Multivariate integration over Rs with exponential rate of convergence

摘要

In this paper we analyze the approximation of multivariate integrals over the Euclidean space 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 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 Rs, as which were recently obtained in the unit cube by Dick, Larcher, Pillichshammer and Woźniakowski, see Dick et al. (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 Rs by taking the truncation error into account and extending them to take the anisotropy of the function space into account.