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

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 nonsmooth 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.

Publications

Marc Herrmann, Roland Herzog, Heiko Kröner, Stephan Schmidt, Josè Vidal: Analysis and an Interior Point Approach for TV Image Reconstruction Problems on Smooth Surfaces , SIAM J. Imaging Sci., 11(2), 889–922, 2018 (SPP1962-019).

Marc Herrmann, Roland Herzog, Stephan Schmidt, Josè Vidal, Gerd Wachsmuth: Discrete Total Variation with Finite Elements and Applications to Imaging, J Math Imaging Vis, 2018 (SPP1962-054).

Preprints

Tommy Etling, Roland Herzog, Estefanía Loayza, Gerd Wachsmuth: First and Second Order Shape Optimization Based on Restricted Mesh Deformations (SPP1962-088, 10/2018, [bib])

Marc Herrmann, Roland Herzog, Stephan Schmidt, Josè Vidal, Gerd Wachsmuth: Discrete Total Variation with Finite Elements and Applications to Imaging (SPP1962-054, 04/2018, [bib])

Ronny Bergmann, Roland Herzog: Intrinsic Formulation of KKT Conditions and Constraint Qualifications on Smooth Manifolds (SPP1962-052, 04/2018, [bib])

Marc Herrmann, Roland Herzog, Heiko Kröner, Stephan Schmidt, Josè Vidal: Analysis and an Interior Point Approach for TV Image Reconstruction Problems on Smooth Surfaces (SPP1962-019, 04/2017, [bib])

Research Area

Modeling, problem analysis, algorithm design and convergence analysis

The focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.

Realization of algorithms, adaptive discretization and model reduction

As the target applications of this SPP involve non-smooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper model-reduction techniques need to be developed.

Incorporation of parameter dependencies and robustness

In many applications the robustness of solutions with respect to a given parameter range (uncertainty set) is highly relevant. Correspondingly, in this research area of the SPP, bi- or multilevel optimization approaches will be studied in order to robustify problem solutions against uncertain parameters.

Members

  • member's portrait

    Prof. Roland Herzog

    Technische Universität Chemnitz
    Principal Investigator
  • member's portrait

    Dr. Stephan Schmidt

    Julius-Maximilians-Universität Würzburg
    Principal Investigator
  • member's portrait

    Marc Herrmann

    Julius-Maximilians-Universität Würzburg
    Research Assistant
  • member's portrait

    Dr. José Vidal Núñez

    Technische Universität Chemnitz
    Research Assistant

Project Related News

  • Oct 24, 2018 : New preprint submitted

    Gerd Wachsmuth submitted the preprint SPP1962-088, First and Second Order Shape Optimization Based on Restricted Mesh Deformations

  • Jan 01, 2017 : Welcome to our new project member

    José Vidal Núñez joins project 8.

  • Oct 10, 2016 : Welcome to our new project member

    Marc Hermann joins project 8