Quasi-Monte Carlo for an Integrand with a Singularity along a Diagonal in the Square

TL;DR

This paper investigates quasi-Monte Carlo methods for integrating functions with singularities along the diagonal in a square, proposing and analyzing three approaches with specific error bounds.

Contribution

It introduces and compares three quadrature methods for singular integrands along the diagonal, including a triangle splitting approach and integrand transformation techniques.

Findings

Triangle splitting method achieves error of O((log(n)/n)^{(1-A)/2}) for singularities of order A<1.

Transforming the integrand can improve error rates to O(n^{-1+ε+A}) under stronger assumptions.

Method extensions into singular regions do not outperform the triangle-based approach.

Abstract

Quasi-Monte Carlo methods are designed for integrands of bounded variation, and this excludes singular integrands. Several methods are known for integrands that become singular on the boundary of the unit cube $[0,1]^d$ or at isolated possibly unknown points within $[0,1]^d$. Here we consider functions on the square $[0,1]^2$ that may become singular as the point approaches the diagonal line $x_1=x_2$, and we study three quadrature methods. The first method splits the square into two triangles separated by a region around the line of singularity, and applies recently developed triangle QMC rules to the two triangular parts. For functions with a singularity `no worse than $|x_1-x_2|^{-A}$ for $0<A<1$ that method yields an error of $O( (\log(n)/n)^{(1-A)/2})$. We also consider methods extending the integrand into a region containing the singularity and show that method will not improve up on using two triangles. Finally, we consider transforming the integrand to have a more QMC-friendly singularity along the boundary of the square. This then leads to error rates of $O(n^{-1+\epsilon+A})$ when combined with some corner-avoiding Halton points or with randomized QMC, but it requires some stronger assumptions on the original singular integrand.