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A Fast Solver for an H1 Regularized PDE-Constrained Optimization Problem

Part of: Partial differential equations, boundary value problems Numerical linear algebra Numerical methods in calculus of variations and optimal control Optimality conditions

Published online by Cambridge University Press:  15 January 2016

Andrew T. Barker ,
Tyrone Rees  and
Martin Stoll
Andrew T. Barker
Affiliation:
Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Mail Stop L-561, Livermore, CA 94551, USA
Tyrone Rees*
Affiliation:
Numerical Analysis Group, Scientific Computing Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, United Kingdom
Martin Stoll
Affiliation:
Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
*
*Corresponding author. Email addresses:barker29@llnl.gov (A. T. Barker), tyrone.rees@stfc.ac.uk (T. Rees), stollm@mpi-magdeburg.mpg.de (M. Stoll)
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Abstract

In this paper we consider PDE-constrained optimization problems which incorporate an H1 regularization control term. We focus on a time-dependent PDE, and consider both distributed and boundary control. The problems we consider include bound constraints on the state, and we use a Moreau-Yosida penalty function to handle this. We propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.

Keywords

PreconditioningKrylov methodsPDE-constrained optimizationoptimal control of PDEs

MSC classification

Secondary: 49M25: Discrete approximations 49K20: Problems involving partial differential equations 65F10: Iterative methods for linear systems 65N22: Solution of discretized equations 65F50: Sparse matrices 65N55: Multigrid methods; domain decomposition
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 143 - 167
Copyright
Copyright © Global-Science Press 2016 

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