GRK 1754: Optimisation and Numerical Analysis for Partial Differential Equations with Non-smooth Structures
Final Report Abstract
Many problems originating from physics, chemistry, biology, or engineering sciences are described by mathematical models involving partial differential equations (PDEs). Frequently they involve nonsmooth structures as, for instance, singularities, interfaces, or inequality constraints. There is an ever growing importance of optimization problems involving nonsmooth functionals in diverse fields, including image analysis, optimal control, and the modeling of material interfaces, for example. The mathematical treatment of the corresponding PDE-based models is crucial for the efficient solution of practical problems. There is a significant demand from academia, as well as from industry and business, for highly qualified junior scientists with postgraduate education in this area. The international research training group (IGDK) “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures” aims at meeting this demand. The major goals of the proposed international research training group are: • transfer of cutting-edge research topics to the education of doctoral students in the form of PhD projects, • development and investigation of novel methods for the analysis, numerical treatment, and the optimization of problems involving PDEs and nonsmooth structures. The involved institutions in Munich and Graz contribute to the joint research program through their internationally visible scientific strengths in applied mathematics: the mathematicians at the Technical University of Munich have long term experience in the analytical and numerical treatment of nonlinear phenomena, and the institutes in Graz contribute their expertise in optimization and numerical analysis. They can also build on pre-existing collaborations. The combination of these expertises results in numerous synergy effects which are exploited by this international research training group. Methodological approaches ranging from adaptivity and nonsmooth optimization to the treatment of interfaces and shapes characterize its research and study program. Our concept for a successful education of doctoral students is based upon joint Munich-Graz supervision, mentoring, and performance control, as well as upon encouraging early scientific independence of doctoral students. The study program, consisting of lectures, compact courses, and summer schools, provides the knowledge of state-of-the art methods for numerical analysis and optimization of problems governed by PDEs.
