A stochastic Galerkin approximation scheme is proposed for an optimal control problem governed by a parabolic PDE with random perturbation in its coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. We obtain the necessary and sufficient optimality conditions and establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Galerkin method and with respect to time by the backward Euler scheme. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.Keywords: Priori error estimates; stochastic Galerkin method; optimal control problem; random parabolic PDE; obstacle control constraint. 1650028-1 Int. J. Comput. Methods Downloaded from www.worldscientific.com by UNIVERSITY OF NEW ENGLAND on 03/28/16. For personal use only. B. Gong et al. 1650028-2 Int. J. Comput. Methods Downloaded from www.worldscientific.com by UNIVERSITY OF NEW ENGLAND on 03/28/16. For personal use only.
In this paper, we try to recover an unknown source in a time-fractional diffusion equation. In order to overcome the influence of boundary conditions on source conditions, we introduce the Jacobi polynomials to construct the approximation and a modified Tikhonov regularization method is proposed to deal with the illposedness. Error estimates are obtained under a discrepancy principle as the parameter choice rule. Numerical results are also presented to demonstrate the effectiveness of the proposed method.
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