Stochastic Discontinuous Galerkin Methods for Robust Deterministic Control of Convection Diffusion Equations with Uncertain Coefficients

We investigate a numerical behaviour of robust deterministic optimal control problem governed by a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem into a system of deterministic problems, is used to handle the stochastic domain, whereas a discontinuous Galerkin method is used to discretize the spatial domain due to its better convergence behaviour for convection dominated optimal control problems. A priori error estimates are derived for the state and adjoint in the energy norm and for the deterministic control in $L^2$-norm. To handle the curse of dimensionality of the stochastic Galerkin method, we take advantage of the low-rank variant of GMRES method, which reduces both the storage requirements and the computational complexity by exploiting a Kronecker-product structure of the system matrices. The efficiency of the proposed methodology is illustrated by numerical experiments on the benchmark problems with and without control constraints.