Sub-optimality of Gauss--Hermite quadrature and optimality of trapezoidal rule for functions with finite smoothness

Yoshihito Kazashi (U Heidelberg)

May 12. 2022, 15:30 — 16:00

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).

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Computational Uncertainty Quantification: Mathematical Foundations, Methodology & Data (Thematic Programme)
Clemens Heitzinger (TU Vienna)
Fabio Nobile (EPFL Lausanne)
Robert Scheichl (U Heidelberg)
Christoph Schwab (ETH Zürich)
Sara van de Geer (ETH Zürich)
Karen Willcox (U of Texas, Austin)