This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.

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In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R d (d = 1, 2, 3), with diffusion coefficient a(x, ω) given as a lognormal random field, i.e., a(x, ω) = exp(Z (x, ω)) where x is the spatial variable and Z (x, ·) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ 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

We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Petrov-Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in the numerical solution of differential and integral equations with random field inputs. We analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field. If p ∈ (0, 1] denotes the "summability exponent" corresponding to the fluctuations in affine-parametric families of operators, then we prove that deterministic "interlaced polynomial lattice rules" of order α = 1/p +1 in s dimensions with N points can be constructed using a fast component-by-component algorithm, in O(α s N log N + α 2 s 2 N ) operations, to achieve a convergence rate of O(N −1/p ), with the implied constant independent of s. This dimension-independent convergence rate is superior to the rate O(N −1/p+1/2 ) for 2/3 ≤ p ≤ 1, which was recently established for randomly shifted lattice rules under comparable assumptions. In our analysis we use a non-standard Banach space setting and introduce "smoothness-driven product and order dependent (SPOD)" weights for which we develop a new fast CBC construction.

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