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

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 is’ 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 upon 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+𝜖+A) when combined with some corner-avoiding Halton points or with randomized QMC but it requires some stronger assumptions on the original singular integrand.

Dedicated to Ian H. Sloan on the occasion of his 80th birthday.