Weighted Reduced Basis Method for Stochastic Optimal Control Problems with Elliptic PDE Constraint

In this paper we develop and analyze an efficient computational method for solving stochastic optimal control problems constrained by an elliptic partial differential equation (PDE) with random input data. We first prove both existence and uniqueness of the optimal solution. Regularity of the optimal solution in the stochastic space is studied in view of the analysis of stochastic approximation error. For numerical approximation, we employ a finite element method for the discretization of physical variables, and a stochastic collocation method for the discretization of random variables. In order to alleviate the computational effort, we develop a model order reduction strategy based on a weighted reduced basis method. A global error analysis of the numerical approximation is carried out, and several numerical tests are performed to verify our analysis.