Year: 2020
Author: Junlong Lyu, Zhiwen Zhang
Communications in Mathematical Research , Vol. 36 (2020), Iss. 2 : pp. 154–192
Abstract
We propose a multiscale multilevel Monte Carlo (MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmr.2020-0009
Communications in Mathematical Research , Vol. 36 (2020), Iss. 2 : pp. 154–192
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 39
Keywords: Random partial differential equations (RPDEs) uncertainty quantification (UQ) multiscale finite element method (MsFEM) multilevel Monte Carlo (MLMC) reduced basis convergence analysis.