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

TL;DR

This paper introduces a data-driven method leveraging intrinsic low-dimensional structures to efficiently solve multiscale elliptic PDEs with random coefficients, significantly reducing computational complexity.

Contribution

The paper presents a novel two-stage approach that extracts low-dimensional bases from data for efficient online solutions of multiscale elliptic PDEs with random coefficients.

Findings

Achieves significant dimension reduction in solution space.

Demonstrates high accuracy and efficiency through numerical examples.

Provides error analysis based on sampling and truncation thresholds.

Abstract

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.