Multivariate integration over $\R^s$ with exponential rate of convergence

TL;DR

This paper demonstrates that multivariate integrals over 0^s of analytic functions can be approximated with exponential convergence rates using regular grid sampling, balancing truncation and discretization errors.

Contribution

It extends exponential convergence results for multivariate integration to 0^s, optimizing mesh sizes and truncation to improve approximation accuracy.

Findings

Achieves exponential convergence rates for multivariate integrals over 0^s.

Provides explicit upper bounds for approximation errors.

Extends previous results to higher dimensions and anisotropic function spaces.

Abstract

In this paper we analyze the approximation of multivariate integrals over the Euclidean plane for functions which are analytic. We show explicit upper bounds which attain the exponential rate of convergence. We use an infinite grid with different mesh sizes and lengths in each direction to sample the function, and then truncate it. In our analysis, the mesh sizes and the truncated domain are chosen by optimally balancing the truncation error and the discretization error. This paper derives results in comparable function space settings, extended to $\R^s$, as which were recently obtained in the unit cube by Dick, Larcher, Pillichshammer and Wo{\'z}niakowski (2011). They showed that both lattice rules and regular grids, with different mesh sizes in each direction, attain exponential rates, hence motivating us to analyze only cubature formula based on regular meshes. We further also amend the analysis of older publications, e.g., Sloan and Osborn (1987) and Sugihara (1987), using lattice rules on $\R^s$ by taking the truncation error into account and extending them to take the anisotropy of the function space into account.