We tackle the problem of efficiently approximating the volume of convex polytopes, when these are given in three different representations: H-polytopes, which have been studied extensively, V-polytopes, and zonotopes (Z-polytopes). We design a novel practical Multiphase Monte Carlo algorithm that leverages random walks based on billiard trajectories, as well as a new empirical convergence tests and a simulated annealing schedule of adaptive convex bodies. After tuning several parameters of our proposed method, we present a detailed experimental evaluation of our tuned algorithm using a rich dataset containing Birkhoff polytopes and polytopes from structural biology. Our open-source implementation tackles problems that have been intractable so far, offering the first software to scale up in thousands of dimensions for H-polytopes and in the hundreds for V- and Z-polytopes on moderate hardware. Last, we illustrate our software in evaluating Z-polytope approximations.

翻译：我们解决了在凸多面体以三种不同表示形式给出时高效逼近其体积的问题：已被广泛研究的H-多面体、V-多面体以及Zonotopes（Z-多面体）。我们设计了一种新颖实用的多阶段蒙特卡洛算法，该算法利用基于台球轨迹的随机游走、新的经验收敛检验以及自适应凸体的模拟退火调度。在调整了所提出方法的若干参数后，我们使用包含Birkhoff多面体和结构生物学多面体的丰富数据集，对调优后的算法进行了详细的实验评估。我们的开源实现解决了迄今难以处理的问题，首次提供了能在中等硬件上扩展至数千维度的H-多面体以及数百维度的V-和Z-多面体的软件。最后，我们通过评估Z-多面体近似展示了该软件的应用。