### A SPARSE GRID STOCHASTIC COLLOCATION AND FINITE VOLUME ELEMENT METHOD FOR CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY RANDOM ELLIPTIC EQUATIONS

Liang Ge1, Tongjun Sun2

1. 1. School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China;
2. School of Mathematics, Shandong University, Jinan, Shandong 250100, China
• Received:2016-06-27 Revised:2016-11-08 Online:2018-03-15 Published:2018-03-15
• Supported by:

The work is partly supported by the Natural Science Foundation of China (Grant No.11271231,11301300,61572297),by the Shandong Province Outstanding Young Scientists Research Award Fund Project (Grant No.BS2013DX010),by the Natural Science Foundation of Shandong Province,China (Grant No.ZR2014FM003),and by the Shandong Academy of Sciences Youth Fund Project (Grant No.2013QN007).

Liang Ge, Tongjun Sun. A SPARSE GRID STOCHASTIC COLLOCATION AND FINITE VOLUME ELEMENT METHOD FOR CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY RANDOM ELLIPTIC EQUATIONS[J]. Journal of Computational Mathematics, 2018, 36(2): 310-330.

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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