  Quantifying the effect of uncertainties in systems where only point
evaluations in the stochastic domain but no regularity conditions are available
is limited to sampling-based techniques. This work presents an adaptive
sequential stratification estimation method that uses Latin Hypercube Sampling
within each stratum. The adaptation is achieved through a sequential
hierarchical refinement of the stratification, guided by previous estimators
using local (i.e., stratum-dependent) variability indicators based on
generalized polynomial chaos expansions and Sobol decompositions. For a given
total number of samples $N$, the corresponding hierarchically constructed
sequence of Stratified Sampling estimators combined with Latin Hypercube
sampling is adequately averaged to provide a final estimator with reduced
variance. Numerical experiments illustrate the procedure's efficiency,
indicating that it can offer a variance decay proportional to $N^{-2}$ in some
cases.
