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)

摘要

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).