Stochastic discontinuous Galerkin methods for robust deterministic control of convection-diffusion equations with uncertain coefficients

• Published in: Advances in computational mathematics Vol. 49; no. 2
• Main Authors: Çi̇loğlu, Pelin, Yücel, Hamdullah
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2023; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Convection-diffusion equation; Diffusion; Error analysis; Galerkin method; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Optimal control; Optimization; Robust control; Visualization; Stochastic discontinuous Galerkin; Error estimates; 35R60; Low-rank approximation; 49J20; PDE-constrained optimization; Uncertainty quantification; 60H15; 60H35
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-023-10015-5

We investigate a numerical behavior 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 behavior for optimization problems governed by convection dominated PDEs. Error analysis is done for the state and adjoint variables in the energy norm, while the estimate 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.