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.
1. Introduction. In the field of computer experiments, maximin and minimax distance-based designs have gained prominence due to their superior space-filling properties [8,16,17,18,31,35]. Conceptually, maximin designs prioritize the separation between sampling points to ensure that they are well-dispersed across the domain, while minimax designs aim to cover the target space as efficiently as possible by minimizing the largest unexplored regions. These two geometric objectives, i.e., separation and coverage, are naturally unified by the concept of quasi-uniformity, which aims to attain both properties simultaneously and has become one of the central criteria for assessing the quality of experimental designs recently [32,33].
The utility of quasi-uniform designs extends across various numerical frameworks, including radial basis function approximation, kernel interpolation, and Gaussian process regression [36,37,44,45,46,47]. To appreciate the necessity of quasiuniformity, consider kernel interpolation with standard Sobolev kernels as a primary example. In this context, the geometric properties of the point set dictate both the approximation accuracy and the numerical stability. When the target function resides within the associated native reproducing kernel Hilbert space, the approximation error is typically bounded by a power of the covering radius. However, minimizing the covering radius is not the sole objective. The separation radius is equally critical, as it governs the condition number of the kernel matrix (Gram matrix); a shrinking separation radius leads to severe ill-conditioning, compromising numerical stability [36]. Furthermore, in scenarios where the target function possesses lower regularity than the kernel implies, i.e., when approximating a “rough” function that lies outside the native space, theoretical error bounds often explicitly depend on the mesh ratio (the ratio of the covering radius to the separation radius), rather than solely on the covering radius [37,44,47]. Consequently, constructing quasi-uniform designs for computer experiments is critical for achieving a balance between high-fidelity convergence and numerical stability. Furthermore, in the context of Bayesian optimization, which frequently employs Gaussian process regression as a surrogate model, the use of quasi-uniform designs in the initial sampling stage has empirically been shown to accelerate the convergence of the optimization process [19].
Quasi-Monte Carlo (QMC) point sets and sequences, also known as lowdiscrepancy point sets and sequences, are widely recognized for their ability to achieve faster convergence rates in numerical integration over the unit hypercube [0, 1] d than random sampling [7,20,30]. Given their excellent equi-distribution properties, they have also been considered prime candidates for space-filling quasi-uniform designs, see, for instance, [47,Section 3.1]. However, the specific geometric property of quasiuniformity in the QMC context has remained relatively under-explored until very recently. Recent studies have begun to fill this gap, revealing highly non-trivial results: many widely used QMC sequences-including the two-dimensional Sobol’, Faure, and Halton sequences-fail to satisfy the requirements for quasi-uniformity (see [13], [5], and [14], respectively). More precisely, the separation radius of these sequences decays at a faster rate than N -1/d , violating the conditions necessary for quasi-uniformity. In contrast, a recent work [4], together with a prior work [12], proves that lattice point sets and Kronecker sequences can achieve quasi-uniformity, provided that their underlying parameters are appropriately selected.
While [4] provided an explicit construction for quasi-uniform Kronecker sequences, the results for lattice point sets remained purely existential; neither an explicit construction nor a constructive algorithm was previously known. Addressing this gap is the primary objective of the present work. In this paper, we propose two distinct methodologies for constructing space-filling, quasi-uniform lattice designs for computer experiments as follows:
- Explicit rank-1 lattice construction: We provide an explicit rank-1 lattice point set construction as a discrete approximation of the quasi-uniform Kronecker sequences introduced in [4]. A characteristic of this method is that the number of sampling points N is not arbitrarily chosen but is determined by the Diophantine properties of the parameters defining the underlying Kronecker sequence. Despite this constraint, the method provides the parameters, i.e., the generating vectors, for rank-1 lattice point sets explicitly. Furthermore, this result yields rank-1 lattice point sets with low isotropic discrepancy, thereby providing a partial solution to an open problem regarding the explicit construction of such sets [6, Corollary 5.27]. 2. Algorithmic Korobov lattice construction: Focusing on a class of Korobov lattice point s
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