Numerical Analysis of the Monte-Carlo Noise for the Resolution of the Deterministic and Uncertain Linear Boltzmann Equation (Comparison of Non-Intrusive gPC and MC-gPC)

Abstract

Monte Carlo-generalized Polynomial Chaos (MC-gPC) has already been thoroughly studied in the literature. MC-gPC both builds a gPC based reduced model of a partial differential equation (PDE) of interest and solves it with an intrusive MC scheme in order to propagate uncertainties. This reduced model captures the behavior of the solution of a set of PDEs subject to some uncertain parameters modeled by random variables. MC-gPC is an intrusive method, it needs modifications of a code in order to be applied. This may be considered a drawback. But, on another hand, important computational gains obtained with MC-gPC have been observed on many applications. The MC-gPC resolution of Boltzmann equation has been investigated in many different ways: the wellposedness of the gPC based reduced model has been proved, the convergence with respect to the truncation order P has been theoretically and numerically studied and the coupling to nonlinear physics has been performed. But the study of the MC noise remains, to our knowledge, to be done. This is the purpose of this paper. We are interested in understanding what can be expected in terms of error estimations with respect to N_MC, the number of MC particles. For this, we estimate the variances of non-intrusive gPC and MC-gPC, theoretically and numerically, and compare them in several configurations for several MC schemes (the semi-analog and the non-analog ones). The results show that the MC schemes of the literature used to solve MC-gPC present an excess of variance with respect to the non-intrusive strategies for comparable particle numbers N_MC (even if this excess of variance remains acceptable and competitive in many situations).

KEYWORDS:

• Monte Carlo
• generalized polynomial chaos
• uncertainty quantification
• transport

Notes

1 It is always possible to come back to such framework, at the cost of more or less tedious pretreatments leading to a controled approximation (Todor and Schwab 2006; Meyer and Matthies 2004; Mercer 1909) and decorrelation (Lebrun and Dutfoy 2009a, 2009b).

2 geometrical, in the cross-sections, in the multiplicity, in the boundary conditions etc.

3 MC-gPC being based on gPC which is sensitive to the curse of dimension, the $P-$truncated reduced models remains exponentially sensitive to P and Q, see (Poëtte 2019a, 2020).

4 The HPC strategy we have in mind is commonly called domain replication, see (Dureau and Poëtte 2013; Martin et al. 1987; Majumdar 2000). It consists in replicating the geometry on several processors and tracking several MC particles populations with different initial seeds in every replicated domains. At the end of the time steps, the contribution of every processors are averaged. This parallel strategy is particularly well suited to MC codes, taking advantage of the independence of the MC particles.

5 Also known as implicit capture.

6 Note that this MC scheme is also intensively used in photonics, see (Poëtte and Valentin 2020; Steinberg and Heizler 2021b; Ahrens and Larsen 2001).

7 Of course, simplexes such as the ones presented in (Blatman 2009) may be used and have less coefficients but studying their effects is beyond the scope of this paper.

8 It is only a renumerotation.

9 Tthese are all detailed in (Poëtte 2019b) and recalled in Appendix C.

10 This is also observable on Figure 1 as the non-intrusive NMC = 100 slope is on the same level as the one for MC-gPC.

11 the Central Limit theorem states that the variance (if obtained from an unbiased estimator (Lapeyre, Pardoux, and Sentis 1998)) is an error estimator.

12 and even some high order moments for the semi-analog and non-analog schemes in sections 3.2.1–3.2.2.

13 where $\mathcal{G}(\mu ,\sigma )$ denotes a gaussian random variable of mean μ and variance ${\sigma }^2$.

14 It only corresponds to a pretreatment of the cross-sections.

15 where $\mathcal{G}(\mu ,\sigma )$ denotes a gaussian random variable of mean μ and variance ${\sigma }^2$.

16 We do not detail the functions compute_cell_exit_time and find_neighbooring_cell as they depend more on the type of grid (cartesian, structured, unstructured) than on the MC resolution scheme.

17 depending on the chosen MC scheme.