Compact gradient tracking in shape optimization
- Eppler, Karsten
- Harbrecht, Helmut
2010 Mathematics Subject Classification
- 49Q10 49M15 65N38 65K10 49K20 65T60
- shape calculus, boundary integral equations, multiscale methods, sufficient second order conditions, ill-posed problem
In the present paper we consider the minimization of gradient tracking functionals defined on a compact and fixed subdomain of the domain of interest. The underlying state is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We prove that, in contrast to other type of objectives, defined on the whole domain, the shape Hessian is not strictly $H^1/2$-coercive at the optimal domain which implies ill-posedness of the shape problem under consideration. Shape functional and gradient require only knowledge of the cauchy data of the state and its adjoint on the boundaries of the domain and the subdomain. These data can be computed in terms of boundary integral equations when reformulating the underlying differential equations as transmission problems. Thanks to fast boundary element techniques, we derive an efficient and accurate computation of the ingredients for optimization. Consequently, difficulties in the solution are related to the ill-posedness of the problem under consideration.