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Investigation of Commutative Properties of Discontinuous Galerkin Methods in PDE Constrained Optimal Control Problems

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Abstract

The aim of this paper is to investigate commutative properties of a large family of discontinuous Galerkin (DG) methods applied to optimal control problems governed by the advection-diffusion equations. To compute numerical solutions of PDE constrained optimal control problems there are two main approaches: optimize-then-discretize and discretize-then-optimize. These two approaches do not always coincide and may lead to substantially different numerical solutions. The methods for which these two approaches do coincide we call commutative. In the theory of single equations, there is a related notion of adjoint or dual consistency. In this paper we classify DG methods both in primary and mixed forms and derive necessary conditions that can be used to develop new commutative methods. We will also derive error estimates in the energy and L 2 norms. Numerical examples reveal that in the context of PDE constrained optimal control problems a special care needs to be taken to compute the solutions. For example, choosing non-commutative methods and discretize-then-optimize approach may result in a badly behaved numerical solution.

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Acknowledgements

We would like to thank the anonymous referee for suggestions that help improve the quality of the paper.

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  1. Department of Mathematics, University of Connecticut, 196 Auditorium Road, Unit 3009, Storrs, CT, 06269-3009, USA

    Dmitriy Leykekhman

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  1. Dmitriy Leykekhman
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Correspondence to Dmitriy Leykekhman.

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This work was supported in part by the National Science Foundation (grant DMS-0811167).

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Leykekhman, D. Investigation of Commutative Properties of Discontinuous Galerkin Methods in PDE Constrained Optimal Control Problems. J Sci Comput 53, 483–511 (2012). https://doi.org/10.1007/s10915-012-9582-y

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  • DOI: https://doi.org/10.1007/s10915-012-9582-y

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