Convergence of Substructuring Methods for Elliptic Optimal Control Problems

Martin J. Gander; Felix Kwok; Bankim C. Mandal

doi:10.1007/978-3-319-93873-8_27

Convergence of Substructuring Methods for Elliptic Optimal Control Problems

Martin J. Gander^*, Felix Kwok, Bankim C. Mandal

^*Corresponding author for this work

Department of Mathematics

Research output: Chapter in book/report/conference proceeding › Conference proceeding

4 Citations (Scopus)

Abstract

We study in this paper Dirichlet–Neumann and Neumann–Neumann methods for the parallel solution of elliptic optimal control problems. Unlike in the case of single linear or non-linear elliptic problems, we need to solve here two coupled elliptic problems that arise as a part of optimality system for the optimal control problem. We present a rigorous convergence analysis for the case of two non-overlapping subdomains, which shows that both methods converge in at most three iterations. We illustrate our findings with numerical results.

Original language	English
Title of host publication	Domain Decomposition Methods in Science and Engineering XXIV
Editors	Petter E. Bjørstad, Susanne C. Brenner, Lawrence Halpern, Hyea Hyun Kim, Ralf Kornhuber, Talal Rahman, Olof B. Widlund
Publisher	Springer Cham
Pages	291-300
Number of pages	10
Edition	1st
ISBN (Electronic)	9783319938738
ISBN (Print)	9783319938721
DOIs	https://doi.org/10.1007/978-3-319-93873-8_27
Publication status	Published - 5 Jan 2019
Event	24th International Conference on Domain Decomposition Methods in Science and Engineering - Svalbard, Norway Duration: 6 Feb 2017 → 10 Feb 2017 http://www.ddm.org/dd24/

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	125
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	24th International Conference on Domain Decomposition Methods in Science and Engineering
Abbreviated title	DD2017
Country/Territory	Norway
City	Svalbard
Period	6/02/17 → 10/02/17
Internet address	http://www.ddm.org/dd24/

Scopus Subject Areas

Modelling and Simulation
General Engineering
Discrete Mathematics and Combinatorics
Control and Optimization
Computational Mathematics

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10.1007/978-3-319-93873-8_27

Cite this

Gander, M. J., Kwok, F., & Mandal, B. C. (2019). Convergence of Substructuring Methods for Elliptic Optimal Control Problems. In P. E. Bjørstad, S. C. Brenner, L. Halpern, H. H. Kim, R. Kornhuber, T. Rahman, & O. B. Widlund (Eds.), Domain Decomposition Methods in Science and Engineering XXIV (1st ed., pp. 291-300). (Lecture Notes in Computational Science and Engineering; Vol. 125). Springer Cham. https://doi.org/10.1007/978-3-319-93873-8_27

Gander, Martin J. ; Kwok, Felix ; Mandal, Bankim C. / Convergence of Substructuring Methods for Elliptic Optimal Control Problems. Domain Decomposition Methods in Science and Engineering XXIV. editor / Petter E. Bjørstad ; Susanne C. Brenner ; Lawrence Halpern ; Hyea Hyun Kim ; Ralf Kornhuber ; Talal Rahman ; Olof B. Widlund. 1st. ed. Springer Cham, 2019. pp. 291-300 (Lecture Notes in Computational Science and Engineering).

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title = "Convergence of Substructuring Methods for Elliptic Optimal Control Problems",

abstract = "We study in this paper Dirichlet–Neumann and Neumann–Neumann methods for the parallel solution of elliptic optimal control problems. Unlike in the case of single linear or non-linear elliptic problems, we need to solve here two coupled elliptic problems that arise as a part of optimality system for the optimal control problem. We present a rigorous convergence analysis for the case of two non-overlapping subdomains, which shows that both methods converge in at most three iterations. We illustrate our findings with numerical results.",

author = "Gander, {Martin J.} and Felix Kwok and Mandal, {Bankim C.}",

note = "Publisher Copyright: {\textcopyright} 2018, Springer International Publishing AG, part of Springer Nature.; 24th International Conference on Domain Decomposition Methods in Science and Engineering, DD2017 ; Conference date: 06-02-2017 Through 10-02-2017",

