Exploring the locally low dimensional structure in solving random elliptic PDEs

TL;DR

This paper introduces a stochastic multiscale finite element method (StoMsFEM) for efficiently solving high-dimensional random elliptic PDEs by exploiting local low stochastic dimensions and regularity, achieving significant computational savings.

Contribution

The paper develops two novel local upscaling methods within StoMsFEM that leverage high regularity and low-rank properties to reduce computational complexity in solving high-dimensional stochastic elliptic PDEs.

Findings

Achieves computational savings of order $(H/h)^{d}/( ext{log}(H/h))^k$ compared to standard FEM.

Demonstrates a factor of 2000 speed-up in high contrast examples.

Shows high accuracy and efficiency through numerical experiments.

Abstract

We propose a stochastic multiscale finite element method (StoMsFEM) to solve random elliptic partial differential equations with a high stochastic dimension. The key idea is to simultaneously upscale the stochastic solutions in the physical space for all random samples and explore the low stochastic dimensions of the stochastic solution within each local patch. We propose two effective methods to achieve this simultaneous local upscaling. The first method is a high order interpolation method in the stochastic space that explores the high regularity of the local upscaled quantities with respect to the random variables. The second method is a reduced-order method that explores the low rank property of the multiscale basis functions within each coarse grid patch. Our complexity analysis shows that compared with the standard FEM on a fine grid, the StoMsFEM can achieve computational saving in the order of $(H/h)^{d}/(\log(H/h))^k$, where $H/h$ is the ratio between the coarse and the fine gird sizes, $d$ is the physical dimension and $k$ is the local stochastic dimension. Several numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed methods. In the high contrast example, we observe a factor of 2000 speed-up.