Quasi-Monte Carlo tractability of high dimensional integration over products of simplices

TL;DR

This paper investigates the conditions under which quasi-Monte Carlo methods achieve different levels of tractability for high-dimensional integrals over products of simplices, extending known results from other domains.

Contribution

It establishes new tractability criteria for QMC methods on product simplices, using Sobolev space and reproducing kernel Hilbert space techniques.

Findings

Strong polynomial tractability iff sum of weights is bounded

Polynomial tractability iff sum of weights grows slower than log(m)

Weak tractability iff sum of weights divided by m tends to zero

Abstract

Quasi-Monte Carlo (QMC) methods for high dimensional integrals over unit cubes and products of spheres are well-studied in literature. We study QMC tractability of integrals of functions defined over the product of $m$ copies of the simplex $T^d \subset \mathbb{R}^{d}$. The domain is a tensor product of $m$ reproducing kernel Hilbert spaces defined by `weights' $\gamma_{m,j}$, for $j = 1,2, \ldots, m$. Similar to the results on the unit cube in $m$ dimensions, and the product of $m$ copies of the $d$-dimensional sphere, we prove that strong polynomial tractability holds iff $\limsup_{m \rightarrow \infty} \sum_{j=1}^m \gamma_{m,j} < \infty$ and polynomial tractability holds iff $\limsup_{m \rightarrow \infty} \frac{\sum_{j=1}^m \gamma_{m,j}}{\log(m + 1 )} < \infty$. We also show that weak tractability holds iff $\lim_{m \rightarrow \infty} \frac{\sum_{j=1}^m \gamma_{m,j}}{m} = 0$. The proofs employ Sobolev space techniques and weighted reproducing kernel Hilbert space techniques for the simplex and products of simplices as domain. Properties of orthogonal polynomials on a simplex are also used extensively.