A Calculus for Non-Smooth Shape Optimization with Applications to Geometric Inverse Problems


Start: 2017-01-01
End: 2019-12-31
Principal Investigators: Roland Herzog, Stephan Schmidt
Staff: José Vidal Núñez, Marc Herrmann
Funded by: DFG within the Priority Program funding scheme
Part of: Non-Smooth and Complementarity-based Distributed Parameter Systems (SPP 1962)

Project Description

We intend to lay the mathematical foundations for a rigorous non-smooth shape calculus. A typical application area are geometric inverse problems, which often involve partial differential equations. Surface fairing with edge preservation as well as the detection of non-smooth inclusions through remote sensing and tomography are typical examples of problems greatly benefiting from this research.

We shall introduce new geometric functionals, which allow a fine-grained control over the non-smooth features of desired shapes. As a particular example, we mention the total surface variation of the normal vector field. To this end, we extend the concept of the total variation semi-norm to non-smooth functions and geometric quantities on non-smooth surfaces. This novel approach will also allow for an anisotropic control of preferred shapes.

Hand in hand with the theoretical considerations above, we will also focus on consistent discrete realizations. In view of the fact that any triangulated surface is essentially non-smooth, we expect considerable improvements of the present state of the art in computational shape optimization. For example, we will address the question of finding the best possible curvature approximation consistent with the tangential Stokes formula.

To exemplify the benefits of our novel approach, we intend to solve a number of prototypical application problems of increasing complexity, in particular problems in surface fairing, inverse obstacle problems, electrical impedance tomography and inverse electro-magnetic scattering problems governed by Maxwell’s equations.

