Numerical Methods for Optimal Control with PDE Constraints

Mathematical optimization has significantly expanded its scope during the last decades. This is, in part, due to the fact that it is increasingly often considered in a function space frame-work which allows for differential equations or variational problems as constraints. It provides the natural setting for parameter estimation and optimal control problems, as well as for the variational formulations of image processing, non-invasive testing, and modeling of complex growth processes as they arise in biosciences. Mathematicians in Graz have contributed significantly to the development of optimization with partial differential equations as constraints. In the future these activities will be expanded including new directions in scientific computing. The recent hiring of two new professors in numerical analysis and scientific computing at TUG and KFU offers a new perspective for collaboration between the fields of optimization in the context of partial differential equations and scientific computing in Graz. The groups on optimization and on numerical methods will be joined by a third group of researchers with expertise in selected branches of the biomedical sciences. The central scheme of the proposed SFB is continuous optimization in the context of differential equations and variational inequalities as well as the development of associated numerical methods. The proposed research includes topics on optimal control based on model reduction techniques, semi-smooth Newton methods, optimization in the context of free boundaries and interfaces, inherent optimizing properties of multigrid cycles, and efficient and robust numerical strategies for solving large-scale optimality systems. These are timely problems within optimization and control per se. But they are also triggered by the applications which are investigated in the biomedical sciences group. The topics treated there include magnetic resonance imaging, special near-field techniques for biomedical imaging, computer models for the heart, the cardio-vascular and the insulin-glucose systems. Advancing the solution strategies for the proposed bio-engineering optimization problems will provide new insight, and has the potential of improving diagnostic methods and tools. The combination of expertise in optimization and control for biomedical sciences involving mathematicians at the KFU and TUG as well as biomedical engineering partners at TUG and MUG makes this group of researchers unique.