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arXiv · 2004.14158

Secure pseudorandom bit generators and point sets with low star-discrepancy

Ana-Isabel Gómez·Domingo Gómez-Pérez·Friedrich Pillichshammer
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Abstract

The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are nowadays also applied to very high-dimensional integration problems. Hence multi-dimensional point sets of reasonable size with low discrepancy are badly needed. A seminal result from Heinrich, Novak, Wasilkowski and Wo\'{z}niakowski shows the existence of a positive number $C$ such that for every dimension $d$ there exists an $N$-element point set in $[0,1)^d$ with star-discrepancy of at most $C\sqrt{d/N}$. This is a pure existence result and explicit constructions of such point sets would be very desirable. The proofs are based on random samples of $N$-element point sets which are difficult to realize for practical applications. In this paper we propose to use secure pseudorandom bit generators for the generation of point sets with star-discrepancy of order $O(\sqrt{d/N})$. This proposal is supported theoretically and by means of numerical experiments.

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Ana-Isabel Gómez, Domingo Gómez-Pérez, Friedrich Pillichshammer. 2020-04-29. Secure pseudorandom bit generators and point sets with low star-discrepancy. https://arxiv.org/abs/2004.14158

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