Optimal numerical integration and approximation of functions on ℝ d equipped with Gaussian measure

• Published in: IMA journal of numerical analysis 2024-04, Vol.44 (2), p.1242-1267
• Main Authors: Dũng, Dinh, Kien Nguyen, Van
• Format: Article
• Language: English
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/drad051

We investigate the numerical approximation of integrals over $\mathbb{R}^{d}$ equipped with the standard Gaussian measure $\gamma $ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma )$ of mixed smoothness $\alpha \in \mathbb{N}$ for $1 < p < \infty $. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling $n$-widths in the Gaussian-weighted space $L_{q}(\mathbb{R}^{d}, \gamma )$ of the unit ball of $W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma )$ for $1 \leq q < p < \infty $ and $q=p=2$.