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
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Computational Uncertainty Quantification: Mathematical Foundations, Methodology & Data (Thematic Programme)
Organizer(s):
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)