An iterative method based on GMsFEM for optimal control problem with random inputs

Lingling Ma; Lingling Ma

doi:10.3934/era.2026342

Electronic Research Archive

2026, Volume 34, Issue 11: 8000-8025. doi: 10.3934/era.2026342

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Research article

An iterative method based on GMsFEM for optimal control problem with random inputs

Lingling Ma ^,

School of Mathematics and Statistics & Hunan Provincial University Key Laboratory for Big Data Analysis and Application, Hunan First Normal University, Changsha 410205, Hunan Province, China

Received: 16 June 2026 Revised: 22 August 2026 Accepted: 25 August 2026 Published: 21 September 2026

In this paper, we propose an iterative method for optimal control problems governed by elliptic PDEs based on the generalized multiscale finite element method (GMsFEM). In the iterative method, we first reformulate the original optimal control system into a new system with parameter-independent coefficients and parameter-dependent right-hand sides. Subsequently, we adopt the fixed-point iteration method to solve the reformulated system to get the optimal solutions. To quantify the statistics of the optimal control problems, we need to solve a coupled optimality system for a large number of samples in the stochastic space. Therefore, the computational cost is extremely large, and it may even lead to the "curse-of-dimensionality". When the diffusion coefficient exhibits multiscale features, the iterative process can be performed in the reduced multiscale space via GMsFEM to enhance computational efficiency. The proposed iterative method can be split into two stages: an offline stage and an online stage. In the offline stage, local multiscale basis functions are constructed from the deterministic optimality system to build the reduced space. In the online stage, the reformulated model is projected onto the reduced space and solved via fixed-point iteration. Finally, we will present two numerical examples to verify the effectiveness of the proposed iterative method.
- stochastic optimal control problem,
- iterative method,
- generalized multiscale finite element method,
- random inputs
Citation: Lingling Ma. An iterative method based on GMsFEM for optimal control problem with random inputs[J]. Electronic Research Archive, 2026, 34(11): 8000-8025. doi: 10.3934/era.2026342

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Abstract

In this paper, we propose an iterative method for optimal control problems governed by elliptic PDEs based on the generalized multiscale finite element method (GMsFEM). In the iterative method, we first reformulate the original optimal control system into a new system with parameter-independent coefficients and parameter-dependent right-hand sides. Subsequently, we adopt the fixed-point iteration method to solve the reformulated system to get the optimal solutions. To quantify the statistics of the optimal control problems, we need to solve a coupled optimality system for a large number of samples in the stochastic space. Therefore, the computational cost is extremely large, and it may even lead to the "curse-of-dimensionality". When the diffusion coefficient exhibits multiscale features, the iterative process can be performed in the reduced multiscale space via GMsFEM to enhance computational efficiency. The proposed iterative method can be split into two stages: an offline stage and an online stage. In the offline stage, local multiscale basis functions are constructed from the deterministic optimality system to build the reduced space. In the online stage, the reformulated model is projected onto the reduced space and solved via fixed-point iteration. Finally, we will present two numerical examples to verify the effectiveness of the proposed iterative method.

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