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

TL;DR

The paper tackles time-dependent neutron transport by a scatter-limited hybrid that couples Monte Carlo (MC) with a deterministic discrete-ordinates solver ($S_N$) and augments MC with quasi-Monte Carlo (QMC) sampling. The method introduces a tunable scatter cap $N_s$ that limits MC collisions per step, with a relabeling step turning excess high-scatter flux back into MC and a DG discretization for the $S_N$-solved collided flux; QMC replaces pseudorandom draws to improve convergence at modest code changes. Numerical tests on Reed's problem and the Dogleg benchmark show notable improvements in $L^2$ accuracy and convergence rate with negligible additional cost, and the $N_s$-scatter strategy offers load-balancing and parallelization opportunities. The work preserves the diffusion limit and suggests extensions to adaptive, problem-driven $N_s$ schedules and more scalable, fully parallel QMC legs.

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.

Figures (5)

• Figure 1: Summary of numerical experiments for Reed's problem (top row) and the Dogleg problem (bottom row). Depicted are the $L^2$--error as depending on the particle count $N_p$(left column), the runtime as depending on $N_p$ (middle column) and $L^2$--error depending on the runtime(right column) as a measure of efficiency. Runs that appear closer to the bottom left provide higher accuracy per runtime. We compare QMC runs(orange) to MC runs(purple) for various values of the scatter limit $N_s\in \{0,5,10,20\}$ as well as $N_s = \infty$(non-hybrid/regular (Q)MC run).
• Figure 2: Material parameters and geometry for Reed's problem
• Figure 3: Sample outputs of $\langle \Psi \rangle$ for Reed's problem with $N_p = 32000$(top row) and $N_p = 64000$(bottom row). We compare results to a reference (dashed) for QMC--hybrid runs(left), MC--hybrid runs(middle) and non-hybrid runs(right). The scattering limit $N_s$ is set to zero for hybrid runs and $\infty$ for non-hybrid runs.
• Figure 4: Material parameters and geometry for the Dogleg problem (rotated by $90^\circ$)
• Figure 5: Sample outputs of $\langle \Psi \rangle$ for the Dogleg problem with $N_p = 32000$(top row) and $N_p = 64000$(bottom row). We compare results for QMC--hybrid runs(left), MC--hybrid runs(middle) and non-hybrid runs(right). The scattering limit $N_s$ is set to zero for hybrid runs and $\infty$ for non-hybrid runs.