Reduced basis method for parametrized optimal control problems governed by PDEs

This master thesis aims at the development, analysis and computer implementation of effcient numerical methods for the solution of optimal control problems based on parametrized partial differential equations. Our goal isfto develop a new approach based on suitable model reduction paradigm --the reduced basis method (RB)-- for the rapid and reliable solution of control problems which may occur in several engineering contexts. In particular, we develop the methodology for parametrized quadratic optimization problem with either coercive elliptic equations or Stokes equations as constraints. Firstly, we recast the optimal control problem in the framework of mixed variational 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 estimation on the state, control and adjoint variables as well as on the cost functional. The reduction scheme is applied to several numerical tests conrming the theoretical results and demonstrating the efficiency of the proposed technique. Moreover an application to an (idealized) inverse problem in haemodynamics is discussed, showing the versatility and potentiality of the method in tackling parametrized optimal control problems that could arise in a a broad variety of application contexts.

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