Model reduction by adaptive discretization in optimal control

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This project will employ the concept of goal-oriented adaptivity for model reduction in solving optimal control problems governed by partial differential equations (PDE). The underlying framework is the Dual Weighted Residual (DWR) method which was originally developed by R. Becker and the applicant for the adaptive discretization of PDE by the finite element Galerkin method. In this approach residual-based weighted a posteriori error estimates are derived for quantities of interest, where the weights are obtained by numerically solving an associated dual problem . Due to the use of problem inherent sensitivity information, these a posteriori error estimates are tailored to the special needs of the computation. This allows for successively improved control of spatial and time discretization which eventually results in highly economical discretization. In this project the main emphasis is on nonstationary optimal control problems, which pose particularly high requirements on computational resources, and on problems involving additional constraints for controls and states. In these cases model reduction by adaptive discretization may prove most useful. Research on these topics in the context of PDE-constrained optimal control has started only recently and there are still many theoretical as well as practical open questions.