The computational cost in evaluation of the volume of a body using numerical integration grows exponentially with dimension of the space n. Today's most generally applicable algorithms for estimating n-volumes and integrals are based on Markov Chain Monte Carlo (MCMC) methods ; they are suited for convex domains. We present an alternate statistical method for estimating an n-dimensional integral that is suited even for non-convex domains and scales as O(n) in the samples required, given any distribution of extents of the arbitrary domain and the allowed tolerance in relative error. This results due to the possible decomposition of an arbitrary n-volume into an integral of statistically weighted volumes of n-spheres  .
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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