Repelled point processes with application to numerical integration

TL;DR

This paper introduces a Coulomb repulsion operator to improve Monte Carlo numerical integration by reducing point clustering, leading to lower variance estimators, with proven variance reduction for Poisson point processes and scalable algorithms.

Contribution

It proposes a novel repulsion operator that enhances Monte Carlo integration by decreasing estimator variance, applicable to Poisson and regular point processes, with theoretical and empirical validation.

Findings

Variance reduction proven for Poisson point processes

Applying the operator reduces clustering and variance

Algorithm is quadratic in complexity and parallelizable

Abstract

We look at Monte Carlo numerical integration from a stochastic geometry point of view. While crude Monte Carlo estimators relate to linear statistics of a homogeneous Poisson point process (PPP), linear statistics of more regularly spread point processes can yield unbiased estimators with faster-decaying variance, and thus lower integration error. Following this intuition, we introduce a Coulomb repulsion operator, which reduces clustering by slightly pushing the points of a configuration away from each other. Our empirical findings show that applying the repulsion operator to a PPP as well as, intriguingly, to more regular point processes, preserves unbiasedness while reducing the variance of the corresponding Monte Carlo estimator, thus enhancing the method. We prove this variance reduction when the initial point process is a PPP. On the computational side, the complexity of the operator is quadratic and the corresponding algorithm can be parallelized without communication across tasks.