Reduced basis method for parametrized elliptic optimal control problems

45/2012

Reduced basis method for parametrized elliptic optimal control problems

Wednesday 24th October 2012

Negri, F.; Rozza, G.; Manzoni, A.; Quarteroni, A.

We propose a suitable model reduction paradigm -the certied reduced basis method (RB) - for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations (PDEs). In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint. Firstly, 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 provided: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling to perform competitive Offine-Online splitting in the computational procedure; an efficient and rigorous a posteriori error estimate on the state, control and adjoint variables as well as on the cost functional. Finally, the reduction scheme is applied to some numerical tests conrming the theoretical results and showing the efficiency of the proposed technique. Keywords: reduced basis methods, parametrized optimal control problems, saddle-point problems, model order reduction, PDE-constrained optimization, a posteriori error estimate