<abstract xmlns="http://www.w3.org/1999/xhtml"><p> The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear systems with block structure. Correspondingly, when the solution method is a Dirichlet-Dirichlet non-overlapping domain decomposition method, we need to solve interface problems which inherit the block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners. In this paper we describe a generic technique which employs a preconditioner block structure based on the fractional Sobolev norms corresponding to the domains of the boundary operators arising in the matrix interface problem, some of which may include a dependence on the control regularization parameter. We illustrate our approach on standard linear elliptic control problems. We present analysis which shows that the resulting iterative method converges independently of the size of the problem. We include numerical results which indicate that performance is also independent of the control regularization parameter and exhibits only a mild dependence on the number of the subdomains.</p></abstract>

The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear systems with block structure. Correspondingly, when the solution method is a Dirichlet-Dirichlet non-overlapping domain decomposition method, we need to solve interface problems which inherit the block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners. In this paper we describe a generic technique which employs a preconditioner block structure based on the fractional Sobolev norms corresponding to the domains of the boundary operators arising in the matrix interface problem, some of which may include a dependence on the control regularization parameter. We illustrate our approach on standard linear elliptic control problems. We present analysis which shows that the resulting iterative method converges independently of the size of the problem. We include numerical results which indicate that performance is also independent of the control regularization parameter and exhibits only a mild dependence on the number of the subdomains.

Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE"}

The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear...

Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE

Published Online: Nov 22, 2018

Page range: 175 - 192

Received: Mar 01, 2017

Accepted: Jul 01, 2017

DOI: https://doi.org/10.2478/auom-2018-0024

Keywords
Domain decomposition, Steklov-Poincaré operator, PDE-constrained optimization, preconditioned Krylov subspace methods

© 2018 Daniel Loghin, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE

Published Online: Nov 22, 2018

Page range: 175 - 192

Received: Mar 01, 2017

Accepted: Jul 01, 2017

DOI: https://doi.org/10.2478/auom-2018-0024

KeywordsDomain decomposition, Steklov-Poincaré operator, PDE-constrained optimization, preconditioned Krylov subspace methods

© 2018 Daniel Loghin, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Domain decomposition, Steklov-Poincaré operator, PDE-constrained optimization, preconditioned Krylov subspace methods