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

Çiloğlu P., Yücel H.

ADVANCES IN COMPUTATIONAL MATHEMATICS, vol.49, pp.16, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 49
  • Publication Date: 2023
  • Doi Number: 10.1007/s10444-023-10015-5
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, MathSciNet, zbMATH
  • Page Numbers: pp.16
  • Keywords: PDE-constrained optimization, Uncertainty quantification, Stochastic discontinuous Galerkin, Error estimates, Low-rank approximation, FINITE-ELEMENT APPROXIMATIONS, LOW-RANK, POLYNOMIAL CHAOS, ERROR ANALYSIS, PDE, OPTIMIZATION, COLLOCATION, ALGORITHM, FLOW
  • Middle East Technical University Affiliated: Yes


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.