Abstract

A standard objective in computer experiments is to approximate the behavior of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) best linear unbiased estimator (BLUE) for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-filling properties.

Keywords

  1. Bayesian quadrature
  2. BLUE
  3. energy minimization
  4. potential
  5. discrepancy
  6. space-filling design

MSC codes

  1. 62K99
  2. 65D30
  3. 65D99

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Information & Authors

Information

Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 959 - 1011
ISSN (online): 2166-2525

History

Submitted: 28 August 2018
Accepted: 3 May 2020
Published online: 11 August 2020

Keywords

  1. Bayesian quadrature
  2. BLUE
  3. energy minimization
  4. potential
  5. discrepancy
  6. space-filling design

MSC codes

  1. 62K99
  2. 65D30
  3. 65D99

Authors

Affiliations

Funding Information

Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-18-CE91-0007
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/K032208/1

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