Show Summary Details
More options …

# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

IMPACT FACTOR 2018: 0.881
5-year IMPACT FACTOR: 1.170

CiteScore 2018: 0.91

SCImago Journal Rank (SJR) 2018: 0.430
Source Normalized Impact per Paper (SNIP) 2018: 0.969

Mathematical Citation Quotient (MCQ) 2018: 0.66

Online
ISSN
1569-3945
See all formats and pricing
More options …

# The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

Barbara Kaltenbacher
/ Andrej Klassen
/ Mario Luiz Previatti de Souza
Published Online: 2019-04-06 | DOI: https://doi.org/10.1515/jiip-2018-0093

## Abstract

In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., ${L}^{\mathrm{\infty }}$ as a preimage space. The theoretical findings are illustrated by numerical experiments.

MSC 2010: 65F22; 65N20

## References

• [1]

A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Math. Appl. (New York) 577, Springer, Dordrecht, 2004. Google Scholar

• [2]

K. Bredies and D. A. Lorenz, Regularization with non-convex separable constraints, Inverse Problems 25 (2009), no. 8, Article ID 085011.

• [3]

C. Clason, B. Kaltenbacher and D. Wachsmuth, Functional error estimators for the adaptive discretization of inverse problems, Inverse Problems 32 (2016), no. 10, Article ID 104004.

• [4]

C. Clason and A. Klassen, Quasi-solution of linear inverse problems in non-reflexive Banach spaces, J. Inverse Ill-Posed Probl. 26 (2018), no. 5, 689–702.

• [5]

I. N. Dombrovskaja and V. K. Ivanov, On the theory of certain linear equations in abstract spaces, Sibirsk. Mat. Ž. 6 (1965), 499–508. Google Scholar

• [6]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. Google Scholar

• [7]

J. Flemming, Generized Tikhonov regularization: Basic theory and comprehensive results on convergence rates, PhD thesis, Technische Universität Chemnitz, 2011. Google Scholar

• [8]

M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods, Inverse Problems 26 (2010), no. 11, Article ID 115014.

• [9]

C. W. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg Math. Sci. Eng., Friedrich Vieweg & Sohn, Braunschweig, 1993.

• [10]

M. Hanke, A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems 13 (1997), no. 1, 79–95.

• [11]

M. Hintermüller and R. H. W. Hoppe, Goal-oriented adaptivity in control constrained optimal control of partial differential equations, SIAM J. Control Optim. 47 (2008), no. 4, 1721–1743.

• [12]

M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2002), no. 3, 865–888.

• [13]

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), no. 3, 987–1010.

• [14]

B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems 28 (2012), no. 10, Article ID 104006.

• [15]

T. Hohage and F. Weidling, Characterizations of variational source conditions, converse results, and maxisets of spectral regularization methods, SIAM J. Numer. Anal. 55 (2017), no. 2, 598–620.

• [16]

V. K. Ivanov, On linear problems which are not well-posed, Dokl. Akad. Nauk SSSR 145 (1962), 270–272. Google Scholar

• [17]

V. K. Ivanov, On ill-posed problems, Mat. Sb. (N.S.) 61 (103) (1963), 211–223. Google Scholar

• [18]

V. K. Ivanov, V. V. Vasin and V. P. Tanana, Theory of Linear Ill-posed Problems and its Applications, 2nd ed., Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2002. Google Scholar

• [19]

Q. Jin and H. Yang, Levenberg–Marquardt method in Banach spaces with general convex regularization terms, Numer. Math. 133 (2016), no. 4, 655–684.

• [20]

B. Kaltenbacher, P. Hungerländer and F. Rendl, Regularization of inverse problems via box constrained minimization, preprint (2018), https://arxiv.org/abs/1807.11316.

• [21]

B. Kaltenbacher and M. L. Previatti de Souza, Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space, Numer. Math. 140 (2018), no. 2, 449–478.

• [22]

B. Kaltenbacher, A. Kirchner and S. Veljović, Goal oriented adaptivity in the IRGNM for parameter identification in PDEs: I. reduced formulation, Inverse Problems 30 (2014), no. 4, Article ID 0450011.

• [23]

B. Kaltenbacher and A. Klassen, On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs, Inverse Problems 34 (2018), no. 5, Article ID 055008.

• [24]

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. Google Scholar

• [25]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Appl. Math. Sci. 120, Springer, New York, 1996.

• [26]

D. Lorenz and N. Worliczek, Necessary conditions for variational regularization schemes, Inverse Problems 29 (2013), no. 7, Article ID 075016.

• [27]

A. K. Louis, Inverse und schlecht gestellte Probleme, Teubner Studienbücher Math., B. G. Teubner, Stuttgart, 1989. Google Scholar

• [28]

V. A. Morozov, Regularization Methods for Ill-posed Problems, CRC Press, Boca, 1993. Google Scholar

• [29]

A. Neubauer and R. Ramlau, On convergence rates for quasi-solutions of ill-posed problems, Electron. Trans. Numer. Anal. 41 (2014), 81–92. Google Scholar

• [30]

A. Rieder, On convergence rates of inexact Newton regularizations, Numer. Math. 88 (2001), no. 2, 347–365.

• [31]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, Radon Ser. Comput. Appl. Math. 10, Walter de Gruyter, Berlin, 2012. Google Scholar

• [32]

T. I. Seidman and C. R. Vogel, Well-posedness and convergence of some regularisation methods for nonlinear ill posed problems, Inverse Problems 5 (1989), no. 2, 227–238.

• [33]

A. N. Tikhonov and V. A. Arsenin, Methods for Solving Ill-posed Problems, Nauka, Moscow, 1979. Google Scholar

• [34]

M. Ulbrich, Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces, MOS-SIAM Ser. Optim. 11, Society for Industrial and Applied Mathematics, Philadelphia, 2011. Google Scholar

• [35]

V. V. Vasin and A. L. Ageev, Ill-posed Problems with A Priori Information, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 1995. Google Scholar

• [36]

G. M. Vaĭnikko and A. Y. Veretennikov, Iteration Procedures in Ill-posed Problems, (in Russian), “Nauka”, Moscow, 1986. Google Scholar

• [37]

B. Vexler and W. Wollner, Adaptive finite elements for elliptic optimization problems with control constraints, SIAM J. Control Optim. 47 (2008), no. 1, 509–534.

## About the article

Revised: 2019-01-01

Accepted: 2019-02-18

Published Online: 2019-04-06

Published in Print: 2019-08-01

Funding Source: Austrian Science Fund

Award identifier / Grant number: I2271

Award identifier / Grant number: P30054

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Cl 487/1-1

Supported by the FWF under grants I2271 and P30054 and by the DFG under grant Cl 487/1-1, as well as partially by the Karl Popper Kolleg “Modeling-Simulation-Optimization”, funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF).

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 27, Issue 4, Pages 539–557, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.