This paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties-with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates-and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.

%B Computers and Mathematics with Applications %V 69 %P 319–336 %G eng %R 10.1016/j.camwa.2014.12.010 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2013 %T Reduced basis method for parametrized elliptic optimal control problems %A Federico Negri %A Gianluigi Rozza %A Andrea Manzoni %A Alfio Quarteroni %X We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-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. %B SIAM Journal on Scientific Computing %V 35 %P A2316–A2340 %G eng %R 10.1137/120894737 %0 Conference Proceedings %B European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012 %D 2012 %T Reduction strategies for PDE-constrained oprimization problems in Haemodynamics %A Gianluigi Rozza %A Andrea Manzoni %A Federico Negri %K inverse problems %X Solving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems. This framework is based on a suitable saddle-point formulation of the optimal control problem and exploits the reduced basis method for the rapid and reliable solution of parametrized PDEs, leading to a relevant computational reduction with respect to traditional discretization techniques such as the finite element method. This allows a very efficient evaluation of state solutions and cost functionals, leading to an effective solution of repeated optimal control problems, even on domains of variable shape, for which a further (geometrical) reduction is pursued, relying on flexible shape parametrization techniques. This setting is applied to the solution of two problems arising from haemodynamics, dealing with both data reconstruction and data assimilation over domains of variable shape,\\r\\nwhich can be recast in a common PDE-constrained optimization formulation. %B European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012 %G en %U http://hdl.handle.net/1963/6338 %1 6268 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T16:30:48Z\\r\\nNo. of bitstreams: 1\\r\\npaper_ECCOMAS2012_AM_SISSAreport.pdf: 4599005 bytes, checksum: 014a5a01276f25177386ad374a5917c8 (MD5)