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.

翻译：本文从随机几何的角度研究蒙特卡洛数值积分问题。原始蒙特卡洛估计量对应于齐次泊松点过程的线性统计量，而分布更规则的点过程的线性统计量可产生方差衰减更快、积分误差更低的无偏估计量。基于这一思路，我们引入了一种库仑排斥算子，该算子通过轻微推动构型中的点相互远离来减少聚集现象。实验结果表明，将排斥算子应用于泊松点过程（有趣的是，也适用于更规则的点过程）能在保持无偏性的同时降低相应蒙特卡洛估计量的方差，从而提升方法性能。我们证明了当初始点过程为泊松点过程时该方差减少效应成立。在计算层面，该算子的复杂度为二次方，且相应算法可实现跨任务无通信的并行化。