A Sparse Grid Stochastic Collocation and Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Elliptic Equations (Journal of Computational Mathematics, 2019)
In this paper, a hybird approximation scheme for an optimal control problem governedby an elliptic equation with random field in its coefficients is considered. The randomcoefficients are smooth in the physical space and depend on a large number of randomvariables in the probability space. The necessary and sufficient optimality conditions forthe optimal control problem are obtained. The scheme is established to approximate theoptimality system through the discretization by using finite volume element method forthe spatial space and a sparse grid stochastic collocation method based on the Smolyakapproximation for the probability space, respectively. This scheme naturally leads to thediscrete solutions of an uncoupled deterministic problem. The existence and uniquenessof the discrete solutions are proved. A priori error estimates are derived for the state,the co-state and the control variables. Numerical examples are presented to illustrate ourtheoretical results.
Keywords: Optimal control problem; Random elliptic equations; Finite volume element; Sparse grid; Smolyak approximation; A priori error estimates.