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Numerical Algorithms

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21-05-2019 | Original Paper

Parameter-robust preconditioning for the optimal control of the wave equation

Authors: Jun Liu, John W. Pearson

Published in: Numerical Algorithms | Issue 3/2020

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Abstract

In this paper, we propose and analyze a new matching-type Schur complement preconditioner for solving the discretized first-order necessary optimality conditions that characterize the optimal control of wave equations. Coupled with this is a recently developed second-order implicit finite difference scheme used for the full space-time discretization of the optimality system of PDEs. Eigenvalue bounds for the preconditioned system are derived, which provide insights into the convergence rates of the preconditioned Krylov subspace method applied. Numerical examples are presented to validate our theoretical analysis and demonstrate the effectiveness of the proposed preconditioner, in particular its robustness with respect to very small regularization parameters, and all mesh sizes in the spatial variables.

previous article A posteriori error estimates of hp spectral element methods for optimal control problems with L2-norm state constraint

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The term “matching” refers to the fact that both terms of the exact Schur complement are captured within the approximation. In more detail, the multiplication of \((\check I_{h}^{1/2}\hat I_{h}^{1/2})\hat I_{h}^{-1} (\hat I_{h}^{1/2} \check I_{h}^{1/2})\) leads to the first term \(\check I_{h}\) on the right of the expression (6), with the second term \(\gamma L_{h}^{{\intercal }} \hat I_{h}^{-1} L_{h}\) obtained by the multiplication of \((\sqrt {\gamma } L_{h}^{{\intercal }}) \hat I_{h}^{-1} (\sqrt {\gamma } L_{h})\).

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Title: Parameter-robust preconditioning for the optimal control of the wave equation
Authors: Jun Liu
John W. Pearson
Publication date: 21-05-2019
Publisher: Springer US
Published in: Numerical Algorithms / Issue 3/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI: https://doi.org/10.1007/s11075-019-00720-y

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