High dimensional finite elements for elliptic problems with multiple scales and stochastic data

TL;DR

The paper addresses elliptic PDEs with multiple scales and stochastic inputs by recasting them as high-dimensional unfolded problems. It demonstrates that sparse tensor product finite elements can solve these high-dimensional problems with complexity comparable to single-scale problems in the physical domain, without requiring explicit homogenization parameter computation. The approach extends to stochastic data by formulating deterministic high-dimensional problems for moments and correlations, achieving near-optimal convergence rates and log-linear complexity for statistical quantities. This framework enables efficient, scalable simulations of multiscale and stochastic elliptic problems, with potential extensions to multiple scales and other PDE models.

Abstract

Multiple scale homogenization problems are reduced to single scale problems in higher dimension. It is shown that sparse tensor product Finite Element Methods (FEM) allow the numerical solution in complexity independent of the dimension and of the length scale. Problems with stochastic input data are reformulated as high dimensional deterministic problems for the statistical moments of the random solution. Sparse tensor product FEM give a deterministic solution algorithm of log-linear complexity for statistical moments.