Optimal Boundary Control in Energy Spaces

In this thesis we study optimal boundary control problems in energy spaces, their construction of robust preconditioners and applications to arterial blood flow. More precisely we first consider the unconstrained optimal Dirichlet and Neumann boundary control problems for the Poisson equation as a model problem. In both cases it turns out that the control can be eliminated and thus a variational formulation in saddle point structure is obtained. The existence and uniqueness of a solution is investigated and for the finite element discretization optimal error estimates are shown. In the particular case of the Laplace equation as a constraint we are able to prove that the primal states of Dirichlet and Neumann boundary control problem coincide. Further, the construction of corresponding robust preconditioners for optimal boundary control problems is investigated. We observe that the optimal boundary control problems are related to biharmonic equation of first kind. For the preconditioner we consider either a preconditioner motivated from boundary element methods, resulting in an optimal condition number, or a multilevel preconditioner of BPX type, where the condition number depends on a logarithmic factor of the mesh size. For both, the related spectral equivalent estimates are proven. Several numerical examples illustrate the obtained theoretical results. Moreover, we study the application of the optimal Dirichlet boundary control problem to arterial blood flow. In particular, we are interested in the optimal inflow profile into an arterial system, motivated for instance by an artificial heart pump. Also, we investigate on hemodynamic indicators, for showing potential risk factors for aneurysms. Here a comparison of two commonly used approaches is considered, where it is shown by several numerical simulations that these can lead to significant differences in the solution. This model problem motivates also the optimization of hemodynamic indicators. Finally, several numerical examples are presented.