Kathrin Welker

Efficient PDE Constrained Shape Optimization Based on Steklov--Poincaré-Type Metrics

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which instead require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov--Poincare-type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor involved in the derivation of shape derivatives.

STRUCTURED INVERSE MODELING IN PARABOLIC DIFFUSION PROBLEMS

Often, the unknown diffusivity in diffusive processes is structured by piecewise constant patches. This paper is devoted to efficient methods for the determination of such structured diffusion parameters by exploiting shape calculus. A novel shape gradient is derived in parabolic processes. Furthermore, quasi-Newton techniques are used in order to accelerate shape gradient based iterations in shape space. Numerical investigations support the theoretical results.

Towards a Lagrange–Newton Approach for PDE Constrained Shape Optimization

The novel Riemannian view on shape optimization developed in [27] is extended to a Lagrange–Newton approach for PDE constrained shape optimization problems. The extension is based on optimization on Riemannian vector space bundles and exemplified for a simple numerical example.

Algorithmic Aspects of Multigrid Methods for Optimization in Shape Spaces

We examine the interaction of multigrid methods and shape optimization in appropriate shape spaces. The impact of discrete approximations of geometrical quantities, like the mean curvature, on a multigrid shape optimization algorithm with quasi-Newton updates is investigated. Both multigrid and quasi-Newton methods are necessary to achieve mesh-independent convergence and, thus, scalable algorithms for supercomputers. For the purpose of illustration, we consider a complex model for the identification of cellular structures in biology with minimal compliance in terms of elasticity and diffusion equations.

PDE Constrained Shape Optimization as Optimization on Shape Manifolds

The novel Riemannian view on shape optimization introduced in [14] is extended to a Lagrange–Newton as well as a quasi–Newton approach for PDE constrained shape optimization problems.

On Optimization Transfer Operators in Shape Spaces

We discuss several approaches towards shape optimization problems in the context of partial differential equations. The general framework used is that of optimization on shape manifolds. Here, transfer operators between tangent spaces and the manifold are crucial and in general non-trivial.

Optimization in the Space of Smooth Shapes

The theory of shape optimization problems constrained by partial differential equations is connected with the differential-geometric structure of the space of smooth shapes.