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

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≥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.