  In this paper, we propose a model reduction method for solving multiscale
elliptic PDEs with random coefficients in the multiquery setting using an
optimization approach. The optimization approach enables us to construct a set
of localized multiscale data-driven stochastic basis functions that give
optimal approximation property of the solution operator. Our method consists of
the offline and online stages. In the offline stage, we construct the localized
multiscale data-driven stochastic basis functions by solving an optimization
problem. In the online stage, using our basis functions, we can efficiently
solve multiscale elliptic PDEs with random coefficients with relatively small
computational costs. Therefore, our method is very efficient in solving target
problems with many different force functions. The convergence analysis of the
proposed method is also presented and has been verified by the numerical
simulation.
