SPP 1253: Optimisation with Partial Differential Equations

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Solving optimisation problems subject to constraints involving distributed parameter systems (DPS) is one of the most challenging problems in the context of industrial, medical and economical applications. In particular, in the design of aircraft, "moving bed" processes in chemical engineering, crystal growth etc. the forward simulation followed by the variation of the optimisation variables has proved to be inefficient. Instead, the optimisation of design and topology of structures and the control of processes involving partial differential equations (PDEs), interpreted as DPS, has to be treated simultaneously such that modern mathematical methods for optimisation with PDEs are interlinked with adaptive goal-oriented simulation tools. After proper structure respecting discretisation, the number of optimisation variables varies typically in the range of up to several millions. It is only very recently that the enormous advances in computing power have made it possible to attack problems of this size. However, in order to accomplish this task it is crucial to utilise and further explore the specific mathematical structure of prototype applications and to develop new mathematical approaches concerning structure exploiting algorithms, model reduction, parallelisability, adaptivity of numerical schemes for the corresponding optimality systems based on a posteriori error estimates and the optimisation with PDEs involving control and state constraints. Methods of automatic differentiation (AD) will turn out to be very important in handling the massive data involved in problems of optimal design, shape and topology as well as in time-dependent problems.