POD-Based Mixed-Integer Optimal Control of the Heat Equation
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2019
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Journal of Scientific Computing. 2019, 81(1), pp. 48-75. ISSN 0885-7474. eISSN 1573-7691. Available under: doi: 10.1007/s10915-019-00924-3
Zusammenfassung
In the present paper an optimal control problem governed by the heat equation is considered, where continuous as well as discrete controls are involved. To obtain the discrete controls the branch-and-bound method is utilized, where in each node a relaxed control constrained optimal control problem has to be solved involving only continuous controls. However, the solutions to many relaxed optimal control problems have to be computed numerically. For that reason model-order reduction is applied to speed-up the branch-and-bound method. In this work the method of proper orthogonal decomposition (POD) is used. A posteriori error estimation in each node ensures that the calculated solutions are sufficiently accurate. Numerical experiments illustrate the efficiency of the proposed strategy.
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Fachgebiet (DDC)
510 Mathematik
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Mixed-integer programming, Optimal control, PDE-constrained optimization, Proper orthogonal decomposition, Error analysis
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BACHMANN, Freya, Dennis BEERMANN, Jianjie LU, Stefan VOLKWEIN, 2019. POD-Based Mixed-Integer Optimal Control of the Heat Equation. In: Journal of Scientific Computing. 2019, 81(1), pp. 48-75. ISSN 0885-7474. eISSN 1573-7691. Available under: doi: 10.1007/s10915-019-00924-3BibTex
@article{Bachmann2019-10PODBa-39073.2,
year={2019},
doi={10.1007/s10915-019-00924-3},
title={POD-Based Mixed-Integer Optimal Control of the Heat Equation},
number={1},
volume={81},
issn={0885-7474},
journal={Journal of Scientific Computing},
pages={48--75},
author={Bachmann, Freya and Beermann, Dennis and Lu, Jianjie and Volkwein, Stefan}
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<dcterms:abstract xml:lang="eng">In the present paper an optimal control problem governed by the heat equation is considered, where continuous as well as discrete controls are involved. To obtain the discrete controls the branch-and-bound method is utilized, where in each node a relaxed control constrained optimal control problem has to be solved involving only continuous controls. However, the solutions to many relaxed optimal control problems have to be computed numerically. For that reason model-order reduction is applied to speed-up the branch-and-bound method. In this work the method of proper orthogonal decomposition (POD) is used. A posteriori error estimation in each node ensures that the calculated solutions are sufficiently accurate. Numerical experiments illustrate the efficiency of the proposed strategy.</dcterms:abstract>
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