Leapfrog multigrid methods for parabolic optimal control problems

Buyang Li; Jun Liu; Mingqing Xiao

doi:10.1109/CCDC.2015.7161680

Leapfrog multigrid methods for parabolic optimal control problems

Buyang Li, Jun Liu, Mingqing Xiao

Research output: Chapter in book / Conference proceeding › Conference article published in proceeding or book › Academic research › peer-review

Abstract

A second-order leapfrog finite difference scheme in time is proposed to solve the first-order necessary optimality systems arising from parabolic optimal control problems. Different from classical approximation, the proposed leapfrog scheme appears to be unconditionally stable. More importantly, the developed leapfrog scheme provides a well-structured discrete algebraic system and allows us to establish a fast linear solver under the multigrid framework. The unconditional stability of the scheme is proved under the L2 norm. Numerical results show that our presented scheme significantly outperforms the widely used Crank-Nicolson scheme and the resultant fast solver demonstrates a mesh-independent convergence rate as well as a desirable feature of linear time complexity.

Original language	English
Title of host publication	Proceedings of the 2015 27th Chinese Control and Decision Conference, CCDC 2015
Publisher	IEEE
Pages	137-143
Number of pages	7
ISBN (Electronic)	9781479970179
DOIs	https://doi.org/10.1109/CCDC.2015.7161680
Publication status	Published - 1 Jan 2015
Externally published	Yes
Event	27th Chinese Control and Decision Conference, CCDC 2015 - Qingdao Haiqing Hotel, Qingdao, China Duration: 23 May 2015 → 25 May 2015

Conference

Conference	27th Chinese Control and Decision Conference, CCDC 2015
Country/Territory	China
City	Qingdao
Period	23/05/15 → 25/05/15

Keywords

Finite difference
Leapfrog scheme
Multigrid method
Parabolic optimal control

ASJC Scopus subject areas

Control and Systems Engineering
Control and Optimization
Management Science and Operations Research

More information

10.1109/CCDC.2015.7161680

Cite this

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title = "Leapfrog multigrid methods for parabolic optimal control problems",

abstract = "A second-order leapfrog finite difference scheme in time is proposed to solve the first-order necessary optimality systems arising from parabolic optimal control problems. Different from classical approximation, the proposed leapfrog scheme appears to be unconditionally stable. More importantly, the developed leapfrog scheme provides a well-structured discrete algebraic system and allows us to establish a fast linear solver under the multigrid framework. The unconditional stability of the scheme is proved under the L2 norm. Numerical results show that our presented scheme significantly outperforms the widely used Crank-Nicolson scheme and the resultant fast solver demonstrates a mesh-independent convergence rate as well as a desirable feature of linear time complexity.",

keywords = "Finite difference, Leapfrog scheme, Multigrid method, Parabolic optimal control",

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doi = "10.1109/CCDC.2015.7161680",

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T1 - Leapfrog multigrid methods for parabolic optimal control problems

AU - Li, Buyang

AU - Liu, Jun

AU - Xiao, Mingqing

PY - 2015/1/1

Y1 - 2015/1/1

N2 - A second-order leapfrog finite difference scheme in time is proposed to solve the first-order necessary optimality systems arising from parabolic optimal control problems. Different from classical approximation, the proposed leapfrog scheme appears to be unconditionally stable. More importantly, the developed leapfrog scheme provides a well-structured discrete algebraic system and allows us to establish a fast linear solver under the multigrid framework. The unconditional stability of the scheme is proved under the L2 norm. Numerical results show that our presented scheme significantly outperforms the widely used Crank-Nicolson scheme and the resultant fast solver demonstrates a mesh-independent convergence rate as well as a desirable feature of linear time complexity.

AB - A second-order leapfrog finite difference scheme in time is proposed to solve the first-order necessary optimality systems arising from parabolic optimal control problems. Different from classical approximation, the proposed leapfrog scheme appears to be unconditionally stable. More importantly, the developed leapfrog scheme provides a well-structured discrete algebraic system and allows us to establish a fast linear solver under the multigrid framework. The unconditional stability of the scheme is proved under the L2 norm. Numerical results show that our presented scheme significantly outperforms the widely used Crank-Nicolson scheme and the resultant fast solver demonstrates a mesh-independent convergence rate as well as a desirable feature of linear time complexity.

KW - Finite difference

KW - Leapfrog scheme

KW - Multigrid method

KW - Parabolic optimal control

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U2 - 10.1109/CCDC.2015.7161680

DO - 10.1109/CCDC.2015.7161680

M3 - Conference article published in proceeding or book

SP - 137

EP - 143

BT - Proceedings of the 2015 27th Chinese Control and Decision Conference, CCDC 2015

PB - IEEE

T2 - 27th Chinese Control and Decision Conference, CCDC 2015

Y2 - 23 May 2015 through 25 May 2015

ER -

Leapfrog multigrid methods for parabolic optimal control problems

Abstract

Conference

Keywords

ASJC Scopus subject areas

More information

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