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.

翻译：使用数字整合体体积的评估计算成本随着空间的大小美元而成倍增长。估算美元体积和元件的最普遍适用的算法基于Markov Caincle Monte Carlo(MCMCC) 方法,并适合 convex 域。我们分析了一种不太为人所知的用于估算美元体积的代用方法,这种方法对于体体的共性和粗糙性是不可知的。其结果是,任意的美元体积可能分解成统计加权体积($-sphees)的有机体。我们建立了其维度缩放,并将其用于评估非convex 域的任意整体体积。我们的结果还表明,这种方法比MCMC方法效率要高得多,即使限于Convex 域,也只有$sim < 100.美元。重要的取样可能将这一优势扩大到更大的维度。