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]

Plamen Stefanov, Yang Yang. Multiwave tomography with reflectors: Landweber's iteration. Inverse Problems & Imaging, 2017, 11 (2) : 373-401. doi: 10.3934/ipi.2017018
[2]

T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems & Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040
[3]

Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial & Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247
[4]

David Maxwell. Kozlov-Maz'ya iteration as a form of Landweber iteration. Inverse Problems & Imaging, 2014, 8 (2) : 537-560. doi: 10.3934/ipi.2014.8.537
[5]

Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015
[6]

Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271
[7]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[8]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[9]

Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita. Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems & Imaging, 2016, 10 (3) : 741-764. doi: 10.3934/ipi.2016019
[10]

Armin Lechleiter, Marcel Rennoch. Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media. Inverse Problems & Imaging, 2017, 11 (1) : 151-176. doi: 10.3934/ipi.2017008
[11]

R. Baier, M. Dellnitz, M. Hessel-von Molo, S. Sertl, I. G. Kevrekidis. The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 2014, 1 (1) : 39-69. doi: 10.3934/jcd.2014.1.39
[12]

Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143
[13]

Robert Denk, Leonid Volevich. A new class of parabolic problems connected with Newton's polygon. Conference Publications, 2007, 2007 (Special) : 294-303. doi: 10.3934/proc.2007.2007.294
[14]

Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685
[15]

Matthew A. Fury. Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space. Conference Publications, 2013, 2013 (special) : 259-272. doi: 10.3934/proc.2013.2013.259
[16]

Irene Benedetti, Nguyen Van Loi, Valentina Taddei. An approximation solvability method for nonlocal semilinear differential problems in Banach spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2977-2998. doi: 10.3934/dcds.2017128
[17]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[18]

Andy M. Yip, Wei Zhu. A fast modified Newton's method for curvature based denoising of 1D signals. Inverse Problems & Imaging, 2013, 7 (3) : 1075-1097. doi: 10.3934/ipi.2013.7.1075
[19]

Bernadette N. Hahn. Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization. Inverse Problems & Imaging, 2015, 9 (2) : 395-413. doi: 10.3934/ipi.2015.9.395
[20]

Bruno Sixou, Cyril Mory. Kullback-Leibler residual and regularization for inverse problems with noisy data and noisy operator. Inverse Problems & Imaging, 2019, 13 (5) : 1113-1137. doi: 10.3934/ipi.2019050

 Impact Factor: 

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences