Preconditioning for partial differential equation constrained optimization with control constraints

Martin Stoll, Andy Wathen

Research output: Contribution to journalArticlepeer-review

31 Scopus citations

Abstract

Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the control poses a significant additional challenge for optimization methods. In this paper, we propose preconditioners for the saddle point problems that arise when a primal-dual active set method is used. We also show for this method that the same saddle point system can be derived when the method is considered as a semismooth Newton method. In addition, the projected gradient method can be employed to solve optimization problems with simple bounds, and we discuss the efficient solution of the linear systems in question. In the case when an acceleration technique is employed for the projected gradient method, this again yields a semismooth Newton method that is equivalent to the primal-dual active set method. We also consider the Moreau-Yosida regularization method for control constraints and efficient preconditioners for this technique. Numerical results illustrate the competitiveness of these approaches. © 2011 John Wiley & Sons, Ltd.
Original languageEnglish (US)
Pages (from-to)53-71
Number of pages19
JournalNumerical Linear Algebra with Applications
Volume19
Issue number1
DOIs
StatePublished - Oct 18 2011
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: The first author would like to thank Tyrone Rees and Nick Gould for sharing their knowledge. The authors would also like to thank the anonymous referee for helping to improve this publication. This publication is partially based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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