Multigrid solution of a distributed optimal control problem constrained by the Stokes equations
Abstract
We construct multigrid preconditioners to accelerate the solution process of a linearquadratic optimal control problem constrained by the Stokes system. The first order optimality conditions of the control problem form a linear system (the KKT system) connecting the state, adjoint, and control variables. Our approach is to eliminate the state and adjoint variables by essentially solving two Stokes systems, and to construct efficient multigrid preconditioners for the Schurcomplement of the block associated with the state and adjoint variables. These multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the Stokes system. In particular, the number of conjugate gradient iterations is shown to decrease as the resolution increases, a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems.
 Authors:

 Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States). Dept. of Mathematics and Statistics
 Publication Date:
 Research Org.:
 Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
 OSTI Identifier:
 1493158
 Grant/Contract Number:
 SC0005455; DMS1016177
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Applied Mathematics and Computation
 Additional Journal Information:
 Journal Volume: 219; Journal Issue: 10; Journal ID: ISSN 00963003
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; multigrid methods; PDEconstrained optimization; Stokes equations; finite elements; optimal order preconditioners
Citation Formats
Drăgănescu, Andrei, and Soane, Ana Maria. Multigrid solution of a distributed optimal control problem constrained by the Stokes equations. United States: N. p., 2013.
Web. doi:10.1016/j.amc.2012.11.070.
Drăgănescu, Andrei, & Soane, Ana Maria. Multigrid solution of a distributed optimal control problem constrained by the Stokes equations. United States. doi:https://doi.org/10.1016/j.amc.2012.11.070
Drăgănescu, Andrei, and Soane, Ana Maria. Thu .
"Multigrid solution of a distributed optimal control problem constrained by the Stokes equations". United States. doi:https://doi.org/10.1016/j.amc.2012.11.070. https://www.osti.gov/servlets/purl/1493158.
@article{osti_1493158,
title = {Multigrid solution of a distributed optimal control problem constrained by the Stokes equations},
author = {Drăgănescu, Andrei and Soane, Ana Maria},
abstractNote = {We construct multigrid preconditioners to accelerate the solution process of a linearquadratic optimal control problem constrained by the Stokes system. The first order optimality conditions of the control problem form a linear system (the KKT system) connecting the state, adjoint, and control variables. Our approach is to eliminate the state and adjoint variables by essentially solving two Stokes systems, and to construct efficient multigrid preconditioners for the Schurcomplement of the block associated with the state and adjoint variables. These multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the Stokes system. In particular, the number of conjugate gradient iterations is shown to decrease as the resolution increases, a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems.},
doi = {10.1016/j.amc.2012.11.070},
journal = {Applied Mathematics and Computation},
number = 10,
volume = 219,
place = {United States},
year = {2013},
month = {1}
}
Web of Science
Works referencing / citing this record:
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