A new approximation of the Schur complement in preconditioners for PDE-constrained optimization

Pearson, John W and Wathen, Andrew J (2012) A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numerical Linear Algebra with Applications, 19 (5). pp. 816-829. ISSN 1099-1506. (doi:https://doi.org/10.1002/nla.814) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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Official URL
http://dx.doi.org/10.1002/nla.814

Abstract

Saddle point systems arise widely in optimization problems with constraints. The utility of Schur complement approximation is now broadly appreciated in the context of solving such saddle point systems by iteration. In this short manuscript, we present a new Schur complement approximation for PDE-constrained optimization, an important class of these problems. Block diagonal and block triangular preconditioners have previously been designed to be used to solve such problems along with MINRES and non-standard Conjugate Gradients, respectively; with appropriate approximation blocks, these can be optimal in the sense that the time required for solution scales linearly with the problem size, however small the mesh size we use. In this paper, we extend this work to designing such preconditioners for which this optimality property holds independently of both the mesh size and the Tikhonov regularization parameter ? that is used. This also leads to an effective symmetric indefinite preconditioner that exhibits mesh and ? independence. We motivate the choice of these preconditioners based on observations about approximating the Schur complement obtained from the matrix system, derive eigenvalue bounds that verify the effectiveness of the approximation and present numerical results that show that these new preconditioners work well in practice.

Item Type: Article
Uncontrolled keywords: PDE-constrained optimization;Poisson control;preconditioning;Schur complement
Subjects: Q Science > QA Mathematics (inc Computing science) > QA297 Numerical analysis
Q Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Depositing User: John Pearson
Date Deposited: 30 Apr 2015 15:47 UTC
Last Modified: 11 Nov 2015 11:33 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/48150 (The current URI for this page, for reference purposes)
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