Treffer: Point processes for numerical integration ; Processus ponctuels pour l'intégration numérique

The Monte Carlo method estimates an integral using pointwise evaluations of the integrand at some points called nodes, which can be chosen as the points of a point process. While crude Monte Carlo relies on a homogeneous Poisson point process (PPP), some more regularly spread point processes yield Monte Carlo methods with faster-decaying variance. In this thesis, we study two families of regular point processes that are potential candidate nodes to speed up the convergence of crude Monte Carlo. The first one is the family of hyperuniform point processes (HUPPs). A HUPP is characterized by the variance of the number of points in a large window scaling slower than the volume of that window. In particular, a HUPP yields a Monte Carlo estimator of volumes with a faster decaying variance than under the PPP. Unfortunately, proving that a point process is hyperuniform is usually difficult.Aiming to provide statistical tools for identifying HUPPs we examine a spectral measure called the structure factor whose decay around zero provides a diagnostic of hyperuniformity. We provide a survey and derivation of natural estimators of the structure factor and contribute an asymptotically valid statistical test of hyperuniformity. We further provide a Python toolbox containing all the estimators and tools that we discuss.The second family of point processes under consideration pertains to repelled point processes which we construct using a so-called repulsion operator. The repulsion operator reduces clustering in a configuration of points by slightly pushing the points away from each other. Our main theoretical result is that applying the repulsion operator to a PPP yields an unbiased Monte Carlo method with lower variance than under the original PPP. Moreover, our numerical investigations shed light on the operator's variance reduction ability, even when applied to more regular point processes than the PPP. This suggests that applying the repulsion operator to the nodes of any Monte Carlo method may decrease its variance and ...