Poisson-Box Sampling algorithms for three-dimensional Markov binary mixtures

Abstract

Particle transport in Markov mixtures can be addressed by the so-called Chord Length Sampling (CLS) methods, a family of Monte Carlo algorithms taking into account the effects of stochastic media on particle propagation by generating on-the-fly the material interfaces crossed by the random walkers during their trajectories. Such methods enable a significant reduction of computational resources as opposed to reference solutions obtained by solving the Boltzmann equation for a large number of realizations of random media. CLS solutions, which neglect correlations induced by the spatial disorder, are faster albeit approximate, and might thus show discrepancies with respect to reference solutions. In this work we propose a new family of algorithms (called 'Poisson Box Sampling', PBS) aimed at improving the accuracy of the CLS approach for transport in d-dimensional binary Markov mixtures. In order to probe the features of PBS methods, we will focus on three-dimensional Markov media and revisit the benchmark problem originally proposed by Adams, Pomraning Adams et al. (1989) and extended by Brantley Brantley (2011): for these configurations we will compare reference solutions, standard CLS solutions and the new PBS solutions for scalar particle flux, transmission and reflection coefficients. PBS will be shown to perform better than CLS at the expense of a reasonable increase in computational time.

Cauchy formulas for linear transport in random media

“…More surprisingly, numerical evidence based on Monte Carlo simulations (see below) shows that eq. ( 31) holds true also for a broader class of annealed disorder models, such as the Poisson-Box Sampling (PBS) [28] and the Local Realization Preserving (LRP) [29] methods, which have been recently introduced on a heuristic basis in order to alleviate the shortcomings of the CLS by including short-and long-range memory effects for the particle trajectories, and thus partially preserve disorder-induced spatial correlations. This seems to suggest that the Cauchy formula in eq.…”

Probability density function for random photon steps in a binary (isotropic-Poisson) statistical mixture

“…The medium The characteristics of the medium considered in the present contribution -a binary (isotropic-Poisson) statistical mixturehave already been presented in other works [12][13][14][15] . For this reason, we will only provide a brief summary of the main points necessary to understand the theory presented in the following sections.…”

“…With this question in mind, many studies have already been proposed in the literature; going from numerical MC simulations to theoretical approaches 5 – 8 . Interestingly, this approach also applies to image rendering 9 .…”

Probability density function for random photon steps in a binary (isotropic-Poisson) statistical mixture

“…1). With this question in mind, many studies have already been proposed in the literature; going from numerical MC simulations to theoretical approaches [5][6][7][8] . Interestingly, this approach also applies to image rendering 9 .…”