QUASI-MONTE CARLO NUMERICAL INTEGRATION ON Rs: DIGITAL NETS AND WORST-CASE ERROR

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 R-s. 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 R-s 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.