DOI: 10.7274/25607787.v1
title: Development of the Iterative Quasi-Monte Carlo (iQMC) Method for Neutron Transport Simulations

description: This work presents the development and evaluation of the iterative-Quasi Monte Carlo (iQMC) method for multigroup neutron transport simulations. iQMC can be viewed as a hybrid between deterministic iterative techniques, Monte Carlo simulation, and Quasi-Monte Carlo techniques. iQMC replaces standard quadrature techniques used in deterministic linear solvers with Quasi-Monte Carlo simulation to obtain accurate and efficient solutions to the neutron transport equation (NTE). Quasi-Monte Carlo (QMC) is the use of low-discrepancy sequences to sample the phase space in place of pseudo-random number generators used by traditional Monte Carlo (MC). QMC techniques decrease the variance in the stochastic transport sweep and therefore increase the accuracy of the iterative method. 

 iQMC holds several algorithmic characteristics that make it desirable for high-performance computing environments including a 1/N convergence scheme, ray tracing transport sweep, multigroup vectorization, and particle parallelization similar to analog MC. These characteristics imply that its performance may benefit greatly from implementation on modern exascale GPU architectures. To investigate iQMC as a potential "next-generation" neutron transport solver iQMC was implemented in the Monte Carlo Dynamic Code to solve steady-state fixed source, time-dependent, and k-eigenvalue problems. Several formulations of iQMC were investigated including a fixed seed and batched approach. In the fixed seed approach particles are reset to their initial position and angular direction of travel at the beginning of each transport sweep. This allows for the use of advanced linear Krylov solvers which converge with far fewer transport sweeps than the typical source iteration or nested power iteration. For the batched approach, Owen randomization is used to randomly perturb the samples of the low-discrepancy sequence before each transport sweep. This scheme necessitates the use of an inactive and active batch approach similar to analog MC k-eigenvalue iterations.
 
 Various results are presented from 1D slab problems to the extended 3D C5G7. The batched approach along with a flattened power iteration, is shown to far outperform the fixed seed approach, requiring an order of magnitude fewer particles for equivalent results. Additionally, a linear discontinuous source tilting scheme was developed which significantly reduces spatial error associated with the use of a uniform Cartesian mesh. Using the batched approach and the linear source tilting method, iQMC was shown to outperform multigroup MC in regions of low flux. iQMC performance was also assessed with several performance benchmarks including strong and weak parallel scaling and initial comparisons to multigroup Monte Carlo. iQMC shows excellent weak scaling for the problems tested and good strong scaling when adjusted for the LLVM Compilation time. This work has identified core components of iQMC and key issues to be addressed. As a result, several avenues for future research and development have emerged from this study.