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

TL;DR

This paper develops a robust numerical framework combining stochastic Galerkin and discontinuous Galerkin methods to efficiently solve control problems governed by convection diffusion equations with uncertain coefficients, including error analysis and computational strategies.

Contribution

It introduces a novel combination of stochastic Galerkin and discontinuous Galerkin methods for robust control of uncertain convection diffusion equations, with efficient low-rank solvers.

Findings

Error estimates for state and adjoint variables in energy norm.

Reduced computational complexity using low-rank GMRES.

Effective handling of control constraints in benchmark examples.

Abstract

We investigate a numerical behaviour of robust deterministic optimal control problem subject to a convection diffusion equation containing uncertain inputs. Stochastic Galerkin approach, turning the original optimization problem containing uncertainties into a large system of deterministic problems, is applied to discretize the stochastic domain, while a discontinuous Galerkin method is preferred for the spatial discretization due to its better convergence behaviour for optimization problems governed by convection dominated PDEs. Error analysis is done for the state and adjoint variables in the energy norm, while the estimates of deterministic control is obtained in the $L^2$--norm. Large matrix system emerging from the stochastic Galerkin method is addressed by the low--rank version of GMRES method, which reduces both the computational complexity and the memory requirements by employing Kronecker--product structure of the obtained linear system. Benchmark examples with and without control constraints are presented to illustrate the efficiency of the proposed methodology.