## Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

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Date

2012-12Type

- Journal Article

Citations

Cited 114 times in
Web of Science

Cited 135 times in
Scopus

ETH Bibliography

yes
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Abstract

In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably infinite number of terms in a Karhunen--Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC method to estimate expected values of linear functionals of the exact or approximate solution of the PDE, with the expected value considered as an infinite dimensional integral in the parameter space corresponding to the randomness induced by the random coefficient. The error analysis, arguably the first rigorous application of the modern theory of QMC in weighted spaces, exploits the regularity with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). In the weighted-space theory of QMC methods, “weights,” describing the varying difficulty of different subsets of the variables, are introduced in order to make sure that the high-dimensional integration problem is tractable. It turns out that the weights arising from the present analysis are of a nonstandard kind, being of neither product nor order dependent form, but instead a hybrid of the two---we refer to these as “product and order dependent weights,” or “POD weights” for short. These POD weights are of a simple enough form to permit a component-by-component construction of a randomly shifted lattice rule that has optimal convergence properties for the given weighted space setting. If the terms in the expansion for the random coefficient have an appropriate decay property, and if we choose POD weights that minimize a certain upper bound on the error, then the solution of the PDE belongs to the joint function space needed for the analysis, and the QMC error (in the sense of a root-mean-square error averaged over shifts) is of order $O(N^{-1 + \delta})$ for arbitrary $\delta$ > 0, where $N$ denotes the number of sampling points in the parameter space. Moreover, this convergence rate and slower convergence rates under more relaxed conditions are achieved under conditions similar to those found recently by Cohen, De Vore, and Schwab [Found. Comput. Math., 10 (2010), pp. 615--646] for the same model by best $N$-term approximation. We analyze the impact of a finite element (FE) discretization on the overall efficiency of the scheme, in terms of accuracy versus overall cost, with results that are comparable to those of the best $N$-term approximation. Show more

Publication status

publishedExternal links

Journal / series

SIAM Journal on Numerical AnalysisVolume

Pages / Article No.

Publisher

SIAMSubject

quasi-Monte Carlo methods; infinite dimensional integration; elliptic partial differential equations with random coefficients; Karhunen-Loève expansion; finite element methodsOrganisational unit

03435 - Schwab, Christoph / Schwab, Christoph
Related publications and datasets

Is new version of: https://doi.org/10.3929/ethz-a-010400794

Is referenced by: https://doi.org/10.3929/ethz-a-010387042

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Show all metadata
Citations

Cited 114 times in
Web of Science

Cited 135 times in
Scopus

ETH Bibliography

yes
Altmetrics