Events

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Event Information:

  • Thu
    03
    Dec
    2026

    SC Seminar: Stephan Schmidt

    11:15Hybrid (Room 32-349 and via Zoom)

    PD Dr. Stephan Schmidt, Department of Mathematics, University of Trier, Germany

    Title: Non-Smooth Geometric Inverse Problems with Partial Differential Equations

    Abstract:

    Many real-world problems benefit from non-smooth PDE-constrained shape optimization and the reconstruction of shapes with kinks. Examples include tomography for geological faults or underground structures, the reconstruction of buildings, the design of hull chines on watercraft, the design of chevrons in aerodynamic applications, the reconstruction of faceted electromagnetic structures, the identification of cracks or fractures in technical and medical applications, and contact problems. Few tailor-made mathematical priors exist for such applications.

    To address this, the total variation (TV) and the total generalized variation (TGV) of the normal are introduced as regularization functionals within PDE-constrained shape optimization. Algorithmically, this leads to L1-type non-smoothness within moving geometries. For this setting, a tailor-made ADMM splitting scheme is developed, which lifts the classical split Bregman algorithm into this manifold setting and decouples the L1 non-smoothness from the smooth part. Both parts are then solved iteratively in alternation. For the non-smooth part, a shrinkage-type solution adapted to the manifold setting is obtained, while the smooth part is addressed through a holistic approach that solves the state equation, the adjoint equation, and the effective geometric flow jointly via a fully coupled Newton-type algorithm.

    Finally, to enable a geometric TGV regularization, a custom adaptation of the Raviart-Thomas finite element is introduced, which discretizes linear mappings between tangential spaces of manifolds. This eliminates the staircasing effect known from TV-based image reconstruction, which otherwise also carries over into three-dimensional reconstructions.

    How to join online

    You can join online via Zoom, using the following link:
    https://uni-kl-de.zoom-x.de/j/69269239534?pwd=Z9UOzMpkhMjrxVhll3d49sNHFe9Fd1.1