  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.
