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Quasi-Monte Carlo rules are equal weight quadrature rules defined over the domain [0, 1]s. Here we introduce quasi-Monte Carlo-type rules for numerical integration of functions defined on ℝs. These rules are obtained by way of some transformation of digital nets such that locally one obtains quasi Monte Carlo rules, but at the same time, globally one also has the required distribution. We prove that these rules are optimal for numerical integration in spaces of bounded fractional variation. The analysis is based on certain tilings of the Walsh phase plane. As a proof of concept, some numerical examples are included for dimensions between 3 and 10. We compare our method with quasi-Monte Carlo rules transformed to ℝs using the inverse cumulative distribution function. In these examples, the new method significantly improves upon both methods for dimensions up to 5; no improvement is seen for dimension 10. [ABSTRACT FROM AUTHOR]

This paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the Korobov form, in which the generating vector for an n-point rule with it prime has the form (1, a,…, ad-1)(mod n). With the parameter a chosen optimally, we establish new error bounds for Korobov lattice rules in weighted Korobov spaces. In particular, we prove that if the weights decay sufficiently fast, the optimal Korobov lattice rule has an error bound of order O(n-α/2+δ) (for arbitrary δ>0), with the implied constant depending at worst polynomially on the dimension. Here α> 1 is the smoothness parameter of the weighted Korobov spaces. We generalize the construction to the case where it is a product of arbitrary distinct prime numbers, with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules. A corresponding result is deduced for weighted Sobolev spaces of nonperiodic functions, using randomly shifted optimal Korobov lattice rules. A comparison of the worst-case errors for Korobov lattice rules and the recent component-by-component constructions is presented. The investigation establishes the usefulness of (shifted) optimal Korobov lattice rules for integration, even in high dimensions, if the weights which characterize the weighted spaces are suitably chosen. [ABSTRACT FROM AUTHOR]