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arXiv · 1701.08562

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

Takashi Goda·Kosuke Suzuki·Takehito Yoshiki
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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.

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Takashi Goda, Kosuke Suzuki, Takehito Yoshiki. 2017-01-30. Quasi-Monte Carlo integration for twice differentiable functions over a triangle. https://doi.org/10.1016/j.jmaa.2017.04.051

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