A Fast Stochastic Galerkin Method for a Constrained Optimal Control Problem Governed by a Random Fractional Diffusion Equation (Journal of Computational Mathematics, 2019)
We develop a fast stochastic Galerkin method for an optimal control problem governed by a random space-fractional diffusion equation with deterministic constrained control. Optimal control problems governed by a fractional diffusion equation tend to provide a better description for transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsened by the large number of random space dimensions to discretize the probability space. We approximate the optimality system by a gradient algorithm combined with the stochastic Galerkin method through the discretization with respect to both the spatial space and the probability space. The resulting linear system can be decoupled for the random and spatial variable, and thus solved separately. A fast preconditioned Bi-Conjugate Gradient Stabilized method is developed to efficiently solve the decoupled systems derived from the fractional diffusion operators in the spatial space. Numerical experiments show the utility of the method.
Keywords: Constrained optimal control; Fractional diffusion; Stochastic Galerkin method; Fast Fourier transform; Preconditioned Bi-Conjugate Gradient Stabilized method.