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Model Order Reduction for PDE Constrained Optimization

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Trends in PDE Constrained Optimization

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 165))

Abstract

The optimization and control of systems governed by partial differential equations (PDEs) usually requires numerous evaluations of the forward problem or the optimality system. Despite the fact that many recent efforts, many of which are reported in this book, have been made to limit or reduce the number of evaluations to 5–10, this cannot be achieved in all situations and even if this is possible, these evaluations may still require a formidable computational effort. For situations where this effort is not acceptable, model order reduction can be a means to significantly reduce the required computational resources. Here, we will survey some of the most popular approaches that can be used for this purpose. In particular, we address the issues arising in the strategies discretize-then-optimize, in which the optimality system of the reduced-order model has to be solved, and optimize-then-discretize, where a reduced-order model of the optimality system has to be found. The methods discussed include versions of proper orthogonal decomposition (POD) adapted to PDE constrained optimization as well as system-theoretic methods.

This work was supported by the DFG Priority Program 1253 “Optimization with Partial Differential Equations”, grants BE 2174/8-2, SA 289/20-1, and the DFG project “A-posteriori-POD Error Estimators for Nonlinear Optimal Control Problems governed by Partial Differential Equations”, grant VO 1658/2-1.

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Notes

  1. 1.

    Not to be confused with the “reduced-order” terminology used in the model reduction context—here, “reduced” means that the cost functional is written in dependence of the control only, using the fact that the weak solution y(u) is uniquely determined by the chosen u!

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Acknowledgements

Our thanks go to Günter Leugering for his well-structured and focused leadership throughout the six years of existence of the DFG Priority Program 1253 “Optimization with Partial Differential Equations”.

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Authors and Affiliations

  1. Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106, Magdeburg, Germany

    Peter Benner

  2. FB 4 – Mathematik, University of Trier, 54286, Trier, Germany

    Ekkehard Sachs

  3. Department of Mathematics and Statistics, University of Konstanz, Universitätsstraße 10, 78464, Konstanz, Germany

    Stefan Volkwein

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  1. Peter Benner
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  2. Ekkehard Sachs
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Editors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

  2. Institut für Dynamik komplexer technisch, Max-Planck-Institut, Magdeburg, Germany

    Peter Benner

  3. Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Dortmund, Germany

    Sebastian Engell

  4. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Andreas Griewank

  5. Mathematisches Institut, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  6. Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  7. Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  8. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

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Benner, P., Sachs, E., Volkwein, S. (2014). Model Order Reduction for PDE Constrained Optimization. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_19

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