Using Supervised Learning to Improve Monte Carlo Integral Estimation

Monte Carlo techniques are used to estimate the integrals of a function using randomly generated samples. The interest in uncertainty quantification and robust design makes calculating the expected values of such functions (e.g., performance measures) important. Recent developments in scramjets, aircraft technology forecasting, structural reliability, and robust low-boom aircraft designs use Monte Carlo techniques to ensure the appropriate quantification of uncertainties. Because of high variance and slow convergence, Monte Carlo techniques require a large number of function evaluations, limiting the fidelity of the tools that can be used to predict performance. Stacked Monte Carlo is presented, which is a new method for postprocessing an existing set of Monte Carlo samples to improve integral estimation. Stacked Monte Carlo is based on combining fitting functions with cross-validation and should reduce the variance of any type of Monte Carlo integral estimate (importance sampling, quasi-Monte Carlo, etc.) without adding bias. An extensive set of experiments is reported, confirming that the stacked Monte Carlo estimate is more accurate than both the unprocessed Monte Carlo estimate and the estimate from a functional fit. Stacked Monte Carlo is applied to estimate the fuel-burn metrics of future commercial aircraft and sonic boom loudness measures, and the efficiency of Monte Carlo is compared with that of more standard methods. It is shown that for negligible, additional, computational cost, significant increases in accuracy are gained.