A Nonlocal Functional Promoting Low-Discrepancy Point Sets

2019/02/01 by Stefan Steinerberger, Steinerberger, Stefan
Computer Science · Mathematics · #Cryptography and Residue Arithmetic #FOS: Mathematics #Mathematical Approximation and Integration #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.1902.00441

openalex publication_date 2019/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X = \x₁, …, x_N\ ⊂ \mathbbT^d ≅ [0,1]^d be a set of N points in the d-dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce a curious energy functional E(X) = ∑_{1 ≤ m,n ≤ N \atop m ≠ n} ∏_k=1^d (1 - log(2 sin ( π|x_m,k - y_m,k |) )) and to suggest that moving a set X into the direction -∇ E(X) may have the effect of increasing regularity of the set in the sense of decreasing discrepancy. We numerically demonstrate the effect for Halton, Hammersley, Kronecker, Niederreiter and Sobol sets. Lattices in d=2 are critical points of the energy functional, some (possibly all) are strict local minima.

Citations

• On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals

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