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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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# Existence of variational source conditions for nonlinear inverse problems in Banach spaces

Jens Flemming
Published Online: 2017-11-29 | DOI: https://doi.org/10.1515/jiip-2017-0092

## Abstract

Variational source conditions proved to be useful for deriving convergence rates for Tikhonov’s regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.

MSC 2010: 65J20; 47J06

## References

• [1]

S. W. Anzengruber, S. Bürger, B. Hofmann and G. Steinmeyer, Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization, Inverse Problems 32 (2016), no. 3, Article ID 035002.

• [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.

• [3]

M. Burger, J. Flemming and B. Hofmann, Convergence rates in ${\mathrm{\ell }}^{1}$-regularization if the sparsity assumption fails, Inverse Problems 29 (2013), no. 2, Article ID 025013. Google Scholar

• [4]

M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Problems 20 (2004), no. 5, 1411–1421.

• [5]

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.

• [6]

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

• [7]

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

• [8]

J. Flemming, Convergence rates for ${\mathrm{\ell }}^{1}$-regularization without injectivity-type assumptions, Inverse Problems 32 (2016), no. 9, Article ID 095001. Google Scholar

• [9]

J. Flemming and D. Gerth, Injectivity and weak*-to-weak continuity suffice for convergence rates in ${\mathrm{\ell }}^{1}$-regularization, J. Inverse Ill-Posed Probl. (2017), 10.1515/jiip-2017-0008. Google Scholar

• [10]

J. Flemming and B. Hofmann, A new approach to source conditions in regularization with general residual term, Numer. Funct. Anal. Optim. 31 (2010), no. 1–3, 254–284.

• [11]

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

• [12]

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.

• [13]

M. Grasmair, M. Haltmeier and O. Scherzer, The residual method for regularizing ill-posed problems, Appl. Math. Comput. 218 (2011), no. 6, 2693–2710.

• [14]

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.

• [15]

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

• [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.

• [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.

• [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.

• [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.

• [20]

P. Mathé and B. Hofmann, How general are general source conditions?, Inverse Problems 24 (2008), no. 1, Article ID 015009.

• [21]

P. Mathé and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems 19 (2003), no. 3, 789–803.

• [22]

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

• [23]

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

• [24]

C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, 2002. Google Scholar

Accepted: 2017-11-10

Published Online: 2017-11-29

Published in Print: 2018-04-01

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 26, Issue 2, Pages 277–286, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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