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Optimization with PDE Constraints pp 25–63Cite as

Phase Field Methods for Binary Recovery

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 101))

Abstract

We consider the inverse problem of recovering a binary function from blurred and noisy data. Such problems arise in many applications, for example image processing and optimal control of PDEs. Our formulation is based on the Mumford-Shah model, but with a phase field approximation to the perimeter regularisation. We use a double obstacle potential as well as a smooth double well potential. We introduce an iterative method for solving the problem, develop a suitable discretisation of this iterative method, and prove some convergence results. Numerical simulations are presented which illustrate the usefulness of the approach and the relative merits of the phase field models.

Mathematics Subject Classification (2010). Primary 49N45; Secondary 65K10, 68U10

This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant EP/H023364/1 and by the ESF within the Programme OPTPDE.

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Acknowledgements

We are grateful to Carsten Gräser for sharing his Dune-Solvers code for the TNNMG method.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Charles Brett, Charles M. Elliott & Andreas S. Dedner

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  1. Charles Brett
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  2. Charles M. Elliott
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  3. Andreas S. Dedner
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Correspondence to Charles Brett .

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Editors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, Augsburg, Germany

    Ronald Hoppe

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Brett, C., Elliott, C.M., Dedner, A.S. (2014). Phase Field Methods for Binary Recovery. In: Hoppe, R. (eds) Optimization with PDE Constraints. Lecture Notes in Computational Science and Engineering, vol 101. Springer, Cham. https://doi.org/10.1007/978-3-319-08025-3_2

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