Kernel density estimation method for Monte Carlo point detector and surface crossing flux tallies

Monte Carlo point detector and surface crossing flux tallies are two widely used tallies, but they suffer from an unbounded variance. As a result, the central limit theorem cannot be used for these tallies to estimate confidence intervals. By construction, kernel density estimator (KDE) tallies can be directly used to estimate flux at a point, but the variance of this point estimate does not converge as 1/N, which is not unexpected for a point quantity. However, an improved approach is to modify both point detector and surface crossing flux tallies directly by using KDE within a variance reduction approach and taking advantage of the fact that KDE estimates the underlying probability density function. This methodology is illustrated by several numerical examples and shows numerically that both the surface crossing tally and the point detector tally converge as 1/N (in variance), and both are asymptotically unbiased.