Discrepancy of sequences and error estimates for the quasi-Monte Carlo method

Abstract: We present the notions of uniform distribution and discrepancy of sequences contained in the unit interval, as well as an important application of discrepancy in numerical integration by way of the quasi-Monte Carlo method. Some fundamental (and other interesting) results with regards to these notions are presented, along with some detalied and instructive examples and comparisons (some of which not often provided by the literature). We go on to analytical and numerical investigations of the asymptotic behaviour of the discrepancy (in particular for the van der Corput-sequence), and for the general error estimates of the quasi-Monte Carlo method. Using the discoveries from these investigations, we give a conditional proof of the van der Corput theorem. Furthermore, we illustrate that by using low discrepancy sequences (such as the vdC-sequence), a rather fast convergence rate of the quasi-Monte Carlo method may still be achieved, even for situations in which the famous theoretical result, the Koksma inequality, hasbeen rendered unusable.