Iterative Method for Stochastic Optimal Control Problem Constrained by Parabolic Partial Differential Equation

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.