A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients (Communications in Mathematical Research, 2022)
We propose a multiscale multilevel Monte Carlo (MsMLMC) methodto solve multiscale elliptic PDEs with random coefficients in the multi-querysetting. Our method consists of offline and online stages. In the offline stage,we construct a small number of reduced basis functions within each coarse gridblock, 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 thatthe MsMLMC offers considerable savings in solving multiscale elliptic PDEswith random coefficients. Moreover, we provide convergence analysis of theproposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochasticproblems without scale separation.
Keywords: Random partial differential equations (RPDEs); uncertainty quantification (UQ); multiscale finite element method (MsFEM); multilevel Monte Carlo (MLMC); reduced basis; convergence analysis.