On randomization of Halton quasi-random sequences

Abstract

The problem of estimating the error of quasi-Monte Carlo methods by means of randomization is considered. The well known Koksma–Hlawka inequality enables one to estimate asymptotics for the error, but it is not useful in computational practice, since computation of the quantities occurring in it, the variation of the function and the discrepancy of the sequence, is an extremely timeconsuming and impractical process. For this reason, there were numerous attempts to solve the problem mentioned above by the probability theory methods. A common approach is to shift randomly the points of quasi-random sequence. There are known cases of the practical use of this approach, but theoretically it is scantily studied. In this paper, it is shown that the estimates obtained this way are upper estimates. A connection with the theory of cubature formulas with one random node is established. The case of Halton sequences is considered in detail. The van der Corput transformation of a sequence of natural numbers is studied, and the Halton points are constructed with its help. It is shown that the cubature formula with one free node corresponding to the Halton sequence is exact for some class of step functions. This class is explicitly described. The obtained results enable one to use these sequences more effectively for calculating integrals and finding extrema and can serve as a starting point for further theoretical studies in the field of quasi-random sequences.