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Published April 6, 2017 | Published + Submitted
Journal Article Open

Exploring the locally low dimensional structure in solving random elliptic PDEs


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 for achieving 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 savings on the order of (H/h)^d/(log(H/h))^k, where H/h is the ratio between the coarse and the fine grid 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.

Additional Information

© 2017 Society for Industrial and Applied Mathematics. Received by the editors May 31, 2016; accepted for publication (in revised form) November 22, 2016; published electronically April 6, 2017. This research was partially supported by NSF grants DMS-1318377 and DMS-1613861.

Attached Files

Submitted - 1607.00693.pdf

Published - 16m1077611.pdf


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August 19, 2023
August 19, 2023