Adaptive Three-Dimensional Monte Carlo Functional-Expansion Tallies

We describe a method that enables Monte Carlo calculations to automatically achieve a user-prescribed error of representation for numerical results. Our approach is to iteratively adapt Monte Carlo functional-expansion tallies (FETs). The adaptivity is based on assessing the cellwise 2-norm of error due to both functional-expansion truncation and statistical uncertainty. These error metrics have been detailed by others for one-dimensional distributions. We extend their previous work to three-dimensional distributions and demonstrate the use of these error metrics for adaptivity. The method examines Monte Carlo FET results, estimates truncation and uncertainty error, and suggests a minimum-required expansion order and run time to achieve the desired level of error. Iteration is required for results to converge to the desired error. Our implementation of adaptive FETs is observed to converge to reasonable levels of desired error for the representation of four distributions. In practice, some distributions and desired error levels may require prohibitively large expansion orders and/or Monte Carlo run times.