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

Dong T. P. Nguyen, Dirk Nuyens

Published 2016-08-26Version 1

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.