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An Adaptive POD Approximation Method for the Control of Advection-Diffusion Equations

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Book cover Control and Optimization with PDE Constraints

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 164))

Abstract

We present an algorithm for the approximation of a finite horizon optimal control problem for advection-diffusion equations. The method is based on the coupling between an adaptive POD representation of the solution and a Dynamic Programming approximation scheme for the corresponding evolutive Hamilton–Jacobi equation. We discuss several features regarding the adaptivity of the method, the role of error estimate indicators to choose a time subdivision of the problem and the computation of the basis functions. Some test problems are presented to illustrate the method.

The authors wish to acknowledge the support obtained by the following grants: ESF-OPTPDE Network, ITN—Marie Curie Grant n. 264735-SADCO and PRIN 2009 “Metodi Innovativi per il Calcolo Scientifico”.

The authors also wish to thank the CASPUR Consortium for its technical support.

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Authors and Affiliations

  1. Università degli studi di Roma “La Sapienza”, Piazzale Aldo Moro, 2, 0010, Roma, Italy

    Alessandro Alla & Maurizio Falcone

Authors
  1. Alessandro Alla
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  2. Maurizio Falcone
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Corresponding author

Correspondence to Maurizio Falcone .

Editor information

Editors and Affiliations

  1. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Kristian Bredies

  2. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Christian Clason

  3. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Karl Kunisch  & Gregory von Winckel  & 

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Alla, A., Falcone, M. (2013). An Adaptive POD Approximation Method for the Control of Advection-Diffusion Equations. In: Bredies, K., Clason, C., Kunisch, K., von Winckel, G. (eds) Control and Optimization with PDE Constraints. International Series of Numerical Mathematics, vol 164. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0631-2_1

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