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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.

We define a Walsh space which contains all functions whose partial mixed derivatives up to order δ ≥ 1 exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on digital (t, α, s)-sequences achieve the optimal rate of convergence of the worst-case error for numerical integration. This rate of convergence is also optimal for the subspace of smooth functions. Explicit constructions of digital (t, α, s)-sequences are given hence providing explicit quasi-Monte Carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions. Walsh functions over groups.In this section we give the definition of Walsh functions over groups and present some essential properties. Walsh functions in base 2 were first introduced by Walsh [33], though a similar but non-complete set of functions has already been studied by Rademacher [27]. Further important results were obtained in [10]. We follow [26] in our presentation.2.1. Definition of Walsh functions over groups. An essential tool for the investigation of digital nets are Walsh functions. A very general definition, corresponding to the most general construction of digital nets over finite rings, was given

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.

In this paper we give explicit constructions of point sets in the s dimensional unit cube yielding quasi-Monte Carlo algorithms which achieve the optimal rate of convergence of the worst-case error for numerically integrating high dimensional periodic functions. In the classical measure Pα of the worst-case error introduced by Korobov the convergence is of O(N − min(α,d) (log N ) sα−2 ) for every even integer α ≥ 1, where d is a parameter of the construction which can be chosen arbitrarily large and N is the number of quadrature points. This convergence rate is known to be best possible up to some log N factors. We prove the result for the deterministic and also a randomized setting. The construction is based on a suitable extension of digital (t, m, s)-nets over the finite field Z b .