Free Boundary Problems and Level-Set Methods

• Applicant: Professor Dr. Michael Hintermüller
• Subject Area: Mathematics
• Term: from 2015 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 271730094

Three major research directions are pursued: (i) Automated regularization parameter choices; (ii) Riemannian optimization for low-rank and sparse tensor decomposition; (iii) contact problems in the heart. The research on integrated automated regularization parameter choice rules in mathematical image processing pursues two approaches: (1) Bilevel optimization problems, which contain the image restoration problem in the constraint set, are formulated, analyzed and numerically solved. The upper level objective encodes statistically relevant information and it involves the regularization parameter and the unknown image as variables. Such problems are notoriously degenerate and require advanced tools from set-valued analysis for deriving sharp stationarity conditions, and sophisticated solvers. (2) More general quasi-equilibrium formulations are considered whenever the formulation of an upper level objective is not possible. Then the restoration problem is augmented by a parameter update step, which involves a confidence technique applied to a statistical quantity (such as local image residuals). In a joint research effort with INVERSE 2.0 and IMAGE local regularization will be applied to diffusion tensor imaging.The research on Riemannian optimization for higher-order low-rank and sparse tensor decomposition requires an appropriate rank concept and a description of the manifold of tensors of multilinear rank. The latter is an important building block in an alternating minimization algorithm for solving a variational problem modeling the low-rank/sparse decomposition without any convex relaxation. To model non/rigid motions, the model contains a parameter, which may be related to the pixel-wise displacement field as in optical flow problems. The tensor decomposition will lead to several cooperations within the SFB: (i) With INVERSE 2.0 it is planned to compare numerically and analytically with PCA/ICA; (ii) with IMAGE temporal-spatial regularization of reconstruction or motion correction tasks will be considered; (iii) with MRI the separation of morphological and temporal features and the decomposition of information from biomarkers will be investigated.Finally, to accurately capture the atrio-ventricular plane displacement, the project will develop a solver for contact problems involving the epi- and peri-cardium which is important for the electromechanical model of HEART. The research involves the introduction of a compliant frictionless contact model, where the compliance results from the displacement dependent boundary condition (implied through the pericardium) on the epicardium and the absence of friction is due to the liquor pericardial. The resulting quasi-equilibrium problem is discretized by a Galerkin finite elements for nearly incompressible materials. Our algorithmic solution procedure will be based on a sequential approximation scheme where each subproblem resembles a Signorini type contact problem with a rigid obstacle.

• DFG Programme: Research Grants
• International Connection: Austria
• Cooperation Partners: Professor Dr.-Ing. Gunther Peichl; Professor Dr. Wolfgang Ring