We describe a new algorithm, vegas+, for adaptive multidimensional Monte Carlo integration. The new algorithm adds a second adaptive strategy, adaptive stratified sampling, to the adaptive importance sampling that is the basis for its widely used predecessor vegas. Both vegas and vegas+ are effective for integrands with large peaks, but vegas+ can be much more effective for integrands with multiple peaks or other significant structures aligned with diagonals of the integration volume. We give examples where vegas+ is 2–19× more accurate than vegas. We also show how to combine vegas+ with other integrators, such as the widely available miser algorithm, to make new hybrid integrators. For a different kind of hybrid, we show how to use integrand samples, generated using MCMC or other methods, to optimize vegas+ before integrating. We give an example where preconditioned vegas+ is more than 100× as efficient as vegas+ without preconditioning. Finally, we give examples where vegas+ is more than 10× as efficient as MCMC for Bayesian integrals with $D=3$ and 21 parameters. We explain why vegas+ will often outperform MCMC for small and moderate sized problems.

我们描述了一种新的算法vegas +，用于自适应多维蒙特卡洛积分。新算法为自适应重要性采样添加了第二种自适应策略，即自适应分层采样，这是其广泛使用的前任维加斯的基础。无论拉斯维加斯和拉斯维加斯+是有效的与大峰积，但拉斯维加斯+可与多峰或与集成音量的对角线排列的其他显著结构积有效得多。我们举了一些例子，其中vegas +比vegas精确2-19倍。我们还展示了如何结合vegas +与其他集成商（例如广泛使用的miser算法）共同创建新的混合集成商。对于不同种类的杂种，我们展示了如何使用通过MCMC或其他方法生成的被混物样本在整合之前优化vegas +。我们举一个例子，预处理过的vegas +的效率是未预处理过的vegas +的100倍以上。最后，我们给出了以下示例：vegas +的贝叶斯积分效率是MCMC的10倍以上，$d=3$和21个参数。我们解释了为什么vegas +在中小规模的问题上通常会胜过MCMC。