Space-filling lattice designs for computer experiments

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.

💡 Research Summary

The paper addresses the fundamental problem of constructing space‑filling designs for computer experiments, where the quality of a design is measured by how uniformly its points cover the input domain. Traditional metrics—covering radius (the radius of the largest empty ball) and separation radius (the smallest distance between any two points)—are combined into a single unified criterion called quasi‑uniformity, defined as the ratio of covering to separation radii. A design is quasi‑uniform when this ratio is close to one, indicating that the points are neither too clustered nor too sparse.

Focusing on quasi‑Monte Carlo (QMC) lattice point sets, the authors propose two concrete construction algorithms. The first algorithm generates rank‑1 lattice point sets that approximate Kronecker sequences. The generating vector g is given explicitly: each component (g_j = \lfloor N^{j/d}\rfloor), where (N) is the number of points and (d) the dimension. Points are then formed as (\mathbf{x}_i = {i\mathbf{g}/N}). The authors prove that this construction attains an isotropic discrepancy of order (O(N^{-1/d})), which improves on the classic (O(N^{-1/d}\log N)) bounds for generic lattice rules. Moreover, direct calculations of covering and separation radii show that the quasi‑uniformity ratio converges to (1 + O(N^{-1/d})).

The second algorithm works with Korobov lattices, whose generating vector has the form (\mathbf{a} = (1, a, a^2, \dots, a^{d-1})). The key contribution is the use of the Lenstra–Lenstra–Lovász (LLL) lattice‑basis reduction algorithm to select the integer (a) that minimizes the quasi‑uniformity. By constructing the lattice basis matrix and applying LLL, the method finds a short, nearly orthogonal basis, which simultaneously maximizes the separation radius and minimizes the covering radius. The resulting Korobov lattice therefore exhibits superior quasi‑uniformity, especially in dimensions greater than ten where naïve choices of (a) become ineffective.

The theoretical results are validated through extensive numerical experiments. For dimensions (d = 2, 3, 5, 10, 20) and sample sizes ranging from (N = 2^5) to (2^{12}), the authors compute covering radius, separation radius, quasi‑uniformity, and isotropic discrepancy for the two proposed designs and for standard QMC sequences (Sobol’, Halton, Faure). The new lattice designs consistently achieve lower quasi‑uniformity values and smaller discrepancy, with the advantage becoming more pronounced as (N) grows.

To demonstrate practical relevance, the paper applies the designs to Gaussian Process Regression (GPR) tasks, a common surrogate‑modeling technique in computer experiments. Using high‑dimensional benchmark functions, the authors compare predictive mean squared error (RMSE) and predictive variance calibration across designs. The rank‑1 lattice and LLL‑optimized Korobov lattice reduce RMSE by roughly 15–25 % relative to Sobol’ and Halton points, and they provide tighter, more reliable uncertainty estimates. These gains are attributed to the balanced coverage (small covering radius) and sufficient spacing (large separation radius) inherent in quasi‑uniform designs.

In summary, the paper makes three principal contributions: (1) it formalizes quasi‑uniformity as a unified metric for space‑filling designs; (2) it introduces an explicit, analytically tractable rank‑1 lattice construction with provable (O(N^{-1/d})) discrepancy; (3) it leverages LLL basis reduction to obtain Korobov lattices that are provably quasi‑uniform. The combination of rigorous theoretical analysis, algorithmic innovation, and empirical validation positions these lattice designs as strong candidates for high‑dimensional computer experiments, Bayesian optimization, and uncertainty quantification tasks where uniform coverage of the input space is critical.