### 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 crosssections. We also describe the theory and practice of algorithms for quantifying the uncertainty of a 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 ε- cost of up to two 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.

Original language | English |
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Title of host publication | Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan |

Publisher | Springer International Publishing |

Pages | 455-481 |

Number of pages | 27 |

ISBN (Electronic) | 9783319724560 |

ISBN (Print) | 9783319724553 |

DOIs | |

Publication status | E-pub ahead of print - 23 May 2018 |

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan*(pp. 455-481). Springer International Publishing. https://doi.org/10.1007/978-3-319-72456-0_22

**Modern Monte Carlo variants for uncertainty quantification in neutron transport.** / Graham, Ivan G.; Parkinson, Matthew J.; Scheichl, Robert.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan.*Springer International Publishing, pp. 455-481. https://doi.org/10.1007/978-3-319-72456-0_22

}

TY - CHAP

T1 - Modern Monte Carlo variants for uncertainty quantification in neutron transport

AU - Graham, Ivan G.

AU - Parkinson, Matthew J.

AU - Scheichl, Robert

PY - 2018/5/23

Y1 - 2018/5/23

N2 - 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 crosssections. We also describe the theory and practice of algorithms for quantifying the uncertainty of a 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 ε- cost of up to two 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.

AB - 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 crosssections. We also describe the theory and practice of algorithms for quantifying the uncertainty of a 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 ε- cost of up to two 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.

UR - http://www.scopus.com/inward/record.url?scp=85049875184&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-72456-0_22

DO - 10.1007/978-3-319-72456-0_22

M3 - Chapter

SN - 9783319724553

SP - 455

EP - 481

BT - Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

PB - Springer International Publishing

ER -