TY - JOUR
T1 - Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs
AU - Nary Kim & Hyung-Chun Lee
JO - East Asian Journal on Applied Mathematics
VL - 2
SP - 166
EP - 188
PY - 2018
DA - 2018/02
SN - 4
DO - http://doi.org/10.4208/eajam.041013.180314a
UR - https://global-sci.org/intro/article_detail/eajam/10830.html
KW - Sparse grid collocation, stochastic partial differential equation, distributed control, finite element methods.
AB - <p style="text-align: justify;">In this article, we propose and analyse a sparse grid collocation method to
solve an optimal control problem involving an elliptic partial differential equation with
random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive
an optimality system. A Galerkin approximation in physical space and a sparse grid
collocation in the probability space is used. Error estimates for a fully discrete solution
using an appropriate norm are provided, and we analyse the computational efficiency.
Computational evidence complements the present theory, to show the effectiveness of
our stochastic collocation method.</p>