Higher-Order Monte Carlo through Cubic Stratification

SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 229-247, February 2024.
Abstract. We propose two novel unbiased estimators of the integral [math] for a function [math], which depend on a smoothness parameter [math]. The first estimator integrates exactly the polynomials of degrees [math] and achieves the optimal error [math] (where [math] is the number of evaluations of [math]) when [math] is [math] times continuously differentiable. The second estimator is also optimal in terms of convergence rate and has the advantage of being computationally cheaper, but it is restricted to functions that vanish on the boundary of [math]. The construction of the two estimators relies on a combination of cubic stratification and control variates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of [math].