Scrambled geometric net integration over general product spaces

TL;DR

This paper develops a new randomized QMC method using geometric transformations for integration over complex product spaces like triangles and spheres, achieving better accuracy than traditional Monte Carlo methods.

Contribution

It introduces a construction of point sets for numerical integration over product spaces using recursive geometric partitions, extending QMC applicability beyond simple hypercubes.

Findings

Variance of estimates is o(1/n) for L^2 integrands.

Under smoothness, variance is O(n^{-1 - 2/d} (log n)^{s-1}).

Method outperforms Monte Carlo in accuracy for complex domains.

Abstract

Quasi-Monte Carlo (QMC) sampling has been developed for integration over $[0,1]^s$ where it has superior accuracy to Monte Carlo (MC) for integrands of bounded variation. Scrambled net quadrature gives allows replication based error estimation for QMC with at least the same accuracy and for smooth enough integrands even better accuracy than plain QMC. Integration over triangles, spheres, disks and Cartesian products of such spaces is more difficult for QMC because the induced integrand on a unit cube may fail to have the desired regularity. In this paper, we present a construction of point sets for numerical integration over Cartesian products of $s$ spaces of dimension $d$, with triangles ($d=2$) being of special interest. The point sets are transformations of randomized $(t,m,s)$-nets using recursive geometric partitions. The resulting integral estimates are unbiased and their variance is $o(1/n)$ for any integrand in $L^2$ of the product space. Under smoothness assumptions on the integrand, our randomized QMC algorithm has variance $O(n^{-1 - 2/d} (\log n)^{s-1})$, for integration over $s$-fold Cartesian products of $d$-dimensional domains, compared to $O(n^{-1})$ for ordinary Monte Carlo.