Combining sparse grids, multilevel MC and QMC for elliptic PDEs with random coefficients
For researchers in uncertainty quantification, this work offers a potential computational speedup for solving high-dimensional PDEs with random coefficients.
This paper combines sparse grids, multilevel Monte Carlo, and Quasi-Monte Carlo methods for elliptic PDEs with random coefficients, achieving computational cost of O(ε^{-r}) with r<2 for O(ε) accuracy, independent of spatial dimension.
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.