论文标题
用于估计卷和其他积分$ n $ dimensions的算法
An algorithm for estimating volumes and other integrals in $n$ dimensions
论文作者
论文摘要
使用数值集成评估身体体积的计算成本随着空间$ n $的尺寸而成倍增长。估计$ n $ wolumes和积分的最普遍适用的算法基于马尔可夫链蒙特卡洛（MCMC）方法，它们适用于凸域。我们分析了一种鲜为人知的替代方法，用于估计$ n $维的体积，这对人体的凸和粗糙度不可知。由于可能将任意$ n $ - 体积分解成$ n $ spheres的统计加权量的积分，因此结果可能会导致。我们建立了其维度缩放，并将其扩展以评估非凸面域上的任意积分。我们的结果还表明，即使仅限于凸域，此方法也比MCMC方法要高得多，因为$ n $ $ \ sim <$ <$ 100。重要的采样可能将此优势扩展到更大的维度。
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.
