A Sparse Grid Stochastic Collocation and Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Elliptic Equations
Year: 2018
Author: Liang Ge, Tongjun Sun
Journal of Computational Mathematics, Vol. 36 (2018), Iss. 2 : pp. 310–330
Abstract
In this paper, a hybird approximation scheme for an optimal control problem governed by an elliptic equation with random field in its coefficients is considered. The random coefficients are smooth in the physical space and depend on a large number of random variables in the probability space. The necessary and sufficient optimality conditions for the optimal control problem are obtained. The scheme is established to approximate the optimality system through the discretization by using finite volume element method for the spatial space and a sparse grid stochastic collocation method based on the Smolyak approximation for the probability space, respectively. This scheme naturally leads to the discrete solutions of an uncoupled deterministic problem. The existence and uniqueness of the discrete solutions are proved. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1703-m2016-0692
Journal of Computational Mathematics, Vol. 36 (2018), Iss. 2 : pp. 310–330
Published online: 2018-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Optimal control problem Random elliptic equations Finite volume element Sparse grid Smolyak approximation A priori error estimates.
Author Details
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A Sparse Grid Stochastic Collocation Discontinuous Galerkin Method for Constrained Optimal Control Problem Governed by Random Convection Dominated Diffusion Equations
Ge, Liang
Sun, Tongjun
Numerical Functional Analysis and Optimization, Vol. 40 (2019), Iss. 7 P.763
https://doi.org/10.1080/01630563.2018.1508034 [Citations: 3]