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Reduced-Order Modeling (ROM) for Simulation and Optimization pp 47–72Cite as

Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition

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Abstract

In this chapter, we combine a global, derivative-free subdivision algorithm for multiobjective optimization problems with a posteriori error estimates for reduced-order models based on Proper Orthogonal Decomposition in order to efficiently solve multiobjective optimization problems governed by partial differential equations. An error bound for a semilinear heat equation is developed in such a way that the errors in the conflicting objectives can be estimated individually. The resulting algorithm constructs a library of locally valid reduced-order models online using a Greedy (worst-first) search. Using this approach, the number of evaluations of the full-order model can be reduced by a factor of more than 1000.

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Notes

  1. 1.

    In a scalar optimization problem, it would not be advisable to give up rigorosity in order to reduce the error by such a factor. However, when dealing with multiple objectives and ‘steep’ Pareto fronts (cf. Remark 4), this factor results in huge computational savings since the number of dominated boxes is drastically reduced in every step.

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Acknowledgements

This work is supported by the Priority Programme SPP 1962 Non-smooth and Complementarity-based Distributed Parameter Systems of the German Research Foundation (DFG) and by the project Hybrides Planungsverfahren zur energieeffizienten Wärme- und Stromversorgung von städtischen Verteilnetzen funded by the German Ministry for Economic Affairs and Energy.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany

    Dennis Beermann & Stefan Volkwein

  2. Department of Mathematics, Paderborn University, Paderborn, Germany

    Michael Dellnitz & Sebastian Peitz

Authors
  1. Dennis Beermann
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  2. Michael Dellnitz
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  3. Sebastian Peitz
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  4. Stefan Volkwein
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Corresponding author

Correspondence to Stefan Volkwein .

Editor information

Editors and Affiliations

  1. Department of Corporate Research, Robert Bosch GmbH, Renningen, Germany

    Winfried Keiper

  2. Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg, Germany

    Anja Milde

  3. Fachbereich Mathematik, Universität Konstanz, Konstanz, Germany

    Stefan Volkwein

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Beermann, D., Dellnitz, M., Peitz, S., Volkwein, S. (2018). Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition. In: Keiper, W., Milde, A., Volkwein, S. (eds) Reduced-Order Modeling (ROM) for Simulation and Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-75319-5_3

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