Comparison of approaches for random PDE optimization problems based on different matching functionals

In this article, we consider an optimal control problem for an elliptic partial differential equation with random inputs. To determine an applicable deterministic control fˆ(x), we consider the four cases which we compare for efficiency and feasibility. We prove the existence of optimal states, adjoint states and optimality conditions for each cases. We also derive the optimality systems for the four cases. The optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space, respectively. The numerical experiments are performed for their efficiency and feasibility.