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Space-filling lattice designs for computer experiments

Naoki Sakai, Takashi Goda

TL;DR

This paper focuses on a special class of designs, known as quasi-Monte Carlo (QMC) lattice point sets, and proposes two construction algorithms, employing the Lenstra--Lenstra--Lov\'{a}sz (LLL) basis reduction algorithm to identify the generating vector that ensures quasi-uniformity.

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.

Space-filling lattice designs for computer experiments

TL;DR

This paper focuses on a special class of designs, known as quasi-Monte Carlo (QMC) lattice point sets, and proposes two construction algorithms, employing the Lenstra--Lenstra--Lov\'{a}sz (LLL) basis reduction algorithm to identify the generating vector that ensures quasi-uniformity.

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 . 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.
Paper Structure (21 sections, 5 theorems, 54 equations, 16 figures, 3 tables, 2 algorithms)

This paper contains 21 sections, 5 theorems, 54 equations, 16 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-uniform designs
  4. Lattices
  5. Successive minima and quasi-uniformity
  6. Construction of space-filling lattice designs
  7. Explicit rank-1 lattice construction
  8. Algorithmic Korobov lattice construction
  9. Numerical experiments
  10. Empirical properties of two constructions
  11. Construction parameters and the number of points
  12. Bound on mesh ratio
  13. Separation radius
  14. Gaussian process regression
  15. Experimental setup
  16. ...and 6 more sections

Key Result

Lemma 2.4

Let $\Lambda \subset \mathbb{R}^d$ be a lattice with a generating matrix $T$, and $\Lambda^{\perp}$ be its dual lattice. Then, with $C_d=2^d\Gamma(1+d/2)/\pi^{d/2}$ where $\Gamma$ denotes the gamma function.

Figures (16)

  • Figure 1: $d=2$
  • Figure 2: $d=3$
  • Figure 3: $d=5$
  • Figure 4: $d=7$
  • Figure 6: $d=2$
  • ...and 11 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proof 1
  • Remark 2.5
  • Lemma 2.6
  • Proof 2
  • Remark 2.7
  • Remark 3.1
  • ...and 7 more