Reduced basis methods for optimal control of advection-diffusion problems ∗

The reduced basis (RB) method is proposed for the approximation of multi-parametrized equations governing an optimal control problem. The idea behind the RB method is to project the so- lution 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 proc edures) 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 emis- sions of industrial chimneys in order to keep the pollutant concentration be low 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 geome- trical input parameters) so that we are able to solve the control problem f rom a global and decisional point of view.

[1]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[2]  S. Ravindran,et al.  A Reduced Basis Method for Control Problems Governed by PDEs , 1998 .

[3]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[4]  Rolf Rannacher,et al.  Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept , 2000, SIAM J. Control. Optim..

[5]  K. ITO,et al.  Reduced Basis Method for Optimal Control of Unsteady Viscous Flows , 2001 .

[6]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[7]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[8]  Annalisa Quaini,et al.  Numerical Approximation of a Control Problem for Advection-Diffusion Processes , 2005, System Modelling and Optimization.

[9]  Gianluigi Rozza,et al.  Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity , 2005 .

[10]  Gianluigi Rozza.  Shape design by optimal flow control and reduced basis techniques , 2005 .

[11]  Alfio Quarteroni,et al.  Optimal Control and Numerical Adaptivity for Advection-Diffusion Equations , 2005 .

[12]  M. Grepl.  Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations , 2005 .

[13]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[14]  Kirsten Morris,et al.  Control of Systems Governed by Partial Differential Equations , 2018, The Control Systems Handbook.