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

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(ε) r.m.s. accuracy to be O(ε-r) with r<2, independently of the spatial dimension of the PDE.