Preconditioning for partial differential equation constrained optimization with control constraints

Handle URI:
http://hdl.handle.net/10754/599380
Title:
Preconditioning for partial differential equation constrained optimization with control constraints
Authors:
Stoll, Martin; Wathen, Andy
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.
Citation:
Stoll M, Wathen A (2011) Preconditioning for partial differential equation constrained optimization with control constraints. Numerical Linear Algebra with Applications 19: 53–71. Available: http://dx.doi.org/10.1002/nla.823.
Publisher:
Wiley-Blackwell
Journal:
Numerical Linear Algebra with Applications
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
18-Oct-2011
DOI:
10.1002/nla.823
Type:
Article
ISSN:
1070-5325
Sponsors:
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).
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorStoll, Martinen
dc.contributor.authorWathen, Andyen
dc.date.accessioned2016-02-28T05:49:59Zen
dc.date.available2016-02-28T05:49:59Zen
dc.date.issued2011-10-18en
dc.identifier.citationStoll M, Wathen A (2011) Preconditioning for partial differential equation constrained optimization with control constraints. Numerical Linear Algebra with Applications 19: 53–71. Available: http://dx.doi.org/10.1002/nla.823.en
dc.identifier.issn1070-5325en
dc.identifier.doi10.1002/nla.823en
dc.identifier.urihttp://hdl.handle.net/10754/599380en
dc.description.abstractOptimal 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.en
dc.description.sponsorshipThe 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).en
dc.publisherWiley-Blackwellen
dc.subjectKrylov subspace solveren
dc.subjectNewton methoden
dc.subjectPDE-constrained optimizationen
dc.subjectPreconditioningen
dc.subjectSaddle point systemsen
dc.titlePreconditioning for partial differential equation constrained optimization with control constraintsen
dc.typeArticleen
dc.identifier.journalNumerical Linear Algebra with Applicationsen
dc.contributor.institutionOxford Centre for Collaborative Applied Mathematics; Mathematical Institute; 24-29 St Giles'; Oxford; OX1 3LB; U.K.en
dc.contributor.institutionNumerical Analysis Group; Mathematical Institute; 24-29 St Giles'; Oxford; OX1 3LB; U.K.en
kaust.grant.numberKUK-C1-013-04en
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