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New Trends in Parameter Identification for Mathematical Models pp 31–51Cite as

Birkhäuser

Convex Regularization of Discrete-Valued Inverse Problems

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Abstract

This work is concerned with linear inverse problems where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex but nondifferentiable regularization term. This allows applying standard approaches to show well-posedness and convergence rates in Bregman distance. Using the specific properties of the regularization term, it can be shown that convergence (albeit without rates) actually holds pointwise. Furthermore, the resulting Tikhonov functional can be minimized efficiently using a semi-smooth Newton method. Numerical examples illustrate the properties of the regularization term and the numerical solution.

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Acknowledgements

This work was supported by the German Science Fund (DFG) under grant CL 487/1-1. The authors also wish to thank Daniel Wachsmuth for several helpful remarks.

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Authors and Affiliations

  1. Faculty of Mathematics, University Duisburg-Essen, 45117, Essen, Germany

    Christian Clason & Thi Bich Tram Do

Authors
  1. Christian Clason
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  2. Thi Bich Tram Do
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Correspondence to Christian Clason .

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Editors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany

    Bernd Hofmann

  2. Department of Mathematics, Federal University of Santa Catarin, Florianopolis, Brazil

    Antonio Leitão

  3. Pura e Aplicada, Instituto Nacional de Matematica, Rio de Janeiro, Rio de Janeiro, Brazil

    Jorge P. Zubelli

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Clason, C., Do, T.B.T. (2018). Convex Regularization of Discrete-Valued Inverse Problems. In: Hofmann, B., Leitão, A., Zubelli, J. (eds) New Trends in Parameter Identification for Mathematical Models. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70824-9_2

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