Repelled point processes with application to numerical integration

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.

Figures (15)

• Figure 1: A sample from a homogeneous Poisson point process of intensity $\rho=1000$ and the corresponding repelled sample.
• Figure 2: PPP sample (first column) and the corresponding RPPP samples, obtained using \ref{['eq:alternative_def_F']} (second column) and using $F^{(0,2)}$ (third column). The first row corresponds to $d=2$ and the second row to $d=3$, with $\varepsilon$ set in each row to the value $\varepsilon_0 = \varepsilon_0(d)$ in \ref{['eq:epsilon_0']}. The last column shows the two RPPP samples superimposed.
• Figure 3: Estimated standard deviations of $\widehat{I}_{s, \Pi_\varepsilon \mathcal{P} \cap K }$ with respect to $\varepsilon$, for $f_1, \, f_2$, and $f_3$, in $d=3$.
• Figure 4: Estimated standard deviations of various Monte Carlo methods for $f_1$, $f_2$, and $f_3$ across different dimensions $d \in \{2,3, 4, 5,7\}$.
• Figure 5: Illustration of the gravitational allocation from Lebesgue to a realization (black points) of a PPP in a disk. Each set of curves sharing the same color illustrates a gravitational cell, which indicates the points of the space allocated to the point of the PPP that belongs to that particular colored region. The code to generate this picture can be found in https://github.com/dhawat/MCRPPy. Note that the image is purely for illustrative purposes, and no claims are being made regarding the existence of a gravitational allocation from the Lebesgue measure to a PPP in dimension $d=2$.

Theorems & Definitions (43)

• Remark 2.1
• Proposition 3.1: Motion-invariance
• Proposition 3.2
• Remark 3.3
• Proposition 3.4: Existence of the moments
• Theorem 3.5: Variance reduction
• Remark 3.6
• Remark 3.7
• Remark 3.8
• Remark 3.9