Adaptive multidimensional integration: VEGAS enhanced

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 2-19x 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 100x as efficient as VEGAS+ without preconditioning. Finally, we give examples where VEGAS+ is more than 10x 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The VEGAS+ algorithm is more accurate than VEGAS in certain cases, especially for integrands with multiple peaks or structures aligned with diagonals of the integration volume. It can be combined with other integrators for better performance.