Stratified Monte Carlo quadrature for continuous random fields

TL;DR

This paper analyzes the accuracy of stratified Monte Carlo methods for approximating integrals of continuous random fields, focusing on locally stationary and isotropic classes, and proposes designs to improve approximation quality.

Contribution

It introduces asymptotic accuracy results for stratified Monte Carlo quadrature on locally stationary fields and designs that mitigate singularity effects.

Findings

Asymptotic approximation accuracy for large sample sizes

Upper bounds for approximation error in Hölder class

Designs that eliminate singularity effects in isotropic fields

Abstract

We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition (or strata) generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behavior is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the H\"{o}lder class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.