Multivariate integration for analytic functions with Gaussian kernels

We study multivariate integration of analytic functions defined on Rd. These functions are assumed to belong to a reproducing kernel Hilbert space whose kernel is Gaussian, with nonincreasing shape parameters. We prove that a tensor product algorithm based on the univariate Gauss-Hermite quadrature rules enjoys exponential convergence and computes an ε-approximation for the d-variate integration using an order of (ln ε−1)d function values as ε goes to zero. We prove that the exponent d is sharp by proving a lower bound on the minimal (worst case) error of any algorithm based on finitely many function values. We also consider four notions of tractability describing how the minimal number n(ε, d) of function values needed to find an ε-approximation in the d-variate case behaves as a function of d and ln ε−1. One of these notions is new. In particular, we prove that for all positive shape parameters, the minimal number n(ε, d) is larger than any polynomial in d and ln ε−1 as d and ε−1 go to infinity. However, it is not exponential in d t and ln ε−1 whenever t > 1.