Fast high-dimensional node generation with variable density

TL;DR

This paper introduces a novel algorithm combining quasi-Monte Carlo methods and energy minimization to generate high-dimensional point distributions with controlled local density, useful for meshless numerical methods and atmospheric modeling.

Contribution

The paper presents a new algorithm that efficiently produces high-dimensional node sets with prescribed local density and regularity, improving upon existing methods.

Findings

Effective in generating high-dimensional node sets with variable density

Applicable to meshless solvers and atmospheric modeling

Produces well-distributed point clouds with controlled separation and coverage

Abstract

We present an algorithm for producing discrete distributions with a prescribed nearest-neighbor distance function. Our approach is a combination of quasi-Monte Carlo (Q-MC) methods and weighted Riesz energy minimization: the initial distribution is a stratified Q-MC sequence with some modifications; a suitable energy functional on the configuration space is then minimized to ensure local regularity. The resulting node sets are good candidates for building meshless solvers and interpolants, as well as for other purposes where a point cloud with a controlled separation-covering ratio is required. Applications of a three-dimensional implementation of the algorithm, in particular to atmospheric modeling, are also given.