Background on Randomized QMC

Abstract

First, we show the intuitive basis for randomizing a deterministic (t, m, s)-net. Section 15.1 gives the details, tailored to base 2 and optionally to digital nets. It uses much less storage and is much faster than a naive implementation. Second, we briefly discuss randomized “lattice” rules and why we have chosen not to use them. Third, we define Latin hyper-cubes, discuss their one-dimensional and multidimensional coordinate projections, display their associated variance, and show how to simultaneously increase dynamically their size and the size of the (t, m, s)-net(s) corresponding to X without throwing anything away. Finally, we deal with Latin supercubes, with an extended example, and their relation to a functional ANOVA.