Maximin design on non hypercube domain and kernel interpolation

TL;DR

This paper explores maximin design strategies for non-hypercube domains in computer experiments, providing theoretical justification and algorithms for optimal experimental design using kernel interpolation.

Contribution

It extends maximin design methodology beyond hypercube domains and introduces simulated annealing algorithms with convergence proofs.

Findings

Theoretical justification of maximin criterion for kernel interpolation.

Development of simulated annealing algorithms for maximin design in arbitrary domains.

Proof of convergence for the proposed algorithms.

Abstract

In the paradigm of computer experiments, the choice of an experimental design is an important issue. When no information is available about the black-box function to be approximated, an exploratory design have to be used. In this context, two dispersion criteria are usually considered: the minimax and the maximin ones. In the case of a hypercube domain, a standard strategy consists of taking the maximin design within the class of Latin hypercube designs. However, in a non hypercube context, it does not make sense to use the Latin hypercube strategy. Moreover, whatever the design is, the black-box function is typically approximated thanks to kernel interpolation. Here, we first provide a theoretical justification to the maximin criterion with respect to kernel interpolations. Then, we propose simulated annealing algorithms to determine maximin designs in any bounded connected domain. We prove the convergence of the different schemes.