We propose a generalized multiscale finite element method (GMsFEM) based on
clustering algorithm to study the elliptic PDEs with random coefficients in the
multi-query setting. Our method consists of offline and online stages. In the
offline stage, we construct a small number of reduced basis functions within
each coarse grid block, which can then be used to approximate the multiscale
finite element basis functions. In addition, we coarsen the corresponding
random space through a clustering algorithm. In the online stage, we can obtain
the multiscale finite element basis very efficiently on a coarse grid by using
the pre-computed multiscale basis. The new GMsFEM can be applied to multiscale
SPDE starting with a relatively coarse grid, without requiring the coarsest
grid to resolve the smallest-scale of the solution. The new method offers
considerable savings in solving multiscale SPDEs. Numerical results are
presented to demonstrate the accuracy and efficiency of the proposed method for
several multiscale stochastic problems without scale separation
