Multi-Level Hybrid Monte Carlo / Deterministic Methods for Particle Transport Problems

This paper presents multi-level hybrid transport (MLHT) methods for solving the neutral particle Boltzmann transport equation. The proposed MLHT methods are formulated on a sequence of spatial grids using a multi-level Monte Carlo (MLMC) approach. The general MLMC algorithm is defined by the recursive estimation of the expected value of a solution functional's correction with respect to a neighboring grid. MLMC theory optimizes the total computational cost for estimating a functional to within a target accuracy. The proposed MLHT algorithms are based on the quasidiffusion (Variable Eddington Factor) and second-moment methods. For these methods, the low-order equations for the angular moments of the high-order transport solution are discretized in space. Monte Carlo techniques compute the closures for the low-order equations; then, the equations are solved, yielding a single realization of the global flux solution. The ensemble average of the realizations yields the level solution. The results for 1-D slab transport problems demonstrates weak convergence of the functionals considered. We observe that the variance of the correction factors decreases faster than the increase in computational costs of generating an MLMC sample. In the problems considered, the variance and costs of the MLMC solution are driven by the coarse grid calculations.