A polynomial chaos approach for uncertainty quantification of Monte Carlo transport codes

TL;DR

The paper tackles uncertainty quantification for Monte Carlo radiation transport by building non-intrusive Polynomial Chaos surrogates that account for solver randomness and re-sampling costs. It develops a cost-aware estimator framework, introduces an unbiased variance estimator, and proposes PC expansion trimming plus a PC-variability analysis to quantify surrogate uncertainty. Numerical results on an attenuation test case show that bias correction and expansion trim significantly improve variance estimates and suppress spurious high-order terms, especially when MC sampling is under-resolved; global sensitivity analysis is demonstrated via Sobol' indices derived from PC coefficients. These advances enable efficient, reliable UQ for MC RT codes, with practical impact in designing budgets for surrogate-based analyses and guiding future multi-fidelity extensions.

Abstract

In this contribution, we discuss the construction of Polynomial Chaos surrogates for Monte Carlo radiation transport applications via non-intrusive spectral projection. This contribution focuses on improvements with respect to the approach that we previously introduced in previous work. We focus on understanding the impact of re-sampling cost on the algorithm performance and provide algorithm refinements, which allow to obtain unbiased estimators for the variance, estimate the PC variability due to limited samples, and adapt the expansion. An attenuation-only test case is provided to illustrate and discuss the results.

Figures (5)

• Figure 1: Three PC repetitions for the 1D attenuation problem (dashed red) with (bottom) and without (top) the expansion trim. All the results are obtained with $N_{\xi}=2000$ and $N_{\xi}=50$. The exact attenuation profile is reported in black.
• Figure 2: MSE for the estimated variance obtained with $1\,500$ independent repetitions with an increasing number of UQ samples $N_{\xi}$ and $N_{\eta}=\left[ 1, 2, 10, 50, 100\right]$.
• Figure 3: MSE for the estimated variance obtained with $1\,500$ independent repetitions with an increasing number of particles $N_{\eta}$ (per UQ sample) and $N_{\xi}=\left[ 25, 50, 100, 500, 1\,000, 2\,000\right]$.
• Figure 4: Probability density functions for the estimated variance with PCE and variance deconvolution.
• Figure 5: Sensitivity index $S_1$ (\ref{['fig:S1']}) and total sensitivity index $S_{T_1}$ (\ref{['fig:TS1']}) obtained with the PC with bias correction and bias correction and expansion trim.