Iterative Method for Stochastic Optimal Control Problem Constrained by Parabolic Partial Differential Equation
- 1School of Mathematics and Statistics & Hunan Provincial University Key Laboratory for Big Data Analysis and Application, Hunan First Normal University, Changsha 410205, China
Abstract
In this paper, an efficient iterative method based on the generalized multiscale finite element (Iter-GMs) method is applied to solve the distributed optimal control problems governed by stochastic parabolic equations. Within this framework, we firstly reformulate the time-dependent optimality system into an equivalent form, which contains the parameter-independent diffusion coefficients and the parameter-dependent right-hand side. Then, the fixed-point iteration method is used to solve the reformulated optimality system. Quantifying the statistics of stochastic optimal control problems requires solving a coupled optimality system across numerous samples in the stochastic space, which incurs a prohibitively high computational cost and may even trigger the so-called “curse of dimensionality.” To mitigate this issue, we exploit the multiscale and high-contrast nature of the physical parameters by invoking the generalized multiscale finite element method (GMsFEM) to construct a reduced-order space. We divide the proposed iterative method into two stages: an offline stage and an online stage. In the offline stage, we construct local multiscale basis functions for each coarse-grid region based on the deterministic multiscale optimality system, and then build the corresponding offline spaces. In the online stage, we project the reformulated model onto the reduced multiscale space and apply the fixed-point iteration method to obtain the optimal solutions of the reduced system. Finally, two numerical experiments are presented to validate the rationality and effectiveness of the proposed method.
Related
- Oscillation of a Second-Order Differential Equation as a Consequence of Iterative Monotonicities
- Existence of weak solutions to two-phase Stefan problems for parabolic partial differential equation system
- Null Controllability of a Stochastic Parabolic Equation with a Space-Dependent Analytic Noise Coefficient
- Ergodicity of stochastic functional differential equation with jumps and finite delay
- Pseudo-differential noise and nonlocal singularity formation in the stochastic Córdoba--Córdoba--Fontelos equation
BibTeX
@article{ma2026iterative,
title = {{Iterative Method for Stochastic Optimal Control Problem Constrained by Parabolic Partial Differential Equation}},
author = {Lingling Ma and Xiangyan Qiu},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100786},
url = {https://www.mdpi.com/2075-1680/15/10/786},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Iterative Method for Stochastic Optimal Control Problem Constrained by Parabolic Partial Differential Equation
AU - Lingling Ma
AU - Xiangyan Qiu
PY - 2026
DA - 2026/10/20
JO - Axioms
DO - 10.3390/axioms15100786
UR - https://www.mdpi.com/2075-1680/15/10/786
AB - In this paper, an efficient iterative method based on the generalized multiscale finite element (Iter-GMs) method is applied to solve the distributed optimal control problems governed by stochastic parabolic equations. Within this framework, we firstly reformulate the time-dependent optimality system into an equivalent form, which contains the parameter-independent diffusion coefficients and the parameter-dependent right-hand side. Then, the fixed-point iteration method is used to solve the reformulated optimality system. Quantifying the statistics of stochastic optimal control problems requires solving a coupled optimality system across numerous samples in the stochastic space, which incurs a prohibitively high computational cost and may even trigger the so-called “curse of dimensionality.” To mitigate this issue, we exploit the multiscale and high-contrast nature of the physical parameters by invoking the generalized multiscale finite element method (GMsFEM) to construct a reduced-order space. We divide the proposed iterative method into two stages: an offline stage and an online stage. In the offline stage, we construct local multiscale basis functions for each coarse-grid region based on the deterministic multiscale optimality system, and then build the corresponding offline spaces. In the online stage, we project the reformulated model onto the reduced multiscale space and apply the fixed-point iteration method to obtain the optimal solutions of the reduced system. Finally, two numerical experiments are presented to validate the rationality and effectiveness of the proposed method.
SN - 2075-1680
ER -