  Monte Carlo (MC) techniques are often used to estimate integrals of a
multivariate function using randomly generated samples of the function. In
light of the increasing interest in uncertainty quantification and robust
design applications in aerospace engineering, the calculation of expected
values of such functions (e.g. performance measures) becomes important.
However, MC techniques often suffer from high variance and slow convergence as
the number of samples increases. In this paper we present Stacked Monte Carlo
(StackMC), a new method for post-processing an existing set of MC samples to
improve the associated integral estimate. StackMC is based on the supervised
learning techniques of fitting functions and cross validation. It should reduce
the variance of any type of Monte Carlo integral estimate (simple sampling,
importance sampling, quasi-Monte Carlo, MCMC, etc.) without adding bias. We
report on an extensive set of experiments confirming that the StackMC estimate
of an integral is more accurate than both the associated unprocessed Monte
Carlo estimate and an estimate based on a functional fit to the MC samples.
These experiments run over a wide variety of integration spaces, numbers of
sample points, dimensions, and fitting functions. In particular, we apply
StackMC in estimating the expected value of the fuel burn metric of future
commercial aircraft and in estimating sonic boom loudness measures. We compare
the efficiency of StackMC with that of more standard methods and show that for
negligible additional computational cost significant increases in accuracy are
gained.
