标题：STOCHASTIC SPLINE-COLLOCATION METHOD FOR CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY RANDOM ELLIPTIC PDE
作者：Gong, Benxue; Ge, Liang; Sun, Tongjun; Shen, Wanfang; Liu, W. B.
作者机构：[Gong, Benxue; Ge, Liang] Shandong Univ Technol, Sch Sci, Zibo 255049, Peoples R China.; [Sun, Tongjun] Shandong Univ, Sch Math, Jinan 250100, Shand 更多
通讯作者地址：[Gong, BX]Shandong Univ Technol, Sch Sci, Zibo 255049, Peoples R China.
来源：INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
关键词：Random elliptic PDE; priori error estimates; stochastic; spline-collocation method; Smolyak approximation; optimal control; problem; deterministic constrained control
摘要：In this paper, we investigate a stochastic spline-collocation approximation scheme for an optimal control problem governed by an elliptic PDE with random field coefficients. We obtain the necessary and sufficient optimality conditions for the optimal control problem and establish a scheme to approximate the optimality system through the discretization with respect to the spatial space by finite elements method and the probability space by stochastic spline-collocation method. We further investigate Smolyak approximation schemes, which are effective collocation strategies for smooth problems that depend on a moderately large number of random variables. For more general control problems where the state may be non-smooth with respect to the random variables in some areas, we adopt a domain decomposition strategy to partition the random space into smooth and non-smooth parts and then apply Smolyak scheme and spline approximation respectively. 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.