joint work with Julian Ortiz
We present a composite step method, designed for equality constrained optimization on differentiable manifolds. The use of retractions allows us to pullback the involved mappings to linear spaces and use tools such as cubic regularization of the objective function and affine covariant damped Newton method for feasibility. We show fast local convergence when different chart retractions are considered. We test our method on equilibrium problems in finite elasticity where the stable equilibrium position of an inextensible transversely isotropic elastic rod under dead load is searched.
joint work with Kristian Bredies
We propose and study a novel optimal transport based regularization of linear dynamic inverse problems. The considered inverse problems aim at recovering a measure valued curve and are dynamic in the sense that (i) the measured data takes values in a time dependent family of Hilbert spaces, and (ii) the forward operators are time dependent and map, for each time, Radon measures into the corresponding data space. The variational regularization we propose bases on dynamic optimal transport. We apply this abstract framework to variational reconstruction in dynamic undersampled MRI
We introduce a class of finite elements for the discretization of energy minimization problems for manifold-valued maps. The discretizations are conforming in the sense that their images are contained in the manifold and invariant under isometries. Based on this desireable invariance, we developed an a priori error theory.