MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

Nguyen, Dong T. P.; Nuyens, Dirk

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 4, pp. 1461-1505

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 y j ϕ j with y j :𝒩(0,1) and a given sequence of functions {ϕ j } 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$$ and ε$$) 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 {ϕ j } j≥1.

Classification : 65D30, 65D32, 65N30
Keywords: Elliptic PDE, stochastic diffusion coefficient, lognormal case, infinite-dimensional integration, multivariate decomposition method, finite element method, higher-order quasi-Monte Carlo, high dimensional quadrature/cubature, complexity bounds

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