In this talk, I will be talking about numerical integration with respect to the Gaussian measure. Our focus is on the fundamental, one-dimensional case.
The methods of interest are Gauss--Hermite quadrature and the trapezoidal rule. It turns out Gauss–Hermite quadrature attains merely of the order $n^{−\alpha/2}$ with $n$ function evaluations, in the sense of worst-case error in the weighted Sobolev spaces of square-integrable functions of order $\alpha\in\mathbb{N}$. In contrast, we show that a suitably truncated trapezoidal rule achieves the optimal rate $n^{−\alpha}$, up to a logarithmic factor.
This is jont work with Yuya Suzuki (NTNU, Norway) and Takashi Goda (University of Tokyo, Japan).