Relationship between low-discrepancy sequence and static solution to multi-bodies problem

TL;DR

This paper explores the connection between low-discrepancy sequences and static solutions in multi-bodies problems, proposing a model that links minimal potential energy to uniform distribution, verified through numerical experiments.

Contribution

It introduces a dynamical evolutionary model that generates low-discrepancy sequences based on potential energy minimization, demonstrating improved sequence quality and applicability.

Findings

Significant positive correlation between potential energy and discrepancy.

DEM-generated sequences outperform other low-discrepancy sequences.

Sequences are effective in both cube and non-cube distributions.

Abstract

The main interest of this paper is to study the relationship between the low-discrepancy sequence and the static solution to the multi-bodies problem in high-dimensional space. An assumption that the static solution to the multi-bodies problem is a low-discrepancy sequence is proposed. Considering the static solution to the multi-bodies problem corresponds to the minimum potential energy principle, we further assume that the distribution of the bodies is the most uniform when the potential energy is the smallest. To verify the proposed assumptions, a dynamical evolutionary model (DEM) based on the minimum potential energy is established to find out the static solution. The central difference algorithm is adopted to solve the DEM and an evolutionary iterative scheme is developed. The selection of the mass and the damping coefficient to ensure the convergence of the evolutionary iteration is discussed in detail. Based on the DEM, the relationship between the potential energy and the discrepancy during the evolutionary iteration process is studied. It is found that there is a significant positive correlation between them, which confirms the proposed assumptions. We also combine the DEM with the restarting technique to generate a series of low-discrepancy sequences. These sequences are unbiased and perform better than other low-discrepancy sequences in terms of the discrepancy, the potential energy, integrating eight test functions and computing the statistical moments for two practical stochastic problems. Numerical examples also show that the DEM can not only generate uniformly distributed sequences in cubes, but also in non-cubes.