Free Boundary Problems and Level-Set Methods
Final Report Abstract
Driven by applications in the biomedical sciences, the project pursued three major research directions: (i) Automated regularization parameter choice rules in medical image processing; (ii) Riemannian optimization for low-rank and sparse tensor decomposition related to dynamic imaging with applications in biomedicine; (iii) some basic aspects of the mathematics of contact problems in the (human) heart. Concerning (i) a novel duality based bilevel optimization approach was proposed, analyzed and numerically tested. In the context of the bilevel optimization problem, the upper level objective aimed at quantifying the quality of an associated distributed regularization function and used methods from the statistics of extremes. The lower level optimization problem accepted the regularization weight as a parameter and yielded an associated reconstruction. In the project work, first total variation (TV) regularization was used as the underlying prior. Later, this was extended to total generalized variation. The project work turned out fruitful in terms of unexpected abstract results on the density of convex intersections and also on variable step mollifiers. Moreover, our analytic insights into weighted TV allowed us to establish a rather general framework for structural TV regularization in inverse problems. Our Riemannian manifold approach to (ii) was motivated by an analytical and numerical comparison of independent / principal component analysis in imaging of cine data in medical applications, temporal-spatial regularization of reconstruction or motion correction tasks, and the separation of morphological and temporal features and the decomposition of information from biomarkers. We proposed, analyzed and tested an alternating minimization method where the first subproblem amounts to solving a sparse projection problem and the second one uses a dogleg trust region step to compute an update direction in the tangent space to a fixed rank manifold. The new method compared favorably against the state-of-the-art solver. Finally, work on a specific contact problem motivated by applications in biomedicine was studied analytically as well as numerically. For a 2D model problem, which turned out to be a quasi-variational inequality of compliant obstacle type including a volume constraint, existence of solutions, adaptive finite element solvers and numerical tests were performed successfully.
