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 existence uniqueness of solution then derive optimality system. For numerical approximation, we use Galerkin spectral methods, which apply method discretization random variables employ spectral-Galerkin approach approximation spatial variables. To solve large coupled saddle-point system resulted from discretization, adopt most commonly used MINRES a more effective PPCG in low-rank matrix iteration format. Specially, develop based on decomposition virtual variable method. also study eigenvalue distribution corresponding preconditioned matrix. Besides, discretized state equation, mean-based Ullmann handle different values variance inputs. Finally, present experiments demonstrate effectiveness our preconditioners.