Optimal order multigrid preconditioners for the distributed control of parabolic equations with coarsening in space and time
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Author/Creator
Author/Creator ORCID
Date
2022-02-17
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Citation of Original Publication
Andrei Drăgănescu & Mona Hajghassem (2022) Optimal order multigrid preconditioners for the distributed control of parabolic equations with coarsening in space and time, Optimization Methods and Software, DOI: 10.1080/10556788.2021.2022145
Rights
This is an Accepted Manuscript of an article published by Taylor & Francis in Optimization Methods and
Software on February 17, 2022, available
online: http://www.tandfonline.com/10.1080/10556788.2021.2022145.
Access to this item will begin on 2-17-2023
Access to this item will begin on 2-17-2023
Subjects
Abstract
We devise multigrid preconditioners for linear-quadratic space-time distributed
parabolic optimal control problems. While our method is rooted in earlier work
on elliptic control, the temporal dimension presents new challenges in terms of algorithm design and quality. Our primary focus is on the cG(s)dG(r) discretizations
which are based on functions that are continuous in space and discontinuous in
time, but our technique is applicable to various other space-time finite element discretizations. We construct and analyse two kinds of multigrid preconditioners: the
first is based on full coarsening in space and time, while the second is based on
semi-coarsening in space only. Our analysis, in conjunction with numerical experiments, shows that both preconditioners are of optimal order with respect to the discretization in case of cG(1)dG(r) for r = 0; 1, and exhibits a suboptimal behavior
in time for Crank-Nicolson. We also show that, under certain conditions, the pre-
conditioner using full space-time coarsening is more efficient than the one involving
semi-coarsening in space, a phenomenon that has not been observed previously. Our
numerical results confi rm the theoretical fi ndings.