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Volume 36, Issue 2
A Sparse Grid Stochastic Collocation and Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Elliptic Equations

Liang Ge & Tongjun Sun

DOI: 10.4208/jcm.1703-m2016-0692

J. Comp. Math., 36 (2018), pp. 310-330.

Published online: 2018-04

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  • 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.

  • Keywords

Optimal control problem, Random elliptic equations, Finite volume element, Sparse grid, Smolyak approximation, A priori error estimates.

  • AMS Subject Headings

65N06, 65B99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

gel@sdas.org (Liang Ge)

tjsun@sdu.edu.cn (Tongjun Sun)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-310, author = {Ge , Liang and Sun , Tongjun}, title = {A Sparse Grid Stochastic Collocation and Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Elliptic Equations}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {2}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1703-m2016-0692}, url = {http://global-sci.org/intro/article_detail/jcm/12260.html} }
TY - JOUR T1 - A Sparse Grid Stochastic Collocation and Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Elliptic Equations AU - Ge , Liang AU - Sun , Tongjun JO - Journal of Computational Mathematics VL - 2 SP - 310 EP - 330 PY - 2018 DA - 2018/04 SN - 36 DO - http://doi.org/10.4208/jcm.1703-m2016-0692 UR - https://global-sci.org/intro/article_detail/jcm/12260.html KW - Optimal control problem, Random elliptic equations, Finite volume element, Sparse grid, Smolyak approximation, A priori error estimates. AB -

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.

Ge , Liang and Sun , Tongjun. (2018). A Sparse Grid Stochastic Collocation and Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Elliptic Equations. Journal of Computational Mathematics. 36 (2). 310-330. doi:10.4208/jcm.1703-m2016-0692
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