Adaptive perturbation method for optimal control problem governed by stochastic elliptic PDEs

In this paper, we apply the stochastic perturbation technique to solve the optimal control problem governed by elliptic partial differential equation with small uncertainty in the random input. We first use finite-dimensional noise assumption and perturbation technique to establish the first-order and second-order deterministic optimality systems, and then discretize the two systems by standard finite-element method. Furthermore, we derive a posteriori error estimators for the finite-element approximation of the state, co-state and control in two different norms, respectively. These error estimators are then used to build our adaptive algorithm. Finally, some numerical examples are presented to verify the effectiveness of the derived estimators.