A Universal Algorithm for Multivariate Integration

TL;DR

This paper introduces a universal, unbiased algorithm for multivariate integration over cubes that achieves optimal convergence rates across various Sobolev spaces, applicable in both randomized and worst-case scenarios.

Contribution

The paper proposes a new algorithm that is universally applicable and optimal for multivariate integration in multiple Sobolev spaces, improving upon existing methods.

Findings

Achieves optimal convergence rates in Sobolev spaces

Works in both randomized and worst-case settings

Unbiased algorithm for multivariate integration

Abstract

We present an algorithm for multivariate integration over cubes that is unbiased and has optimal order of convergence (in the randomized sense as well as in the worst case setting) for all Sobolev spaces $H^{r, mix}([0,1]^d)$ and $H^s([0,1]^d)$ for $s>d/2$.