Efficient solution methods for constrained heat phenomena

The project is concerned with the design of fast and robust optimization algorithms for the optimal control of nonlinear heat phenomena. Also reaction-diffusion equations which arise, for example, in chemical engineering and mathematical biology are considered. The main focus is an problems which include constraints an the control action or an the temperature, i.e. state, of the system. For instance, constraints an the control in form of upper and lower bounds frequently occur in practical applications for technical reasons. lt also may happen that the temperature must be bounded, e.g., in order to prevent material phase transition. In other applications, volume preservation or bounds are an issue. While the modeling of these types of constraints is rather accessible, the efficient numerical treatment and many theoretical issues -- especially when including constraints an the state of the system -- in an optimization context require a substantial amount of research work. Typically the algorithmic development is based an theoretical investigations of optimality conditions related to the sysem under consideration. This aspect leads to appropriate discretizations, the efficient solution of certain subproblems and the analysis of convergence properties. Besides the design and analysis of the algorithms, as a first step towards a decision tool for the engineering practice the algorithms are implemented in a modular way so that it will be possible to combine different constraints by choice.