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

Arun I.; Murugesan Venkatapathi

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arXiv · 2007.06808

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

Arun I.·Murugesan Venkatapathi

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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.

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Arun I., Murugesan Venkatapathi. 2020-07-14. An algorithm for estimating volumes and other integrals in $n$ dimensions. https://arxiv.org/abs/2007.06808

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