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Efficient Stochastic Galerkin Spectral Methods for Optimal Control Problems Constrained by Fractional PDEs with Uncertain Inputs

This paper is devoted to designing fast solvers and efficient preconditioners for the optimal control problems (OCPs) constrained by stochastic fractional elliptic equations. We first prove the existence and uniqueness of the stochastic optimal control solution and then derive the stochastic optimality system. For the numerical approximation, we use the stochastic Galerkin spectral methods, which apply the stochastic Galerkin method for the discretization of random variables and employ the spectral-Galerkin approach for the approximation of spatial variables. To solve the large coupled saddle-point system resulted from discretization, we adopt the most commonly used MINRES method and a more effective PPCG method in low-rank matrix iteration format. Specially, we develop the efficient preconditioners based on the matrix decomposition method and the virtual variable method. We also study the eigenvalue distribution of the corresponding preconditioned matrix. Besides, for the approximation of the discretized state equation, we use the mean-based approximation and the Ullmann approximation to handle different values of the variance of the random inputs. Finally, we present numerical experiments to demonstrate the effectiveness of our solvers and preconditioners.