Combining sparse grids, multilevel MC and QMC for elliptic PDEs with random coefficients

TL;DR

This paper explores combining sparse grids, multilevel Monte Carlo, and Quasi-Monte Carlo methods to efficiently solve elliptic PDEs with uncertain coefficients, achieving near-optimal computational cost independent of spatial dimension.

Contribution

It introduces a novel integrated approach that leverages all three methods for elliptic PDEs with finite-dimensional uncertainty, improving computational efficiency.

Findings

Potential to achieve $O( ext{error})$ cost with $O( ext{error}^{-r})$, $r<2$

Cost independence from spatial dimension

Enhancement over previous separate methods

Abstract

Building on previous research which generalized multilevel Monte Carlo methods using either sparse grids or Quasi-Monte Carlo methods, this paper considers the combination of all these ideas applied to elliptic PDEs with finite-dimensional uncertainty in the coefficients. It shows the potential for the computational cost to achieve an $O(\varepsilon)$ r.m.s. accuracy to be $O(\varepsilon^{-r})$ with $r<2$, independently of the spatial dimension of the PDE.