Scatter-Limited Hybrid Monte Carlo, Deterministic Transport with Quasi-Monte Carlo Sampling

Abstract: We present a hybrid method for time-dependent particle transport that combines Monte Carlo (MC) estimation with a deterministic discrete ordinates ((S_N)) solve, augmented by quasi-Monte Carlo (QMC) sampling. For spatial discretizations, the MC component computes a piecewise-constant (cell-averaged) solution, while the (S_N) stage employs bilinear discontinuous finite elements. By hybridizing the formulation, the MC subproblem after a prescribed scatter limit becomes scattering-free, yielding a simple and efficient streaming/attenuation procedure. Between time steps, a simple scatter-free MC step is run to relabel the $S_N$ solution as an MC solution. A key feature of the approach is a tunable parameter (N_{s}) that controls how many material collisions are handled in the (Q)MC leg before handing off to the deterministic (S_N) solve; (N_s=0) recovers a purely uncollided MC leg, while (N_s>0) produces multi-scatter hybrids. QMC replaces pseudorandom draws with low-discrepancy points in the existing MC sampling maps, enabling a plug-in adoption within the standard MC code with modest, localized changes. We observe significant accuracy and convergence rate improvements through the use of QMC and practically no additional computational cost, which are generally not seen in comparable non-hybrid solves. We believe the multi-scatter approach provides additional flexibility in terms of parallelization and the choice of deterministic solver.