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booktitle = "Domain Decomposition Methods in Science and Engineering XXIV",

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Gander, MJ, Kwok, F & Mandal, BC 2019, Convergence of Substructuring Methods for Elliptic Optimal Control Problems. in PE Bjørstad, SC Brenner, L Halpern, HH Kim, R Kornhuber, T Rahman & OB Widlund (eds), Domain Decomposition Methods in Science and Engineering XXIV. 1st edn, Lecture Notes in Computational Science and Engineering, vol. 125, Springer Cham, pp. 291-300, 24th International Conference on Domain Decomposition Methods in Science and Engineering, Svalbard, Norway, 6/02/17. https://doi.org/10.1007/978-3-319-93873-8_27

Convergence of Substructuring Methods for Elliptic Optimal Control Problems. / Gander, Martin J.; Kwok, Felix; Mandal, Bankim C.
Domain Decomposition Methods in Science and Engineering XXIV. ed. / Petter E. Bjørstad; Susanne C. Brenner; Lawrence Halpern; Hyea Hyun Kim; Ralf Kornhuber; Talal Rahman; Olof B. Widlund. 1st. ed. Springer Cham, 2019. p. 291-300 (Lecture Notes in Computational Science and Engineering; Vol. 125).

Research output: Chapter in book/report/conference proceeding › Conference proceeding

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T1 - Convergence of Substructuring Methods for Elliptic Optimal Control Problems

AU - Gander, Martin J.

AU - Kwok, Felix

AU - Mandal, Bankim C.

PY - 2019/1/5

Y1 - 2019/1/5

N2 - We study in this paper Dirichlet–Neumann and Neumann–Neumann methods for the parallel solution of elliptic optimal control problems. Unlike in the case of single linear or non-linear elliptic problems, we need to solve here two coupled elliptic problems that arise as a part of optimality system for the optimal control problem. We present a rigorous convergence analysis for the case of two non-overlapping subdomains, which shows that both methods converge in at most three iterations. We illustrate our findings with numerical results.

AB - We study in this paper Dirichlet–Neumann and Neumann–Neumann methods for the parallel solution of elliptic optimal control problems. Unlike in the case of single linear or non-linear elliptic problems, we need to solve here two coupled elliptic problems that arise as a part of optimality system for the optimal control problem. We present a rigorous convergence analysis for the case of two non-overlapping subdomains, which shows that both methods converge in at most three iterations. We illustrate our findings with numerical results.

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DO - 10.1007/978-3-319-93873-8_27

M3 - Conference proceeding

AN - SCOPUS:85060271358

SN - 9783319938721

T3 - Lecture Notes in Computational Science and Engineering

SP - 291

EP - 300

BT - Domain Decomposition Methods in Science and Engineering XXIV

A2 - Bjørstad, Petter E.

A2 - Brenner, Susanne C.

A2 - Halpern, Lawrence

A2 - Kim, Hyea Hyun

A2 - Kornhuber, Ralf

A2 - Rahman, Talal

A2 - Widlund, Olof B.

PB - Springer Cham

T2 - 24th International Conference on Domain Decomposition Methods in Science and Engineering

Y2 - 6 February 2017 through 10 February 2017

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Gander MJ, Kwok F, Mandal BC. Convergence of Substructuring Methods for Elliptic Optimal Control Problems. In Bjørstad PE, Brenner SC, Halpern L, Kim HH, Kornhuber R, Rahman T, Widlund OB, editors, Domain Decomposition Methods in Science and Engineering XXIV. 1st ed. Springer Cham. 2019. p. 291-300. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-319-93873-8_27

Convergence of Substructuring Methods for Elliptic Optimal Control Problems

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