Adaptive multidimensional integration: vegas enhanced

• Published in: Journal of computational physics 2021-08, Vol.439, p.110386, Article 110386
• Main Author: Lepage, G. Peter
• Format: Article
• Language: English
• Subjects: Adaptive algorithms; Adaptive sampling; Algorithms; Bayesian statistics; Computational physics; Importance sampling; Integrators; Monte Carlo integration; Multidimensional integration; Preconditioning
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110386

•A new algorithm for adaptive multidimensional integration is proposed.•It adds a new adaptive strategy to that used by the popular VEGAS algorithm.•It can greatly increase accuracy for integrands with diagonal structure.•VEGAS can be combined with other algorithms to make new hybrid algorithms.•The new algorithm can be substantially faster than MCMC. 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.