  Science and engineering problems subject to uncertainty are frequently both
computationally expensive and feature nonsmooth parameter dependence, making
standard Monte Carlo too slow, and excluding efficient use of accelerated
uncertainty quantification methods relying on strict smoothness assumptions. To
remedy these challenges, we propose an adaptive stratification method suitable
for nonsmooth problems and with significantly reduced variance compared to
Monte Carlo sampling. The stratification is iteratively refined and samples are
added sequentially to satisfy an allocation criterion combining the benefits of
proportional and optimal sampling. Theoretical estimates are provided for the
expected performance and probability of failure to correctly estimate essential
statistics. We devise a practical adaptive stratification method with strata of
the same kind of geometrical shapes, cost-effective refinement satisfying a
greedy variance reduction criterion. Numerical experiments corroborate the
theoretical findings and exhibit speedups of up to three orders of magnitude
compared to standard Monte Carlo sampling.
