Shuang Zhao

This paper introduces a new family of hybrid estimators aimed at controlling the efficiency of Monte Carlo computations in particle transport problems. In this context, efficiency is usually measured by the figure of merit (FOM) given by the inverse product of the estimator variance Var[ξ] and the run time T: FOM := (Var[ξ], T)^-1. Previously, we developed a new family of transport-constrained unbiased radiance estimators (T-CURE) that generalize the conventional collision and track length estimators and provide 1--2 orders of magnitude additional variance reduction. However, these gains in variance reduction are partly offset by increases in overhead time, lowering their computational efficiency. Here we show that combining T-CURE estimation with conventional terminal estimation within each individual biography can moderate the efficiency of the resulting "hybrid" estimator without introducing bias in the computation. This is achieved by treating only the refractive interface crossings with the extended next event estimator, and all others by standard terminal estimators. This is because when there are index-mismatched interfaces between the collision location and the detector, the T-CURE computation rapidly becomes intractable due to the large number of refractions and reflections that can arise. We illustrate the gains in efficiency by comparing our hybrid strategy with more conventional estimation methods in a series of multi-layer numerical examples.

Monte Carlo methods provide the "gold standard" computational technique for solving biomedical problems but their use is hindered by the slow convergence of the sample means. An exponential increase in the convergence rate can be obtained by adaptively modifying the sampling and weighting strategy employed. However, if the radiance is represented globally by a truncated expansion of basis functions, or locally by a region-wise constant or low degree polynomial, a bias is introduced by the truncation and/or the number of subregions. The sheer number of expansion coefficients or geometric subdivisions created by the biased representation then partly or entirely offsets the geometric acceleration of the convergence rate. As well, the (unknown amount of) bias is unacceptable for a gold standard numerical method. We introduce a new unbiased estimator of the solution of radiative transfer equation (RTE) that constrains the radiance to obey the transport equation. We provide numerical evidence of the superiority of this Transport-Constrained Unbiased Radiance Estimator (T-CURE) in various transport problems and indicate its promise for general heterogeneous problems.