# Quasi-Monte Carlo integration for twice differentiable functions over a triangle

## TL;DR

This paper develops a quasi-Monte Carlo method for integrating twice differentiable functions over a triangle, achieving near-optimal error bounds with explicit point sequence constructions.

## Contribution

It provides an explicit construction of point sequences for QMC integration over a triangle with near-optimal error bounds, extending previous methods.

## Key findings

- Achieves integration error of order N^{-1} (log N)^3
- Includes a construction by Basu and Owen (2015) as a special case
- Proves the bounds are nearly optimal given known lower bounds

## Abstract

We study quasi-Monte Carlo integration for twice differentiable functions defined over a triangle. We provide an explicit construction of infinite sequences of points including one by Basu and Owen (2015) as a special case, which achieves the integration error of order $N^{-1}(\log N)^3$ for any $N\geq 2$. Since a lower bound of order $N^{-1}$ on the integration error holds for any linear quadrature rule, the upper bound we obtain is best possible apart from the $\log N$ factor. The major ingredient in our proof of the upper bound is the dyadic Walsh analysis of twice differentiable functions over a triangle under a suitable recursive partitioning.