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Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients

by F. Y. Kuo and Ch. Schwab and I. H. Sloan

(Report number 2012-25)

Abstract
This paper is a sequel to our previous work (\emph{Kuo, Schwab, Sloan, SIAM J.\ Numer.\ Anal., 2013}) where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented in a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as an infinite-dimensional integral in the parameter space. Here the (single level) error analysis of our previous work is generalized to a \emph{multi-level} scheme, with the number of QMC points depending on the discretization level, and with a level-dependent dimension truncation strategy. In some scenarios, it is shown that the overall error (i.e., the root-mean-square error averaged over all shifts) is of order ${\cal O}(h^2)$, where $h$ is the finest FE mesh width, or ${\cal O}(N^{-1+\delta})$ for arbitrary $\delta>0$, where $N$ denotes the maximal number of QMC sampling points in the parameter space. For these scenarios, the total work for all PDE solves in the multi-level QMC FE method is essentially of the order of {\em one single PDE solve at the finest FE discretization level}, for spatial dimension $d=2$ with linear elements. The analysis exploits regularity of the parametric solution with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). As in our previous work, families of QMC rules with ``POD weights'' (``product and order dependent weights") which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of QMC errors that are independent of the number of parametric variables.

Keywords: Multi-level, Quasi-Monte Carlo methods, Infinite dimensional integration, Elliptic partial differential equations with random coefficients Karhunen-Loeve expansion, Finite element methods

@Techreport{KSS12_468,
  author = {F. Y. Kuo and Ch. Schwab and I. H. Sloan},
  title = {Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-25},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-25.pdf },
  year = {2012}
}

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