Dr. K. Welker (Universität Trier)
Wednesday, June 13, 2018 - 13:00
Weierstraß-Institut
Mohrenstr. 39, 10117 Berlin, Erhard-Schmidt-Hörsaal, Erdgeschoss
Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization
Shape optimization problems arise frequently in technological processes which are modelled in the form of partial differential equations (PDEs) or variational inequalities (VIs). In many practical circumstances, the shape under investigation is parametrized by finitely many parameters, which on the one hand allows the application of standard optimization approaches, but on the other hand limits the space of reachable shapes unnecessarily. In this talk, the theory of shape optimization is connected to the differential-geometric structure of shape spaces. In particular, efficient algorithms in terms of shape spaces and the resulting framework from infinite dimensional Riemannian geometry are presented. In this context, the space of H^1/2-shapes is defined. The H^1/2-shapes are a generalization of smooth shapes and arise naturally in shape optimization problems. Moreover, VI constrained shape optimization problems are treated from an analytical and numerical point of view in order to formulate approaches aiming at semi-smooth Newton methods on shape vector bundles. Shape optimization problems constrained by VIs are very challenging because of the necessity to operate in inherently non-linear and non-convex shape spaces. In classical VIs, there is no explicit dependence on the domain, which adds an unavoidable source of non-linearity and non-convexity due to the non-linear and non-convex nature of shape spaces.
submitted by lohse (jutta.lohse@wias-berlin.de, 030 20372587)