摘要
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 tends to provide a better description for transport or conduction processes in heterogeneous media. Howev- er, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsen by the large number of random space dimensions to discretize the probability space. We ap- proximate 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.
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 tends to provide a better description for transport or conduction processes in heterogeneous media. Howev- er, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsen by the large number of random space dimensions to discretize the probability space. We ap- proximate 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.
基金
This work was supported by the National Natural Science Foundation of China under grants 11371229, 11571026 and 11501326, and by the China Scholarship Council （File No. 2013083Y0102）.
