A Sparse Grid Stochastic Collocation Discontinuous Galerkin Method for Constrained Optimal Control Problem Governed by Random Convection Dominated Diffusion Equations. (19th May 2019)

• Record Type: Journal Article
• Authors: Ge, Liang; Sun, Tongjun
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 7(2019)
• Page Start: 763
• Page End: 797
• Publication Date: 2019-05-19
• Subjects: Discontinuous Galerkin (DG) -- optimal control problem -- priori error estimates -- random convection diffusion equation -- Smolyak approximation -- sparse grid stochastic collocation (SGSC); 65M55 -- 65N30; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1508034
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: A sparse grid stochastic collocation method combined with discontinuous Galerkin method is developed for solving convection dominated diffusion optimal control problem with random coefficients. By the optimal control theory, an optimality system is obtained for the problem, which consists of a state equation, a co-state equation and an inequality. Based on finite dimensional noise assumption of random field, the random coefficients are assumed to have finite term expansions depending on a finite number of mutually independent random variables in the probability space. An approximation scheme is established by using a discontinuous Galerkin method for the physical space and a sparse grid stochastic collocation method based on the Smolyak construction for the probability space, which leads to the solution of uncoupled deterministic problems. A priori error estimates are derived for the state, co-state and control variables. Numerical experiments are presented to illustrate the theoretical results.