ON EFFECTIVE STOCHASTIC GALERKIN FINITE ELEMENT METHOD FOR STOCHASTIC OPTIMAL CONTROL GOVERNED BY INTEGRAL-DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS

In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integral-differential PDEs with random field. The control problem has the control constraints of obstacle type. A new gradient algorithm based on the pre-conditioner conjugate gradient algorithm (PCG) is developed for this optimal control problem. This algorithm can transform a part of the state equation matrix and co-state equation matrix into block diagonal matrix and then solve the optimal control systems iteratively. The proof of convergence for this algorithm is also discussed. Finally numerical examples of a medial size are presented to illustrate our theoretical results.