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-generalised Polynomial Chaos (MC-gPC) has already been thoroughly studied in the literature [1,2,3,4,5,6,7]. 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 behaviour 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 (linear [1,3] or nonlinear [4,5,6,7]) 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 studied in [2,6], the convergence with respect to the truncation order P has been theoretically and numerically studied [2, 1], the coupling to nonlinear physics has been performed in [7,6]. But the study of the MC noise remains, to our knowledge, to be done. This is the purpose of this paper.

Multigroup-like MC resolution of generalised Polynomial Chaos reduced models of the uncertain linear Boltzmann equation (+discussion on hybrid intrusive/non-intrusive uncertainty propagation)

“…MC-gPC both builds a gPC based reduced model of some kinetic equations of interest and solves it with an MC scheme in order to propagate uncertainties. In [3,4], the kinetic equation is a flocking model; in [5,6,7] it corresponds to the quadratic Boltzmann equation; in [1,2,8,9,10], it is (or is closely related to) the linear Boltzmann equation. We also focus on the latter in this paper.…”

“…Let us summarize the pros and cons of MC-gPC when applied to equation (1). The next points can be considered a summary of what can be found in [1,2,8,9,10]. The main benefits of MC-gPC for (1) are ( 1) spectral convergence: it corresponds to the fast convergence with respect to P (proved in [2]).…”

“…(c) Nonlinear physics needing small time steps/many iterations may not benefit from an MC-gPC implementation if the tallies/parallel reductions are too frequent, see [8,9]. (d) It also has an impact on the memory consumption: each MC particles carries a field X p of size Q in the MC schemes described in [1,8,9,10]. ( * 3 ) Finally, it has recently been proved [10] that the MC schemes solving (3) and described in [1,8,9] are noisier than other strategies (even if the noise remains acceptable).…”

“…

Monte Carlo-generalised Polynomial Chaos (MC-gPC) has already been thoroughly studied in the literature [1,2,3,4,5,6,7,8,9,10]. MC-gPC both builds a gPC based reduced model of some kinetic equations of interest and solves it with an MC scheme in order to propagate uncertainties.

…”

“…Furthermore, MC-gPC can be implemented thanks to simple modifications of an existing MC code [1,9,8]. But MC-gPC also presents some weaknesses: it is sensitive to the curse of dimensionality [1,9] and is noisier than other strategies [10]. The aim of this paper is to present new MC schemes solving the same gPC based reduced model but attenuating the two previous drawbacks.…”