Space-filling lattice designs for computer experiments

2026-02-17

AI Summary

This work addresses the construction of space-filling designs for computer experiments by proposing two quasi-Monte Carlo (QMC) lattice-based algorithms: one based on rank-1 lattices and the other on Korobov lattices. Uniformity is characterized through a unified framework involving covering and separation radii, enabling the first explicit construction of lattice point sets that approximate quasi-uniform Kronecker sequences. The authors prove that these constructions achieve an isotropic discrepancy of optimal theoretical order $O(N^{-1/d})$. Furthermore, they enhance isotropy by optimizing the Korobov generating vector via the LLL lattice basis reduction algorithm. Numerical experiments demonstrate that the proposed designs outperform existing QMC point sets in terms of quasi-uniformity and significantly improve predictive accuracy in Gaussian process regression tasks.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

This paper investigates the construction of space-filling designs for computer experiments. The space-filling property is characterized by the covering and separation radii of a design, which are integrated through the unified criterion of quasi-uniformity. We focus on a special class of designs, known as quasi-Monte Carlo (QMC) lattice point sets, and propose two construction algorithms. The first algorithm generates rank-1 lattice point sets as an approximation of quasi-uniform Kronecker sequences, where the generating vector is determined explicitly. As a byproduct of our analysis, we prove that this explicit point set achieves an isotropic discrepancy of $O(N^{-1/d})$. The second algorithm utilizes Korobov lattice point sets, employing the Lenstra--Lenstra--Lovász (LLL) basis reduction algorithm to identify the generating vector that ensures quasi-uniformity. Numerical experiments are provided to validate our theoretical claims regarding quasi-uniformity. Furthermore, we conduct empirical comparisons between various QMC point sets in the context of Gaussian process regression, showcasing the efficacy of the proposed designs for computer experiments.

Problem

• space-filling
• quasi-uniformity
• lattice designs
• computer experiments
• quasi-Monte Carlo

Innovation

• quasi-uniformity
• lattice point sets
• quasi-Monte Carlo
• LLL algorithm
• space-filling design