Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations

In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. Existence and uniqueness of the stochastic optimal solution is proved by establishing the equivalence between the constrained optimization problem and a stochastic saddle point problem. Analytic regularity of the optimal solution in the probability space is obtained under certain assumptions on the random input data. Finite element method and stochastic collocation method are employed for numerical approximation of the problem in deterministic space and probability space, respectively. A reduced basis method using a multilevel greedy algorithm based on isotropic and anisotropic sparse-grid techniques and weighted a posteriori error estimate is proposed in order to reduce the computational effort. A global error is obtained based on estimate results of error contribution from each method. Numerical experiments are performed with stochastic dimension ranging from 10 to 100, demonstrating that the proposed method is very efficient, especially for high dimensional and large-scale optimization problems.