Associated Publications

  • Ronny Bergmann, Roland Herzog, Maurício Silva Louzeiro, Daniel Tenbrinck and José Vidal-Núñez
    Fenchel duality theory and a primal-dual algorithm on Riemannian manifolds
    Foundations of Computational Mathematics 21(6), p.1465-1504, 2021
    arXiv:1908.02022 doi:10.1007/s10208-020-09486-5
    bibtex 
    @ARTICLE{BergmannHerzogSilvaLouzeiroTenbrinckVidalNunez:2021:1,
  AUTHOR = {Bergmann, Ronny and Herzog, Roland and Silva Louzeiro, Maurício and Tenbrinck, Daniel and Vidal-Núñez, José},
  PUBLISHER = {Springer Science and Business Media LLC},
  DATE = {2021-01},
  DOI = {10.1007/s10208-020-09486-5},
  EPRINT = {1908.02022},
  EPRINTTYPE = {arXiv},
  JOURNALTITLE = {Foundations of Computational Mathematics},
  NUMBER = {6},
  PAGES = {1465--1504},
  TITLE = {Fenchel duality theory and a primal-dual algorithm on Riemannian manifolds},
  VOLUME = {21},
}
  • Ronny Bergmann, Marc Herrmann, Roland Herzog, Stephan Schmidt and José Vidal-Núñez
    Total variation of the normal vector field as shape prior
    Inverse Problems 36(5), 2020
    arXiv:1902.07240 doi:10.1088/1361-6420/ab6d5b
    bibtex 
    @ARTICLE{BergmannHerrmannHerzogSchmidtVidalNunez:2020:1,
  AUTHOR = {Bergmann, Ronny and Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal-Núñez, José},
  DATE = {2020},
  DOI = {10.1088/1361-6420/ab6d5b},
  EPRINT = {1902.07240},
  EPRINTTYPE = {arXiv},
  JOURNALTITLE = {Inverse Problems},
  NUMBER = {5},
  PAGES = {054004},
  TITLE = {Total variation of the normal vector field as shape prior},
  VOLUME = {36},
}
  • Ronny Bergmann, Marc Herrmann, Roland Herzog, Stephan Schmidt and José Vidal-Núñez
    Discrete total variation of the normal vector field as shape prior with applications in geometric inverse problems
    Inverse Problems 36(5), 2020
    arXiv:1908.07916 doi:10.1088/1361-6420/ab6d5c
    bibtex 
    @ARTICLE{BergmannHerrmannHerzogSchmidtVidalNunez:2020:2,
  AUTHOR = {Bergmann, Ronny and Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal-Núñez, José},
  DATE = {2020},
  DOI = {10.1088/1361-6420/ab6d5c},
  EPRINT = {1908.07916},
  EPRINTTYPE = {arXiv},
  JOURNALTITLE = {Inverse Problems},
  NUMBER = {5},
  PAGES = {054003},
  TITLE = {Discrete total variation of the normal vector field as shape prior with applications in geometric inverse problems},
  VOLUME = {36},
}
  • Ronny Bergmann, Marc Herrmann, Roland Herzog, Stephan Schmidt and José Vidal-Núñez
    Geometry processing problems using the total variation of the normal vector field
    Proceedings in Applied Mathematics and Mechanics 19(1), 2019
    doi:10.1002/pamm.201900189
    bibtex 
    @ARTICLE{BergmannHerrmannHerzogSchmidtVidalNunez:2019:3,
  AUTHOR = {Bergmann, Ronny and Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal-Núñez, José},
  PUBLISHER = {Wiley},
  DATE = {2019},
  DOI = {10.1002/pamm.201900189},
  JOURNALTITLE = {Proceedings in Applied Mathematics and Mechanics},
  NUMBER = {1},
  TITLE = {Geometry processing problems using the total variation of the normal vector field},
  VOLUME = {19},
}
  • Ronny Bergmann and Roland Herzog
    Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds
    SIAM Journal on Optimization 29(4), p.2423-2444, 2019
    arXiv:1804.06214 doi:10.1137/18M1181602
    bibtex 
    @ARTICLE{BergmannHerzog:2019:1,
  AUTHOR = {Bergmann, Ronny and Herzog, Roland},
  DATE = {2019},
  DOI = {10.1137/18M1181602},
  EPRINT = {1804.06214},
  EPRINTTYPE = {arXiv},
  JOURNALTITLE = {SIAM Journal on Optimization},
  NUMBER = {4},
  PAGES = {2423--2444},
  TITLE = {Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds},
  VOLUME = {29},
}
  • Marc Herrmann, Roland Herzog, Stephan Schmidt, José Vidal-Núñez and Gerd Wachsmuth
    Discrete total variation with finite elements and applications to imaging
    Journal of Mathematical Imaging and Vision 61(4), p.411-431, 2019
    arXiv:1804.07477 doi:10.1007/s10851-018-0852-7
    bibtex 
    @ARTICLE{HerrmannHerzogSchmidtVidalNunezWachsmuth:2019:1,
  AUTHOR = {Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal-Núñez, José and Wachsmuth, Gerd},
  DATE = {2019},
  DOI = {10.1007/s10851-018-0852-7},
  EPRINT = {1804.07477},
  EPRINTTYPE = {arXiv},
  JOURNALTITLE = {Journal of Mathematical Imaging and Vision},
  NUMBER = {4},
  PAGES = {411--431},
  TITLE = {Discrete total variation with finite elements and applications to imaging},
  VOLUME = {61},
}
  • Marc Herrmann, Roland Herzog, Heiko Kröner, Stephan Schmidt and José Vidal-Núñez
    Analysis and an interior point approach for TV image reconstruction problems on smooth surfaces
    SIAM Journal on Imaging Sciences 11(2), p.889-922, 2018
    doi:10.1137/17M1128022
    bibtex 
    @ARTICLE{HerrmannHerzogKroenerSchmidtVidalNunez:2018:1,
  AUTHOR = {Herrmann, Marc and Herzog, Roland and Kröner, Heiko and Schmidt, Stephan and Vidal-Núñez, José},
  DATE = {2018},
  DOI = {10.1137/17M1128022},
  JOURNALTITLE = {SIAM Journal on Imaging Sciences},
  NUMBER = {2},
  PAGES = {889--922},
  TITLE = {Analysis and an interior point approach for TV image reconstruction problems on smooth surfaces},
  VOLUME = {11},
}

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