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Volume 24, Issue 5
An Alternating Direction Method of Multipliers for the Optimization Problem Constrained with a Stationary Maxwell System

Yongle Hao, Haiming Song, Xiaoshen Wang & Kai Zhang

DOI: 10.4208/cicp.OA-2017-0117

Commun. Comput. Phys., 24 (2018), pp. 1435-1454.

Published online: 2018-06

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  • Abstract

This paper mainly focuses on an efficient numerical method for the optimization problem constrained with a stationary Maxwell system. Following the idea of [32], the edge element is applied to approximate the state variable and the control variable, then the continuous optimal control problem is discretized into a finite dimensional one. The novelty of this paper is the approach for solving the discretized system. Based on the separable structure, an alternating direction method of multipliers (ADMM) is proposed. Furthermore, the global convergence analysis is established in the form of the objective function error, which includes the discretization error by the edge element and the iterative error by ADMM. Finally, numerical simulations are presented to demonstrate the efficiency of the proposed algorithm.

  • Keywords

Optimal control problem, stationary Maxwell's equations, Nédélec element, ADMM.

  • AMS Subject Headings

90C30, 90C33, 65K10, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-1435, author = {Yongle Hao, Haiming Song, Xiaoshen Wang and Kai Zhang}, title = {An Alternating Direction Method of Multipliers for the Optimization Problem Constrained with a Stationary Maxwell System}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {5}, pages = {1435--1454}, abstract = {

This paper mainly focuses on an efficient numerical method for the optimization problem constrained with a stationary Maxwell system. Following the idea of [32], the edge element is applied to approximate the state variable and the control variable, then the continuous optimal control problem is discretized into a finite dimensional one. The novelty of this paper is the approach for solving the discretized system. Based on the separable structure, an alternating direction method of multipliers (ADMM) is proposed. Furthermore, the global convergence analysis is established in the form of the objective function error, which includes the discretization error by the edge element and the iterative error by ADMM. Finally, numerical simulations are presented to demonstrate the efficiency of the proposed algorithm.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0117}, url = {http://global-sci.org/intro/article_detail/cicp/12484.html} }
TY - JOUR T1 - An Alternating Direction Method of Multipliers for the Optimization Problem Constrained with a Stationary Maxwell System AU - Yongle Hao, Haiming Song, Xiaoshen Wang & Kai Zhang JO - Communications in Computational Physics VL - 5 SP - 1435 EP - 1454 PY - 2018 DA - 2018/06 SN - 24 DO - http://doi.org/10.4208/cicp.OA-2017-0117 UR - https://global-sci.org/intro/article_detail/cicp/12484.html KW - Optimal control problem, stationary Maxwell's equations, Nédélec element, ADMM. AB -

This paper mainly focuses on an efficient numerical method for the optimization problem constrained with a stationary Maxwell system. Following the idea of [32], the edge element is applied to approximate the state variable and the control variable, then the continuous optimal control problem is discretized into a finite dimensional one. The novelty of this paper is the approach for solving the discretized system. Based on the separable structure, an alternating direction method of multipliers (ADMM) is proposed. Furthermore, the global convergence analysis is established in the form of the objective function error, which includes the discretization error by the edge element and the iterative error by ADMM. Finally, numerical simulations are presented to demonstrate the efficiency of the proposed algorithm.

Yongle Hao, Haiming Song, Xiaoshen Wang and Kai Zhang. (2018). An Alternating Direction Method of Multipliers for the Optimization Problem Constrained with a Stationary Maxwell System. Communications in Computational Physics. 24 (5). 1435-1454. doi:10.4208/cicp.OA-2017-0117
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