A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation

Abstract

In this paper, we are interested in the resolution of the time-dependent problem of particle transport in a media whose composition is uncertain. The most common resolution strategy consists in running, at prescribed points in the uncertain space (at experimental designs points), a simulation device as a black-box and perform the uncertainty propagation resolution (i.e. resolution of SPDE). This kind of strategy is commonly called non-intrusive. The nonintrusive resolution can be carried with Monte-Carlo, quasi Monte-Carlo, generalized Polynomial Chaos etc. The latter is of interest in this document for its fast convergence rate. After going over and illustrating the main drawbacks of the non-intrusive (gPC or not) uncertainty propagation resolution for the linear Boltzmann equation in a simplified configuration, we build a new gPC based MC scheme solving intrusively the uncertain counterpart of the problem. The paper ends with some numerical examples.

Spectral convergence of the generalized Polynomial Chaos reduced model obtained from the uncertain linear Boltzmann equation

“…Note that a similar approach has been developed for the Fokker-Planck equation in [11]. In [50], fast convergence has been practically observed with respect to the truncation order P . The aim of this paper is to explain this fast convergence and for this, in the following sections, we prove the spectral convergence of the P −truncated gPC based reduced models obtained from (4).…”

“…In section 4, we present few numerical results confirming the theoretical results of section 3. Some are obtained with the resolution schemes presented in [50]. We finally conclude in section 5 with emphasis on the remaining open questions.…”

“…This resolution method is known to be efficient for high (3(x) + 1(t) + 3(v) = 7) dimensional problems but costly. This is emphasized, detailed and illustrated in [50]. In [50], a P -truncated gPC reduced model has been introduced in order to solve (4).…”

“…This is emphasized, detailed and illustrated in [50]. In [50], a P -truncated gPC reduced model has been introduced in order to solve (4). It is solved thanks to an astute Monte-Carlo resolution of the gPC reduced model.…”

“…The paper is organized as follows. In section 2, we present the P -truncated gPC reduced model of interest here (and solved in [50]). In section 3, we prove the spectral convergence of the built reduced model.…”