Incremental space-filling design based on coverings and spacings: improving upon low discrepancy sequences

TL;DR

This paper introduces incremental algorithms for space-filling designs that improve upon low discrepancy sequences by optimizing covering and spacing criteria, with proven bounds and better performance.

Contribution

It presents novel incremental design algorithms with proven optimality bounds based on coverings and spacings, outperforming existing methods.

Findings

Covering-based method outperforms state-of-the-art competitors.

Algorithms have guaranteed 50% optimality gap.

Methods converge to dispersion for large parameters.

Abstract

The paper addresses the problem of defining families of ordered sequences $\{x_i\}_{i\in N}$ of elements of a compact subset $X$ of $R^d$ whose prefixes $X_n=\{x_i\}_{i=1}^{n}$, for all orders $n$, have good space-filling properties as measured by the dispersion (covering radius) criterion. Our ultimate aim is the definition of incremental algorithms that generate sequences $X_n$ with small optimality gap, i.e., with a small increase in the maximum distance between points of $X$ and the elements of $X_n$ with respect to the optimal solution $X_n^\star$. The paper is a first step in this direction, presenting incremental design algorithms with proven optimality bound for one-parameter families of criteria based on coverings and spacings that both converge to dispersion for large values of their parameter. The examples presented show that the covering-based method outperforms state-of-the-art competitors, including coffee-house, suggesting that it inherits from its guaranteed 50\% optimality gap.