An algorithm for estimating volumes and other integrals in $n$ dimensions

TL;DR

This paper introduces a less known, efficient algorithm for estimating volumes and integrals in high-dimensional spaces, which is applicable to both convex and non-convex domains and outperforms MCMC methods for dimensions below 100.

Contribution

It analyzes and extends an alternative volume estimation method based on decomposing n-volumes into integrals of weighted n-spheres, demonstrating its efficiency over MCMC in certain high dimensions.

Findings

The method scales favorably with dimension up to about 100.

It is more efficient than MCMC for convex domains in high dimensions.

Importance sampling can further extend its effectiveness to larger dimensions.

Abstract

The computational cost in evaluation of the volume of a body using numerical integration grows exponentially with dimension of the space $n$. The most generally applicable algorithms for estimating $n$-volumes and integrals are based on Markov Chain Monte Carlo (MCMC) methods, and they are suited for convex domains. We analyze a less known alternate method used for estimating $n$-dimensional volumes, that is agnostic to the convexity and roughness of the body. It results due to the possible decomposition of an arbitrary $n$-volume into an integral of statistically weighted volumes of $n$-spheres. We establish its dimensional scaling, and extend it for evaluation of arbitrary integrals over non-convex domains. Our results also show that this method is significantly more efficient than the MCMC approach even when restricted to convex domains, for $n$ $\sim <$ 100. An importance sampling may extend this advantage to larger dimensions.