A Monte Carlo method for integration of multivariate smooth functions

TL;DR

This paper introduces a Monte Carlo integration method using a lattice point set, achieving optimal convergence rates for smooth functions in various Sobolev spaces, with error bounds independent of dimension.

Contribution

It establishes a new Monte Carlo algorithm with optimal convergence rates for multivariate smooth functions, surpassing deterministic methods in high dimensions.

Findings

Error bound of $n^{-1/2}$ times Fourier transform outside a region

Optimal convergence order $n^{-s-1/2}$ for Sobolev spaces

Dimension-independent convergence rates for functions on the unit cube

Abstract

We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by $n^{-1/2}$ times the $L_2$-norm of the Fourier transform outside a region around the origin, where $n$ is the expected number of function evaluations. As corollaries we obtain the optimal order of convergence for functions from the Sobolev spaces $H^s_p$ with isotropic, anisotropic, or mixed smoothness with given compact support for all values of the parameters. If the region of integration is the unit cube, we obtain the same optimal orders for functions without boundary conditions. This proves, in particular, that the optimal order of convergence in the latter case is $n^{-s-1/2}$ for $p\ge2$, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than $n^{-1/2}$ for a whole class of natural function spaces.