A Quasi-Monte Carlo Method with Krylov Linear Solvers for Multigroup Neutron Transport Simulations

TL;DR

This paper introduces a hybrid Quasi-Monte Carlo and Krylov linear solver approach to improve the accuracy and efficiency of deterministic neutron transport simulations, demonstrating faster convergence than traditional methods.

Contribution

It presents a novel hybrid iterative-QMC method using Krylov solvers for neutron transport, achieving higher accuracy and fewer iterations compared to standard techniques.

Findings

Krylov solvers converge up to 8x faster than Source Iteration.

Hybrid method achieves O(N^{-1}) convergence rate, outperforming traditional MC.

QMC reduces variance, enhancing solution accuracy in neutron transport simulations.

Abstract

In this work we investigate replacing standard quadrature techniques used in deterministic linear solvers with a fixed-seed Quasi-Monte Carlo calculation to obtain more 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. Historically, QMC has largely been ignored by the particle transport community because it breaks the Markovian assumption needed to model scattering in analog MC particle simulations. However, by using iterative methods the NTE can be modeled as a pure-absorption problem. This removes the need to explicitly model particle scattering and provides an application well-suited for QMC. To obtain solutions we experimented with three separate iterative solvers: the standard Source Iteration (SI) and two linear Krylov Solvers, GMRES and BiCGSTAB. The resulting hybrid iterative-QMC (iQMC) solver was assessed on three one-dimensional slab geometry problems. In each sample problem the Krylov Solvers achieve convergence with far fewer iterations (up to 8x) than the Source Iteration. Regardless of the linear solver used, the hybrid method achieved an approximate convergence rate of $O(N^{-1})$, as compared to the expected $O(N^{-1/2})$ of traditional MC simulation, across all test problems.