# Modern Monte Carlo Variants for Uncertainty Quantification in Neutron Transport

## TL;DR

This paper advances Monte Carlo methods for uncertainty quantification in neutron transport, demonstrating significant computational gains through hybrid solvers and multilevel quasi-Monte Carlo techniques in complex stochastic settings.

## Contribution

It introduces novel theoretical convergence results and practical algorithms for UQ in neutron transport with low-regularity random fields, including hybrid solvers and multilevel quasi-Monte Carlo methods.

## Key findings

- Multilevel quasi-Monte Carlo reduces computational cost by up to 100 times.
- Hybrid iterative/direct solver improves efficiency for each realization.
- Numerical experiments confirm theoretical gains in high-dimensional stochastic problems.

## Abstract

We describe modern variants of Monte Carlo methods for Uncertainty Quantification (UQ) of the Neutron Transport Equation, when it is approximated by the discrete ordinates method with diamond differencing. We focus on the mono-energetic 1D slab geometry problem, with isotropic scattering, where the cross-sections are log-normal correlated random fields of possibly low regularity. The paper includes an outline of novel theoretical results on the convergence of the discrete scheme, in the cases of both spatially variable and random cross-sections. We also describe the theory and practice of algorithms for quantifying the uncertainty of a linear functional of the scalar flux, using Monte Carlo and quasi-Monte Carlo methods, and their multilevel variants. A hybrid iterative/direct solver for computing each realisation of the functional is also presented. Numerical experiments show the effectiveness of the hybrid solver and the gains that are possible through quasi-Monte Carlo sampling and multilevel variance reduction. For the multilevel quasi-Monte Carlo method, we observe gains in the computational $\varepsilon$-cost of up to 2 orders of magnitude over the standard Monte Carlo method, and we explain this theoretically. Experiments on problems with up to several thousand stochastic dimensions are included.