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2021 Monte Carlo integration of non-differentiable functions on [0,1]ι, ι=1,,d, using a single determinantal point pattern defined on [0,1]d
Jean-François Coeurjolly, Adrien Mazoyer, Pierre-Olivier Amblard
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Electron. J. Statist. 15(2): 6228-6280 (2021). DOI: 10.1214/21-EJS1929

Abstract

This paper concerns the use of a particular class of determinantal point processes (DPP), a class of repulsive spatial point processes, for Monte Carlo integration. Let d1, Id={1,,d} with ι=|I|. Using a single set of N quadrature points {u1,,uN} defined, once for all, in dimension d from the realization of a specific DPP, we investigate “minimal” assumptions on the integrand in order to obtain unbiased Monte Carlo estimates of μ(fI)=[0,1]ιfI(u)du for any known ι-dimensional integrable function on [0,1]ι. In particular, we show that the resulting estimator has variance with order N1(2s1)d when the integrand belongs to some Sobolev space with regularity s>0. When s>12 (which includes a large class of non-differentiable functions), the variance is asymptotically explicit and the estimator is shown to satisfy a Central Limit Theorem.

Funding Statement

JFC and AM were supported by Natural Sciences and Engineering Research Council, Canada; POA is supported by Grenoble Data Institute (ANR-15-IDEX-02) and LIA CNRS/Melbourne Univ Geodesic.

Acknowledgments

The authors would like to sincerely thank the reviewers for the great interest they have shown in our work, for the important number of suggestions and comments they made which have significantly improved a previous version of the manuscript. The authors would also like to thank Guillaume Gautier and Rémi Bardenet for the fruitful discussions and for sharing their code simulating the DPP defined in Bardenet and Hardy [3]. JF Coeurjolly and A Mazoyer were supported by National Research Council of Canada for this research.

Citation

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Jean-François Coeurjolly. Adrien Mazoyer. Pierre-Olivier Amblard. "Monte Carlo integration of non-differentiable functions on [0,1]ι, ι=1,,d, using a single determinantal point pattern defined on [0,1]d." Electron. J. Statist. 15 (2) 6228 - 6280, 2021. https://doi.org/10.1214/21-EJS1929

Information

Received: 1 January 2020; Published: 2021
First available in Project Euclid: 27 December 2021

Digital Object Identifier: 10.1214/21-EJS1929

Subjects:
Primary: 60G55; 62K99

Keywords: Determinantal point processes , numerical integration , Optimal designs

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Vol.15 • No. 2 • 2021
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