Adaptive Multidimensional Integration: VEGAS Enhanced

TL;DR

VEGAS+ is an advanced adaptive multidimensional Monte Carlo integration algorithm that improves accuracy and efficiency over VEGAS, especially for complex integrands with multiple peaks or structures, and can be combined with other methods for further gains.

Contribution

The paper introduces VEGAS+, a new algorithm that adds adaptive stratified sampling to VEGAS, enhancing performance for complex integrands and enabling hybrid and preconditioned approaches.

Findings

VEGAS+ achieves 2-19 times more accuracy than VEGAS.

Preconditioned VEGAS+ is over 100 times more efficient than non-preconditioned.

VEGAS+ outperforms MCMC for Bayesian integrals with 3 to 21 parameters.

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 times 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 times as efficient as VEGAS+ without preconditio ing. Finally, we give examples where VEGAS+ is more than 10 times 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.