Abstract
We introduce the multivariate decomposition finite element method (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form a = exp(Z) where Z is a Gaussian random field defined by an infinite series expansion Z(y) = ∑j≥1yjϕj with yj ~ N(0,1) and a given sequence of functions {ϕj}j≥1. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the multivariate decomposition method (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using quasi-Monte Carlo (QMC) methods, and for which we use the finite element method (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in terms of error versus cost, i.e., to achieve an accuracy of O(ε) the computational cost is O(ε-1/λ-d′/λ)=O(ε-(p*+d′/τ)/(1-p*)) where ε−1/λ and ε−d′/λ) are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with d′= d (1+δ′) for some δ′ ≥ 0 and d the physical dimension, and 0<p*≤(2+d′/τ)-1 is a parameter representing the sparsity of {ϕj}j≥1.
Highlights
In this paper we are concerned with the application of higher-order quasi-Monte Carlo (QMC) rules and multivariate decomposition methods (MDM) to elliptic PDEs with random diffusion coefficients
We focus on the lognormal diffusion coefficient, the logarithm of which is a Gaussian random field
The multivariate decomposition finite element method (MDFEM) was already analysed in the case of a uniform diffusion coefficient in [28], but in this case higher-order QMC rules are readily available for integration over the unit cube
Summary
In this paper we are concerned with the application of higher-order quasi-Monte Carlo (QMC) rules and multivariate decomposition methods (MDM) to elliptic PDEs with random diffusion coefficients. Most existing QMC methods for integration with respect to the normal density, and most existing QMC based methods for estimating expected values in the PDE context with lognormal diffusion, map the integral to the unit cube by using the inverse of the Gaussian cumulative distribution function and use randomly shifted lattice rules to approximate this integral, see, e.g., [14, 15, 17, 18, 22, 23, 29]. A comparison with two existing methods, the QMCFEM [18] and MLQMCFEM [17], shows the benefit of the MDFEM
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