TY - JOUR
T1 - Stochastic Spline-Collocation Method for Constrained Optimal Control Problem Governed by Random Elliptic PDE
AU - Benxue Gong, Liang Ge, Tongjun Sun, Wanfang Shen & W.B. Liu
JO - International Journal of Numerical Analysis and Modeling
VL - 4-5
SP - 627
EP - 645
PY - 2017
DA - 2017/08
SN - 14
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/10053.html
KW - Random elliptic PDE, priori error estimates, stochastic spline-collocation method, Smolyak approximation, optimal control problem, deterministic constrained control.
AB - <p style="text-align: justify;">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.</p>