Startseite A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations
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A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

  • Jens Flemming EMAIL logo
Veröffentlicht/Copyright: 11. April 2018

Abstract

We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

MSC 2010: 65J22; 47A52

Acknowledgements

The author thanks the two referees for several valuable comments which improved the paper’s quality. Especially Remark 3.1 bases on a referee’s hint.

References

[1] V. Albani, P. Elbau, M. V. de Hoop and O. Scherzer, Optimal convergence rates results for linear inverse problems in Hilbert spaces, Numer. Funct. Anal. Optim. 37 (2016), no. 5, 521–540. 10.1080/01630563.2016.1144070Suche in Google Scholar PubMed PubMed Central

[2] R. I. Boţ and B. Hofmann, An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems, J. Integral Equations Appl. 22 (2010), no. 3, 369–392. 10.1216/JIE-2010-22-3-369Suche in Google Scholar

[3] S. Bürger, J. Flemming and B. Hofmann, On complex-valued deautoconvolution of compactly supported functions with sparse Fourier representation, Inverse Problems 32 (2016), no. 10, Article ID 104006. 10.1088/0266-5611/32/10/104006Suche in Google Scholar

[4] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Suche in Google Scholar

[5] J. Flemming, Generalized Tikhonov Regularization and Modern Convergence Rate Theory in Banach Spaces, Shaker, Aachen, 2012. Suche in Google Scholar

[6] J. Flemming, Solution smoothness of ill-posed equations in Hilbert spaces: Four concepts and their cross connections, Appl. Anal. 91 (2012), no. 5, 1029–1044. 10.1080/00036811.2011.563736Suche in Google Scholar

[7] J. Flemming, Existence of variational source conditions for nonlinear inverse problems in Banach spaces, J. Inverse Ill-Posed Probl. 26 (2018), no. 2, 277–286. 10.1515/jiip-2017-0092Suche in Google Scholar

[8] J. Flemming and D. Gerth, Injectivity and weak-to-weak continuity suffice for convergence rates in 1-regularization, J. Inverse Ill-Posed Probl. 26 (2018), no. 1, 85–94. 10.1515/jiip-2017-0008Suche in Google Scholar

[9] J. Flemming, B. Hofmann and P. Mathé, Sharp converse results for the regularization error using distance functions, Inverse Problems 27 (2011), no. 2, Article ID 025006. 10.1088/0266-5611/27/2/025006Suche in Google Scholar

[10] M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods, Inverse Problems 26 (2010), no. 11, Article ID 115014. 10.1088/0266-5611/26/11/115014Suche in Google Scholar

[11] M. Grasmair, Variational inequalities and higher order convergence rates for Tikhonov regularisation on Banach spaces, J. Inverse Ill-Posed Probl. 21 (2013), no. 3, 379–394. 10.1515/jip-2013-0002Suche in Google Scholar

[12] 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. 10.1088/0266-5611/23/3/009Suche in Google Scholar

[13] B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems 28 (2012), no. 10, Article ID 104006. 10.1088/0266-5611/28/10/104006Suche in Google Scholar

[14] B. Hofmann and P. Mathé, Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales, preprint (2017), https://arxiv.org/abs/1705.03289. 10.1088/1361-6420/aa9b59Suche in Google Scholar

[15] T. Hohage and F. Weidling, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problems 31 (2015), no. 7, Article ID 075006. 10.1088/0266-5611/31/7/075006Suche in Google Scholar

[16] 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. 10.1137/16M1067445Suche in Google Scholar

[17] T. Hohage and F. Werner, Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data, Numer. Math. 123 (2013), no. 4, 745–779. 10.1007/s00211-012-0499-zSuche in Google Scholar

[18] S. Kindermann, Convex Tikhonov regularization in Banach spaces: New results on convergence rates, J. Inverse Ill-Posed Probl. 24 (2016), no. 3, 341–350. 10.1515/jiip-2015-0038Suche in Google Scholar

[19] C. König, F. Werner and T. Hohage, Convergence rates for exponentially ill-posed inverse problems with impulsive noise, SIAM J. Numer. Anal. 54 (2016), no. 1, 341–360. 10.1137/15M1022252Suche in Google Scholar

[20] P. Mathé and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems 19 (2003), no. 3, 789–803. 10.1088/0266-5611/19/3/319Suche in Google Scholar

[21] A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems, SIAM J. Numer. Anal. 34 (1997), no. 2, 517–527. 10.1137/S0036142993253928Suche in Google Scholar

[22] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Appl. Math. Sci. 167, Springer, New York, 2009. Suche in Google Scholar

[23] 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. 10.1515/9783110255720Suche in Google Scholar

[24] B. Sprung and T. Hohage, Higher order convergence rates for Bregman iterated variational regularization of inverse problems, preprint (2017), https://arxiv.org/abs/1710.09244. 10.1007/s00211-018-0987-xSuche in Google Scholar

Received: 2017-12-12
Revised: 2018-03-26
Accepted: 2018-03-26
Published Online: 2018-04-11
Published in Print: 2018-10-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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