A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction

Abstract

We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic approximate low-dimensional structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low-dimensional space and its basis are extracted from solution samples to achieve significant dimension reduction in the solution space. At the online stage, the extracted data-driven basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of approximate low-dimensional structure is established in two scenarios based on (1) high separability of the underlying Green's functions, and (2) smooth dependence of the parameters in the random coefficients. Various online construction methods are proposed for different problem setups. We provide error analysis based on the sampling error and the truncation threshold in building the data-driven basis. Finally, we present extensive numerical examples to demonstrate the accuracy and efficiency of the proposed method.