On the optimal order of integration in Hermite spaces with finite smoothness

TL;DR

This paper investigates the optimal order of numerical integration in Hermite spaces with finite smoothness, demonstrating that transformed digital nets can achieve near-optimal convergence rates for Gaussian integrals.

Contribution

It introduces a method to map higher order digital nets to Hermite spaces and proves they attain near-optimal convergence rates for Gaussian integration.

Findings

Transformed digital nets achieve near-optimal convergence rates.

The decay rate of the Hermite space influences the integration error.

The method applies to integrands with finite smoothness.

Abstract

We study the numerical approximation of integrals over $\mathbb{R}^s$ with respect to the standard Gaussian measure for integrands which lie in certain Hermite spaces of functions. The decay rate of the associated sequence is specified by a single integer parameter which determines the smoothness classes and the inner product can be expressed via $L_2$ norms of the derivatives of the function. We map higher order digital nets from the unit cube to a suitable subcube of $\mathbb{R}^s$ via a linear transformation and show that such rules achieve, apart from powers of $\log N$, the optimal rate of convergence of the integration error.