On the Halton quasi-random sequences randomization

Abstract

The problem of estimating the error of quasi-Monte Carlo methods with the use of randomization is considered. The well-known Koksma Hlawka inequality allows us to estimate the asymptotics for the error, but it is not suitable for practical use in the process of computation, since calculation of the quantities occurring in it is a variation of the function and discrepancy of the sequence, is an extremely time-consuming and impractical process. For this reason, there were numerous attempts to solve the aforementioned problem with the probability theory methods. One common approach is to randomly shift the points of the pseudo-random sequence. Cases of practical use of this approach are known, but theoretically it has been scantily studied. In this paper it is shown that estimates obtained this way are the upper estimates. A connection with the theory of cubature formulas with one random node is established. The case of the Halton sequences is considered in details. The Van der Corput transformation for sequence of natural numbers is analyzed, with its help the Halton points are constructed. It is shown that the cubature formula with one free node corresponding to the Halton sequence is exact for some class of step functions. The class is explicitly described. The obtained results allow us to use the mentioned sequences more effectively for integral calculation and extremum finding, and can also serve as a starting point for further theoretical studies in the field of the quasi-random methods. Refs 6. Fig. 1. Table 1.