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

Abstract

In this paper, we are interested in propagating uncertainties through the linear Boltzmann equation. Such model is intensively used in neutronics, photonics, socio-economics, epidemiology. It is often solved thanks to Monte-Carlo (MC) schemes. MC codes are reliable and accurate but costly and, as a consequence, propagating uncertainties through them remains quite complicated. In order to alleviate the cost of an uncertainty propagation, reduced models are often built. In this paper, we focus on generalised Polynomial Chaos (gPC) reduced models, and especially on their resolution with an MC scheme: such strategy is commonly called Monte Carlo-generalised Polynomial Chaos (MC-gPC) in the literature [1–10]. It allows important computational gains on many applications: in a nutshell, the reasons for its success are spectral convergence [2] plus the fact that it is based on an MC resolution which 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. They are based on multigroup-like resolution methods. The new MC schemes improve the run times of MC-gPC. The resolution scheme is intrusive: this means that modifications of an existing solver are necessary (even if people familiar with multigroup MC resolution will not be intimidated by them). The paper ends with a discussion about taking into account uncertainties at the early stages of the development of a simulation code together with some original and efficient hybrid intrusive/non-intrusive applications.