Randomizing the trapezoidal rule gives the optimal RMSE rate in Gaussian Sobolev spaces

Abstract

Randomized quadratures for integrating functions in Sobolev spaces of order α ≥ 1 \alpha \ge 1 , where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of O ( n − α − 1 / 2 ) O(n^{-\alpha -1/2}) is proven, where n n denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory.