Efficient numerical methods for elliptic optimal control problems with random coefficient

Xiaowei Pang; Haiming Song; Xiaoshen Wang; Jiachuan Zhang; Xiaowei Pang; Haiming Song; Xiaoshen Wang; Jiachuan Zhang

doi:10.3934/era.2020053

Electronic Research Archive

2020, Volume 28, Issue 2: 1001-1022. doi: 10.3934/era.2020053

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Efficient numerical methods for elliptic optimal control problems with random coefficient

1.
School of Mathematics, Jilin University, Changchun 130012, China
2.
Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock 72204, United States
3.
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
4.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

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

Efficient numerical methods for solving Poisson equation constraint optimal control problems with random coefficient are discussed in this paper. By applying the finite element method and the Monte Carlo approximation, the original optimal control problem is discretized and transformed into an optimization problem. Taking advantage of the separable structures, Algorithm 1 is proposed for solving the problem, where an alternating direction method of multiplier is used. Both computational and storage costs of this algorithm are very high. In order to reduce the computational cost, Algorithm 2 is proposed, where the multi-modes expansion is introduced and applied. Further, for reducing the storage cost, we propose Algorithm 3 based on Algorithm 2. The main idea is that the random term is shifted to the objective functional, which could be computed in advance. Therefore, we only need to solve a deterministic optimization problem, which could reduce all the costs significantly. Moreover, the convergence analyses of the proposed algorithms are established, and numerical simulations are carried out to test the performances of them.
- Optimal control problem,
- finite element method,
- Monte Carlo approximation,
- alternating direction method of multiplier,
- multi-modes expansion
Citation: Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient[J]. Electronic Research Archive, 2020, 28(2): 1001-1022. doi: 10.3934/era.2020053

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Abstract

Efficient numerical methods for solving Poisson equation constraint optimal control problems with random coefficient are discussed in this paper. By applying the finite element method and the Monte Carlo approximation, the original optimal control problem is discretized and transformed into an optimization problem. Taking advantage of the separable structures, Algorithm 1 is proposed for solving the problem, where an alternating direction method of multiplier is used. Both computational and storage costs of this algorithm are very high. In order to reduce the computational cost, Algorithm 2 is proposed, where the multi-modes expansion is introduced and applied. Further, for reducing the storage cost, we propose Algorithm 3 based on Algorithm 2. The main idea is that the random term is shifted to the objective functional, which could be computed in advance. Therefore, we only need to solve a deterministic optimization problem, which could reduce all the costs significantly. Moreover, the convergence analyses of the proposed algorithms are established, and numerical simulations are carried out to test the performances of them.

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