A nonlocal functional promoting low-discrepancy point sets

Abstract

Let X = {x 1 , . . . , x N } ⊂ T 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 the energy functionaland 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.2010 Mathematics Subject Classification. 11L03, 42B05, 82C22.

Dynamically defined sequences with small discrepancy

“…1). Our sequence is actually comparable (or even superior) in quality to many of the classical constructions (see also [22]). We compare (see Table 1) the sequence with the Halton set (using base 2 and 3), the Hammersley sequence (using base 2) and the Kronecker-type set…”

“…Results. This paper is a companion paper to [22] where we showed that minimizing a certain functional can decrease the discrepancy of point sets. Here we show that this functional also allows us to construct uniformly distributed sequences in a way that is very different from the usual constructions.…”

Greedy energy minimization can count in binary: point charges and the van der Corput sequence

“…The second statement in the main theorem, i.e. discrepancy being preserved over all possible choices, shows that potential theoretic approaches along the lines of what was proposed by Steinerberger [25,26]…”

“…Motivated by this, Steinerberger [25,26] recently proposed to study whether regular sequences could be constructed via dynamical systems of the type outlined in (1). More precisely, suppose we are given {x 0 , … , x N−1 } ⊂ [0, 1) , then he proposed to construct x N in a greedy manner as and if the minimum is not unique, any choice is admissible.…”

“…The functional studied by Steinerberger [25,26] is an example of an admissible function; for x ∈ (0, 1) see Fig. 2 for an illustration.…”