The objective of this work is to study an efficient approximation of the solution to stochastic Burgers equation driven by an additive space-time noise and a distributed feedback control problem for the stochastic Burgers equation with a random diffusion coefficient. For these goals, we introduce several stochastic tools in uncertainty quantification, which involve the white noise, the stochastic integrals, the Karhunen-Lo_x0012_eve expansion, and the sparse grid stochastic collocation method.
We discuss existence and uniqueness of the stochastic Burgers equation with an additive space-time noise, which is an infinite dimensional random process, through the Orstein-Uhlenbeck (OU) process. To approximate the OU process, we use the Karhunen-Lo_x0012_eve expansion, and sparse grid stochastic collocation method. About spatial discretization of Burgers equation, two separate finite element approximations are presented: the conventional Galerkin method and Galerkin-conservation method. Numerical experiments are provided to demonstrate the efficacy of schemes mentioned above.
Meanwhile, we study a control problem for stochastic Burgers equation with a random diffusion coefficient. Numerical schemes for computing the stochastic Burgers equation are developed. We apply the finite element method in spatial discretization and the sparse grid stochastic collocation method in random parameter space. We also use those schemes to compute closed-loop, suboptimal state feedback control. Several numerical experiments are performed to demonstrate the efficiency and plausibility of proposed approximation methods for the stochastic Burgers equation and the related control problem.
