Adaptive multidimensional integration: vegas enhanced

Abstract

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.