Circulant embedding with QMC

analysis for elliptic PDE with lognormal coefficients

Ivan Graham, Frances Y. Kuo, Dirk Nuyens, Robert Scheichl, Ian H. Sloan

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In a previous paper (Graham et al. in J Comput Phys 230:3668–3694, 2011), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but no convergence theory was provided. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.

Original languageEnglish
Pages (from-to)479-511
Number of pages33
JournalNumerische Mathematik
Volume140
Issue number2
Early online date3 May 2018
DOIs
Publication statusPublished - 1 Oct 2018

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Quasi-Monte Carlo
Elliptic PDE
Monte Carlo methods
Random Field
Sampling
Quasi-Monte Carlo Methods
Coefficient
Covariance matrix
Factorization
Expected Value
Error analysis
Partial differential equations
Flow of fluids
Interpolation
Corner Singularity
Lattice Rules
Grid
Finite element method
Circulant Matrix
Heterogeneous Media

Cite this

Circulant embedding with QMC : analysis for elliptic PDE with lognormal coefficients. / Graham, Ivan; Kuo, Frances Y.; Nuyens, Dirk ; Scheichl, Robert; Sloan, Ian H.

In: Numerische Mathematik, Vol. 140, No. 2, 01.10.2018, p. 479-511.

Research output: Contribution to journalArticle

Graham, Ivan ; Kuo, Frances Y. ; Nuyens, Dirk ; Scheichl, Robert ; Sloan, Ian H. / Circulant embedding with QMC : analysis for elliptic PDE with lognormal coefficients. In: Numerische Mathematik. 2018 ; Vol. 140, No. 2. pp. 479-511.
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