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Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

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Abstract

In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in \(\mathbb {R}^d\) (\(d = 1, 2,3\)), with diffusion coefficient \(a({\varvec{x}},\omega )\) given as a lognormal random field, i.e., \(a({\varvec{x}},\omega ) = \exp (Z({\varvec{x}},\omega ))\) where \({\varvec{x}}\) is the spatial variable and \(Z({\varvec{x}}, \cdot )\) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from \(0\) or \(\infty \) over all possible realizations of \(a\). Focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (1) standard continuous and piecewise linear finite element approximation in physical space; (2) truncated Karhunen–Loève expansion for computing realizations of \(a\) (leading to a possibly high-dimensional parametrized deterministic diffusion problem); and (3) lattice-based quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based on a recent result on randomly shifted lattice rules for high-dimensional integrals over the unbounded domain of Euclidean space, which shows that (under suitable conditions) the quadrature error decays with \(\mathcal {O}(n^{-1+\delta })\) with respect to the number of quadrature points \(n\), where \(\delta >0\) is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration.

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Notes

  1. The assumption on \(a\) made in [5, 36] can be weakened in the boundary case \(t=1\) from \(a(\cdot ,\omega ) \in {\varvec{C}}^{1}(\overline{D})\) to \(a(\cdot ,\omega ) \in {\varvec{C}}^{0,1}(\overline{D})\), since this implies \(\nabla a(\cdot ,\omega ) \in L^{\infty }(D)\) and then the crucial Lemma A.2 in [5] can be replaced by the simple inequality \(\Vert bv\Vert _{H^1(D)} \le C (\Vert \nabla b\Vert _{L^{\infty }(D)} \Vert v\Vert _{L^2(D)} + \Vert b\Vert _{L^{\infty }(D)} \Vert v\Vert _{H^1(D)})\), for all \(b \in {\varvec{C}}^{0,1}(\overline{D})\) and \(v \in H^1(D)\).

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Acknowledgments

The authors would like to thank Tony Shardlow and Helmut Harbrecht for fruitful discussions that lead to Corollary 5 and Proposition 9. We would also like to acknowledge the financial support of the Australian Research Council, of the UK Engineering and Physical Sciences Research Council and of the European Research Council under grant ERC AdG247277.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    I. G. Graham & R. Scheichl

  2. School of Mathematics and Statistics, University of NSW, Sydney, NSW, 2052, Australia

    F. Y. Kuo, J. A. Nichols & I. H. Sloan

  3. Seminar für Angewandte Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Ch. Schwab

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  1. I. G. Graham
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Correspondence to R. Scheichl.

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Graham, I.G., Kuo, F.Y., Nichols, J.A. et al. Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math. 131, 329–368 (2015). https://doi.org/10.1007/s00211-014-0689-y

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