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978-3-8439-2442-9, Reihe Mathematik
Stefanie Marita Bott
Adaptive SQP Method with Reduced Order Models for Optimal Control Problems with Constraints on the State Applied to the Navier-Stokes Equations
233 Seiten, Dissertation Technische Universität Darmstadt (2015), Softcover, B5
In this thesis, we develop a new adaptive multilevel SQP method with reduced order models for optimal control problems with constraints on the control and state. In general, solving these problems is computationally expensive. Reduced order models based on FEM are a remedy. However, the challenge is to combine these models in an appropriate way with an optimization method.
To this end, we extend an adaptive SQP algorithm of Ziems and Ulbrich for control constrained problems by reduced order models using a posteriori error estimators and a goal-oriented error estimate. Our new criteria guarantee that the current reduced order model represents the FEM well. In a next step, this algorithm is connected with the Moreau-Yosida regularization to solve state constrained problems. We show first-order and second-order convergence results.
Finally, we apply the adaptive SQP method for reduced order models to flow control problems. Restrictions on the velocity can reduce recirculations or high shear stresses - relevant in fluid mechanics, stirring tasks, combustion or blood flow. Hence, we derive first-order necessary and second-order sufficient optimality conditions and extend our method for a Banach space setting. Numerical results for a cylinder flow with restrictions on the velocity or its gradient demonstrate the efficiency of our algorithm.