Quasi-Monte Carlo integration for twice differentiable functions over a triangle
This work advances numerical integration over triangles, a fundamental domain in computational geometry and finite element methods, by providing optimal error bounds for twice differentiable functions.
The paper provides an explicit construction of infinite sequences of points for quasi-Monte Carlo integration over a triangle, achieving an integration error of order N^{-1}(log N)^3, which is optimal up to a logarithmic factor. The result is proven via dyadic Walsh analysis under recursive partitioning.
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.