On the treatment of distributed uncertainties in PDE-constrained optimization

Most physical phenomena are significantly affected by uncertainties associated with variations in properties and fluctuations in operating conditions. This has to be reflected also in the design and control of real-application systems. Recent advances in PDE constrained optimization open the possibility of realistic optimization of such systems in the presence of model and data uncertainties. These emerging techniques require only the knowledge of the probability distribution of the perturbations, which is usually available, and provide optimization solutions that are robust with respect to the stochasticity of the application framework. In this paper, some of these methodologies are reviewed. The focus is on PDE constrained optimization frameworks where distributed uncertainties are modeled by random fields and the structures in the underlying optimization problems are exploited in the form of multigrid methods and one-shot methods. Applications are presented, including control problems with uncertain coefficients and erodynamic design under geometric uncertainties. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.