An algorithm for estimating non-convex volumes and other integrals in n 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. They have well-bounded rates of convergence for convex domains, but in general, are suited for domains with smooth boundaries. We analyze a less known alternate method for estimating n-dimensional volumes, that is agnostic to the roughness and non-convexity of the boundaries 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 more efficient than the MCMC approach even when restricted to the typical convex domains, for n $\lesssim$ 100. An importance sampling may extend this advantage to larger n.

Appendix A. Multi-valued extent function

In case the extent function S is multi-valued (see 1b), the volume of a body, whether it is simply connected or not, is

$$\begin{aligned} V = \oint _{S^{n-1}} \left\{ \sum _{\text {odd }j}\int _{S^{j-1}(\hat{s})}^{S^j(\hat{s})} \rho ^{n-1}\textrm{d}\rho \right\} \textrm{d}\hat{s} \end{aligned}$$

(A.1)

with $j = 1, 2, 3, \dotsc$, and the above can again be reduced to the statistical estimate of the volume as

$$\begin{aligned} V = \frac{1}{n} \oint _{S^{n-1}} r^n \textrm{d}\hat{s} = \left\{ \frac{1}{n} \oint _{S^{n-1}} \textrm{d}\hat{s}\right\} \left\{ \int _0^\infty r^n f_R(r) \textrm{d}r\right\} \end{aligned}$$

(A.2)

with the random extent R now generalized as

$$\begin{aligned} R^n = \sum _j (-1)^{j+1} R_j^n \end{aligned}$$

(A.3)

where $R_1< R_2 < R_3\dots$ are the random extents representing multiple values $S^j$ for a given direction $\hat{s}$, and $S^0$=0 always. Note that the largest natural number j representing number of extents in a given direction, is always odd for a closed body defined by a bounding surface S around the origin of reference. In case, the origin is outside the closed body, the number of extents is even valued and this can be treated by a simple negation of signs in the above equation defining the generalized extents. Using bisection to determine the multiple extents in such cases may need more advanced methods that have been described elsewhere (Bernstein 1984; Kearfott 1987; Morgan and Shapiro 1987).