Reduced basis methods for optimal control of advection-diffusion problems


Advanced Numerical Methods for Scientific Computing
MOX 79
Reduced basis methods for optimal control of advection-diffusion problems
Monday 30th January 2006
Quarteroni, Alfio; Rozza, Gianluigi; Quaini, Annalisa
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The reduced basis (RB) method is proposed for the approximation of multiparametrized equations governing an optimal control problem. The idea behind the RB method is to project the solution onto a space of small dimension, specifically designed on the problem at hand, and to decouple the generation and projection stages (off-line/on-line computational procedures) of the approximation process in order to solve parametrized equations in a rapid and not expensive way. The application that we investigate is an air pollution control problem: we aim at regulating the emissions of industrial chimneys in order to keep the pollutant concentration below a certain threshold over an observation area, like a town. Adopting the RB method for both state and adjoint equations of the optimal control problem leads to important computational savings with respect to the use of the Galerkin-finite element method. We consider different parametrization (control, physical and geometrical input parameters) so that we are able to solve the control problem from a global and decisional point of view.