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Research Group Numerical Mathematics (Partial Differential Equations)
Research Group Numerical Mathematics (Partial Differential Equations)

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

General Information

nonsmooth_shapes

Keywords

  • non-smooth shape optimization
  • total-variation semi-norm on surfaces
  • anisotropic geometric functionals
  • surface fairing
  • geometric inverse problems
  • consistent discretization
  • automatic geometric reparametrization

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.

Related Publications

Related Talks

  • R. Herzog: Total Variation Image Reconstruction on Smooth Surfaces, SCAIM Seminar, UBC Vancouver, April 2017
  • R. Herzog: Total Variation Image Reconstruction on Smooth Surfaces, SIAM Conference on Optimization, Vancouver, May 2017
  • R. Herzog: Total Variation Image Reconstruction on Smooth Surfaces, Simula Research Lab, Oslo, May 2017
  • J. Vidal: Total Variation Reconstruction on Smooth Surfaces, Poster in OVA8, Alicante, June 2017
  • J. Vidal: Functions of Bounded Variation on Non-Smooth Surfaces, FGI2017, Paderborn, September 2017
  • S. Schmidt and J. Vidal: Non-Smooth Optimization and Computational 3D Image Processing, Minisymposium in FGI2017, Paderborn, September 2017
  • J. Vidal: Total Variation Image Reconstruction on Smooth Surfaces, Non-Smooth Systems 2017, Darmstadt, October 2017
  • R. Herzog: Total Variation Image Reconstruction on Surfaces, 4th Conference on Optimization Methods and Software, Havana, Cuba, December 2017
  • J. Vidal: An Introduction to TV passing through discretization using FE, Chemnitzer Seminar zur Optimalsteuerung 2018, Haus im Ennstal, February 2018
  • R. Herzog: Discrete Total Variation with Finite Elements and Applications in Imaging, Inverse Problems and Optimal Control, GAMM Annual Scientific Meeting, Munich, Germany, March 2018
  • J. Vidal: Discrete Total Variation with Finite Elements and Applications to Imaging, SIGOPT 2018 International Conference on Optimization, Kloster Irsee, March 2018
  • R. Herzog: Discrete Total Variation with Finite Elements, SIAM Conference on Imaging Sciences, Bologna, Italy, June 2018
  • R. Herzog and S. Schmidt: Images and Finite Elements, Minisymposium in SIAM Conference on Imaging Sciences, Bologna, Italy, June 2018
  • J. Vidal: Discrete Total Variation with Finite Elements and Applications to Imaging , 5th European Conference on Computational Optimization, Trier, September 2018
  • R. Herzog: KKT conditions for optimization problems on manifolds, 5th European Conference on Computational Optimization, Trier, September 2018
  • R. Herzog: Discrete Total Variation with Finite Elements , Finite Element Symposium, Chemnitz, September 2018
  • J. Vidal: Applications of the Discrete Total Variation in Imaging , Finite Element Symposium, Chemnitz, September 2018
  • J. Vidal: Variational Mesh Denoising and Surface Fairing Using the Total Variation of the Normal , Vienna Workshop on Computational Optimization, Vienna, December 2018
  • J. Vidal: SPP 1962 Young Researchers Event (Software Course), in Braunschweig, February 2019
  • R. Herzog: Total Variation of the Normal as a Prior in Geometric Inverse Problems , GAMM 2019, Vienna, February 2019
  • J. Vidal: Geometry Processing Problems Using the Total Variation of the Normal Vector Field , GAMM 2019, Vienna, February 2019

Upcoming Talks

  • R. Herzog: TBA , ICIAM 2019, Valencia, July 2019
  • J. Vidal: TBA , ICIAM 2019, Valencia, July 2019