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Volume 12, Issue 2
An Alternating Direction Method of Multipliers for Optimal Control Problems Constrained with Elliptic Equations

Jinda Yang, Kai Zhang, Haiming Song & Ting Cheng

DOI: 10.4208/aamm.OA-2018-0198

Adv. Appl. Math. Mech., 12 (2020), pp. 336-361.

Published online: 2020-01

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  • Abstract

In this paper, we propose an efficient numerical method for the optimal control problem constrained by elliptic equations. Being approximated by the finite element method (FEM), the continuous optimal control problem is discretized into a finite dimensional optimization problem with separable structures. Furthermore, an alternating direction method of multipliers (ADMM) is applied to solve the discretization problem. The total convergence analysis which includes the discretization error by FEM and iterative error by ADMM is established. Finally, numerical simulations are presented to verify the efficiency of the proposed method.

  • Keywords

Optimal control problem, elliptic equation, finite element method, ADMM.

  • AMS Subject Headings

49M37, 65K10, 65M60, 90C33

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

kzhang@jlu.edu.cn (Kai Zhang)

songhaiming@jlu.edu.cn (Haiming Song)

  • BibTex
  • RIS
  • TXT
@Article{AAMM-12-336, author = {Yang , JindaZhang , KaiSong , Haiming and Cheng , Ting}, title = {An Alternating Direction Method of Multipliers for Optimal Control Problems Constrained with Elliptic Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {2}, pages = {336--361}, abstract = {

In this paper, we propose an efficient numerical method for the optimal control problem constrained by elliptic equations. Being approximated by the finite element method (FEM), the continuous optimal control problem is discretized into a finite dimensional optimization problem with separable structures. Furthermore, an alternating direction method of multipliers (ADMM) is applied to solve the discretization problem. The total convergence analysis which includes the discretization error by FEM and iterative error by ADMM is established. Finally, numerical simulations are presented to verify the efficiency of the proposed method.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0198}, url = {http://global-sci.org/intro/article_detail/aamm/13625.html} }
TY - JOUR T1 - An Alternating Direction Method of Multipliers for Optimal Control Problems Constrained with Elliptic Equations AU - Yang , Jinda AU - Zhang , Kai AU - Song , Haiming AU - Cheng , Ting JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 336 EP - 361 PY - 2020 DA - 2020/01 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2018-0198 UR - https://global-sci.org/intro/article_detail/aamm/13625.html KW - Optimal control problem, elliptic equation, finite element method, ADMM. AB -

In this paper, we propose an efficient numerical method for the optimal control problem constrained by elliptic equations. Being approximated by the finite element method (FEM), the continuous optimal control problem is discretized into a finite dimensional optimization problem with separable structures. Furthermore, an alternating direction method of multipliers (ADMM) is applied to solve the discretization problem. The total convergence analysis which includes the discretization error by FEM and iterative error by ADMM is established. Finally, numerical simulations are presented to verify the efficiency of the proposed method.

Yang , JindaZhang , KaiSong , Haiming and Cheng , Ting. (2020). An Alternating Direction Method of Multipliers for Optimal Control Problems Constrained with Elliptic Equations. Advances in Applied Mathematics and Mechanics. 12 (2). 336-361. doi:10.4208/aamm.OA-2018-0198
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