Preconditioning for partial differential equation constrained optimization with control constraints

Stoll, Martin; Wathen, Andy

Preconditioning for partial differential equation constrained optimization with control constraints

Type

Article

Authors

Stoll, Martin
Wathen, Andy

KAUST Grant Number

KUK-C1-013-04

Online Publication Date

2011-10-18

Print Publication Date

2012-01

Date

2011-10-18

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.

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).