Decrease of the mean of the quasi-random integration error

Abstract

The article is devoted to the study of the behavior of the quasi-random integration remainder in the calculation of high-dimensional integrals. As noted in the previous work of the authors, the asymptotic behavior of its decrease, determined by the Koksma-Hlawka inequality, can be used only with a very large number of integration nodes N, which cannot be implemented on modern computers. The article introduces the concept of a mean order of decreasing remainder, which makes it possible to judge its properties with the N values available for realization and to compare various pseudo-random sequences. A number of numerical examples are given. In all cases, it turned out that the Sobol’ sequences in the sense of this criterion are somewhat better than the Holton sequences.

Keywords:

• Quasi-random sequences
• Quasi-Monte Carlo method
• Randomization

Additional information

Funding

This work was supported by the Russian Foundation for Basic Research, under Grant number 17-01-00267-a.