Abstract

Computer experiments have been widely used in various fields. Among different design

types, space-filling designs stand out as the most common choice for computer experiments

due to their effectiveness in thoroughly exploring the experimental region. Extensive research

has been conducted on the space-filling criteria.

However, there are relatively few studies

developing a systematic framework for relationships among most space-filling criteria. Kernel

functions possess numerous elegant properties, and some space-filling criteria also have inherent

connections with them. This paper establishes links among different space-filling criteria via

their expressions in the form of kernel functions.

Focusing on the designs with Kronecker

product structure, this paper provides explicit expressions and associated theoretical results for

kernel-based space-filling criteria. In addition, construction methods for optimal designs with

Kronecker product structure are also proposed. Moreover, an algorithm is proposed to generate

a series of space-filling designs that have better performance compared with other designs.

Key words and phrases: Discrepancy; Kernel function; Maximin distance; Uniform design 1. Introduction In the past decades, computer experiments have been widely used in industry, 2 ZHENG ET AL

Information

Preprint No.SS-2025-0279
Manuscript IDSS-2025-0279
Complete AuthorsRuonan Zheng, Xinran Zhang, Jian-Feng Yang, Min-Qian Liu
Corresponding AuthorsMin-Qian Liu
Emailsmqliu@nankai.edu.cn

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Acknowledgments

The authors thank Editor John Stufken, an associate editor and two anonymous

referees for their insightful comments and suggestions. This research was supported by

the National Natural Science Foundation of China (Grant Nos. 12131001, 12271270 and

12371260). The four authors are affiliated with the NITFID, LPMC, KLMDASR, and

the School of Statistics and Data Science at Nankai University, and they contributed

equally to this work.

Supplementary Materials

The supplementary material discusses the corresponding results about extending

the mean squared correlation criterion and distance variance criterion to (2.1) respectively, provides the proofs of some theoretical results, the additional comparisons and

simulations, and a large table.

Supplementary materials are available for download.