A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations

Jingshi Li; Jiachuan Zhang; Guoliang Ju; Juntao You; Jingshi Li; Jiachuan Zhang; Guoliang Ju; Juntao You

doi:10.3934/era.2020052

Electronic Research Archive

2020, Volume 28, Issue 2: 977-1000. doi: 10.3934/era.2020052

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A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations

1.
School of Mathematics and Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, China
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
3.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
4.
School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China
5.
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

Received: 01 March 2020 Revised: 01 April 2020
90C30, 65K10, 65C05, 65N30

This paper develops efficient numerical algorithms for the optimal control problem constrained by Poisson equations with uncertain diffusion coefficients. We consider both unconstrained condition and box-constrained condition for the control. The algorithms are based upon a multi-mode expansion (MME) for the random state, the finite element approximation for the physical space and the alternating direction method of multipliers (ADMM) or two-block ADMM for the discrete optimization systems. The compelling aspect of our method is that, target random constrained control problem can be approximated to one deterministic constrained control problem under a state variable substitution equality. Thus, the computing resource, especially the memory consumption, can be reduced sharply. The convergence rates of the proposed algorithms are discussed in the paper. We also present some numerical examples to show the performance of our algorithms.
- Random optimal control problems,
- boundary control,
- random Poisson equations,
- multi-modes representation,
- alternating direction method of multipliers
Citation: Jingshi Li, Jiachuan Zhang, Guoliang Ju, Juntao You. A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations[J]. Electronic Research Archive, 2020, 28(2): 977-1000. doi: 10.3934/era.2020052

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Abstract

This paper develops efficient numerical algorithms for the optimal control problem constrained by Poisson equations with uncertain diffusion coefficients. We consider both unconstrained condition and box-constrained condition for the control. The algorithms are based upon a multi-mode expansion (MME) for the random state, the finite element approximation for the physical space and the alternating direction method of multipliers (ADMM) or two-block ADMM for the discrete optimization systems. The compelling aspect of our method is that, target random constrained control problem can be approximated to one deterministic constrained control problem under a state variable substitution equality. Thus, the computing resource, especially the memory consumption, can be reduced sharply. The convergence rates of the proposed algorithms are discussed in the paper. We also present some numerical examples to show the performance of our algorithms.

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