Affiliation:
1. MOX - Dipartimento di Matematica , Politecnico di Milano , Piazza Leonardo da Vinci 32, I-20133 Milano , Italy
Abstract
Abstract
We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.
Subject
Applied Mathematics,Industrial and Manufacturing Engineering
Reference50 articles.
1. 1. A. Borzì and V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations. SIAM, 2011.
2. 2. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints. Springer, 2009.
3. 3. A. Quarteroni, G. Rozza, and A. Quaini, Reduced basis methods for optimal control of advection- diffusion problem, in Advances in Numerical Mathematics, W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, and Y. Vassilevski, Editors, pp. 193–216, 2007.
4. 4. L. Dedè, Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems, SIAM Journal on Scientific Computing, vol. 32, no. 2, pp. 997–1019, 2010.
5. 5. L. Dedè, Reduced basis method and error estimation for parametrized optimal control problems with control constraints, Journal of Scientific Computing, vol. 50, no. 2, pp. 287–305, 2012.