Schwarz methods for the time-parallel solution of parabolic control problems

Martin J. Gander; Wing Hong Felix Kwok

doi:10.1007/978-3-319-18827-0_19

Schwarz methods for the time-parallel solution of parabolic control problems

Martin J. Gander, Wing Hong Felix Kwok^*

^*Corresponding author for this work

Department of Mathematics

Research output: Chapter in book/report/conference proceeding › Conference contribution › peer-review

12 Citations (Scopus)

Abstract

Discretized parabolic control problems lead to very large systems of equations, because trajectories must be approximated forward and backward in time. It is therefore of interest to devise parallel solvers for such systems, and a natural idea is to apply Schwarz preconditioners to the large space-time discretized problem. The performance of Schwarz preconditioners for elliptic problems is well understood, but how do such preconditioners perform on discretized parabolic control problems? We present a convergence analysis for a class of Schwarz methods applied to a model parabolic optimal control problem. We show that just applying a classical Schwarz method in time already implies better transmission conditions than the ones usually used in the elliptic case, and we propose an even better variant based on optimized Schwarz theory.

Original language	English
Title of host publication	Domain Decomposition Methods in Science and Engineering XXII
Editors	Thomas Dickopf, Martin J. Gander, Laurence Halpern, Rolf Krause, Luca F. Pavarino
Publisher	Springer Cham
Pages	207-216
Number of pages	10
Edition	1st
ISBN (Electronic)	9783319188270
ISBN (Print)	9783319188263, 9783319792606
DOIs	https://doi.org/10.1007/978-3-319-18827-0_19
Publication status	Published - Apr 2016
Event	22nd International Conference on Domain Decomposition Methods, DD 2013 - Lugano, Switzerland Duration: 16 Sept 2013 → 20 Sept 2013 https://link.springer.com/book/10.1007/978-3-319-18827-0 (Conference proceedings)

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	104
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	22nd International Conference on Domain Decomposition Methods, DD 2013
Country/Territory	Switzerland
City	Lugano
Period	16/09/13 → 20/09/13
Internet address	https://link.springer.com/book/10.1007/978-3-319-18827-0 (Conference proceedings)

Scopus Subject Areas

Modelling and Simulation
Engineering(all)
Discrete Mathematics and Combinatorics
Control and Optimization
Computational Mathematics

User-Defined Keywords

Classical and optimized Schwarz methods
Domain decomposition
Parabolic control problems

Access to Document

10.1007/978-3-319-18827-0_19

Cite this

Gander, M. J., & Kwok, W. H. F. (2016). Schwarz methods for the time-parallel solution of parabolic control problems. In T. Dickopf, M. J. Gander, L. Halpern, R. Krause, & L. F. Pavarino (Eds.), Domain Decomposition Methods in Science and Engineering XXII (1st ed., pp. 207-216). (Lecture Notes in Computational Science and Engineering; Vol. 104). Springer Cham. https://doi.org/10.1007/978-3-319-18827-0_19

Gander, Martin J. ; Kwok, Wing Hong Felix. / Schwarz methods for the time-parallel solution of parabolic control problems. Domain Decomposition Methods in Science and Engineering XXII. editor / Thomas Dickopf ; Martin J. Gander ; Laurence Halpern ; Rolf Krause ; Luca F. Pavarino. 1st. ed. Springer Cham, 2016. pp. 207-216 (Lecture Notes in Computational Science and Engineering).

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abstract = "Discretized parabolic control problems lead to very large systems of equations, because trajectories must be approximated forward and backward in time. It is therefore of interest to devise parallel solvers for such systems, and a natural idea is to apply Schwarz preconditioners to the large space-time discretized problem. The performance of Schwarz preconditioners for elliptic problems is well understood, but how do such preconditioners perform on discretized parabolic control problems? We present a convergence analysis for a class of Schwarz methods applied to a model parabolic optimal control problem. We show that just applying a classical Schwarz method in time already implies better transmission conditions than the ones usually used in the elliptic case, and we propose an even better variant based on optimized Schwarz theory.",

keywords = "Classical and optimized Schwarz methods, Domain decomposition, Parabolic control problems",

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Gander, MJ & Kwok, WHF 2016, Schwarz methods for the time-parallel solution of parabolic control problems. in T Dickopf, MJ Gander, L Halpern, R Krause & LF Pavarino (eds), Domain Decomposition Methods in Science and Engineering XXII. 1st edn, Lecture Notes in Computational Science and Engineering, vol. 104, Springer Cham, pp. 207-216, 22nd International Conference on Domain Decomposition Methods, DD 2013, Lugano, Switzerland, 16/09/13. https://doi.org/10.1007/978-3-319-18827-0_19

Schwarz methods for the time-parallel solution of parabolic control problems. / Gander, Martin J.; Kwok, Wing Hong Felix.

Domain Decomposition Methods in Science and Engineering XXII. ed. / Thomas Dickopf; Martin J. Gander; Laurence Halpern; Rolf Krause; Luca F. Pavarino. 1st. ed. Springer Cham, 2016. p. 207-216 (Lecture Notes in Computational Science and Engineering; Vol. 104).

Research output: Chapter in book/report/conference proceeding › Conference contribution › peer-review

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AU - Gander, Martin J.

AU - Kwok, Wing Hong Felix

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N2 - Discretized parabolic control problems lead to very large systems of equations, because trajectories must be approximated forward and backward in time. It is therefore of interest to devise parallel solvers for such systems, and a natural idea is to apply Schwarz preconditioners to the large space-time discretized problem. The performance of Schwarz preconditioners for elliptic problems is well understood, but how do such preconditioners perform on discretized parabolic control problems? We present a convergence analysis for a class of Schwarz methods applied to a model parabolic optimal control problem. We show that just applying a classical Schwarz method in time already implies better transmission conditions than the ones usually used in the elliptic case, and we propose an even better variant based on optimized Schwarz theory.

AB - Discretized parabolic control problems lead to very large systems of equations, because trajectories must be approximated forward and backward in time. It is therefore of interest to devise parallel solvers for such systems, and a natural idea is to apply Schwarz preconditioners to the large space-time discretized problem. The performance of Schwarz preconditioners for elliptic problems is well understood, but how do such preconditioners perform on discretized parabolic control problems? We present a convergence analysis for a class of Schwarz methods applied to a model parabolic optimal control problem. We show that just applying a classical Schwarz method in time already implies better transmission conditions than the ones usually used in the elliptic case, and we propose an even better variant based on optimized Schwarz theory.

KW - Classical and optimized Schwarz methods

KW - Domain decomposition

KW - Parabolic control problems

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Schwarz methods for the time-parallel solution of parabolic control problems

Abstract

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