  We propose a data-driven approach to solve multiscale elliptic PDEs with
random coefficients based on the intrinsic low dimension structure of the
underlying elliptic differential operators. Our method consists of offline and
online stages. At the offline stage, a low dimension space and its basis are
extracted from the data to achieve significant dimension reduction in the
solution space. At the online stage, the extracted basis will be used to solve
a new multiscale elliptic PDE efficiently. The existence of low dimension
structure is established by showing the high separability of the underlying
Green's functions. Different online construction methods are proposed depending
on the problem setup. We provide error analysis based on the sampling error and
the truncation threshold in building the data-driven basis. Finally, we present
numerical examples to demonstrate the accuracy and efficiency of the proposed
method.
