11]

Q. Jin, Inexact Newton-Landweber iteration for solving nonlinear inverse problems in Banach spaces,, Inverse Problems, (2012).   Google Scholar

[12]

Q. Jin and L.Stals, Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces,, Inverse Problems, (2012).   Google Scholar

[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, (2013), 38.   Google Scholar

[14]

B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized gauss-newton method in Banach spaces,, Inverse Problems, (2010).   Google Scholar

[15]

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

[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, (2009).   Google Scholar

[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 (2010), 1729.   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

T. Schuster, B. Kaltenbacher, B. Hofmann, and K. Kazimierski, Regularization Methods in Banach Spaces,, de Gruyter, (2012).   Google Scholar

[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.   Google Scholar

show all references

References:
[1]

A. B. Bakushinsky and M. Yu. Kokurin, Iterative methods for approximate solution of inverse problems,, Springer, (2004).   Google Scholar

[2]

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

[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.   Google Scholar

[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.   Google Scholar

[5]

T. Hein and B. Hofmann, Approximate source conditions for nonlinear ill-posed problems - chances and limitations,, Inverse Problems, (2009).   Google Scholar

[6]

T. Hein and K. S. Kazimierski, Accelerated Landweber iteration in Banach spaces,, Inverse Problems, (2010).   Google Scholar

[7]

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

[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 (2007), 987.   Google Scholar

[9]

B. Kaltenbacher and I. Tomba, Convergence rates for an iteratively regularized Newton-Landweber iteration in Banach space,, Inverse Problems, (2013).   Google Scholar

[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).   Google Scholar

[11]

Q. Jin, Inexact Newton-Landweber iteration for solving nonlinear inverse problems in Banach spaces,, Inverse Problems, (2012).   Google Scholar

[12]

Q. Jin and L.Stals, Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces,, Inverse Problems, (2012).   Google Scholar

[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, (2013), 38.   Google Scholar

[14]

B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized gauss-newton method in Banach spaces,, Inverse Problems, (2010).   Google Scholar

[15]

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

[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, (2009).   Google Scholar

[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 (2010), 1729.   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

T. Schuster, B. Kaltenbacher, B. Hofmann, and K. Kazimierski, Regularization Methods in Banach Spaces,, de Gruyter, (2012).   Google Scholar

[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.   Google Scholar

[1]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074
[2]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
[3]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[4]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[5]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077
[6]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072
[7]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355
[8]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462
[9]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018
[10]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260
[11]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101
[12]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[13]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[14]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[15]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463
[16]

Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105
[17]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[18]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103
[19]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013
[20]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

 Impact Factor: 

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences