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2014 | OriginalPaper | Chapter

Model Order Reduction for PDE Constrained Optimization

Authors : Peter Benner, Ekkehard Sachs, Stefan Volkwein

Published in: Trends in PDE Constrained Optimization

Publisher: Springer International Publishing

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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.

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previous chapter Graded Meshes in Optimal Control for Elliptic Partial Differential Equations: An Overview
next chapter Adaptive Trust-Region POD Methods in PIDE-Constrained Optimization
Footnotes
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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Metadata
Title
Model Order Reduction for PDE Constrained Optimization
Authors
Peter Benner
Ekkehard Sachs
Stefan Volkwein
Copyright Year
2014
Book
Trends in PDE Constrained Optimization

Print ISBN: 978-3-319-05082-9

Electronic ISBN: 978-3-319-05083-6

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-05083-6_19