Quasi–Monte Carlo integration with product weights for elliptic PDEs with log-normal coefficients

Abstract

Abstract Quasi–Monte Carlo (QMC) integration of output functionals of solutions of the diffusion problem with a log-normal random coefficient is considered. The random coefficient is assumed to be given by an exponential of a Gaussian random field that is represented by a series expansion of some system of functions. Graham et al. (2015, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math., 131, 329–368) developed a lattice-based QMC theory for this problem and established a quadrature error decay rate ≈ 1 with respect to the number of quadrature points. The key assumption there was a suitable summability condition on the aforementioned system of functions. As a consequence, product-order-dependent weights were used to construct the lattice rule. In this paper, a different assumption on the system is considered. This assumption, originally considered by Bachmayr et al. (2017c, Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients. ESAIM Math. Model. Numer. Anal., 51, 321–339) to utilise the locality of support of basis functions in the context of polynomial approximations applied to the same type of the diffusion problem, is shown to work well in the same lattice-based QMC method considered by Graham et al.: the assumption leads us to product weights, which enables the construction of the QMC method with a smaller computational cost than Graham et al. A quadrature error decay rate ≈ 1 is established, and the theory developed here is applied to a wavelet stochastic model. By a characterisation of the Besov smoothness, it is shown that a wide class of path smoothness can be treated with this framework.