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Reduced basis method for parametrized elliptic optimal control problems. (English) Zbl 1280.49046

Summary: We propose a suitable model reduction paradigm – the certified Reduced Basis (RB) method – for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.

MSC:

49M30 Other numerical methods in calculus of variations (MSC2010)
49M25 Discrete approximations in optimal control
35Q93 PDEs in connection with control and optimization
65K10 Numerical optimization and variational techniques
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
93C20 Control/observation systems governed by partial differential equations
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