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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Kathrin Welker (Trier University)

Efficient PDE Constrained Shape Optimization

Shape optimization problems arise frequently in technological processes which are modelled by partial differential equations (PDEs). 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. Shape calculus presents a way to circumvent this dilemma. However, so far it is mainly applied in the form of gradient descent methods, which can be shown to converge. The major difference between shape optimization and the standard PDE constrained optimization framework is the lack of a linear space structure on shape spaces. If one cannot use a linear space structure, then the next best structure is a Riemannian manifold structure. We consider optimization problems which are constrained by PDEs and embed these problems in the framework of optimization on Riemannian manifolds to provide efficient techniques for PDE constrained shape optimization problems on shape spaces. The Riemannian geometrical point of view on unconstrained shape optimization established in [1] is extended to a Lagrange-Newton, as well as to a quasi-Newton technique in shape spaces for PDE constrained shape optimization problems. These techniques are based on the so-called Hadamard-form of shape derivatives, i.e., in the form of integrals over the surface of the shape under investigation. It is often a very tedious, not to say painful, process to derive such surface expressions. Along the way, there appear volume formulations in the form of integrals over the whole domain as an intermediate step. Domain integral formulations of shape derivatives are coupled with optimization strategies on shape spaces. Efficient shape algorithms reducing analytical effort and programming work are presented and a novel shape space is proposed in this context.

[1] V. H. Schulz. A Riemannian view on shape optimization. Foundations of Computational Mathematics, 14:483-501, 2014.