Verification method of Monte Carlo codes for transport processes with arbitrary accuracy

Abstract

In this work, we present a robust and powerful method for the verification, with arbitrary accuracy, of Monte Carlo codes for simulating random walks in complex media. Such random walks are typical of photon propagation in turbid media, scattering of particles, i.e., neutrons in a nuclear reactor or animal/humans’ migration. Among the numerous applications, Monte Carlo method is also considered a gold standard for numerically “solving” the scalar radiative transport equation even in complex geometries and distributions of the optical properties. In this work, we apply the verification method to a Monte Carlo code which is a forward problem solver extensively used for typical applications in the field of tissue optics. The method is based on the well-known law of average path length invariance when the entrance of the entities/particles in a medium obeys to a simple cosine law, i.e., Lambertian entrance, and annihilation of particles inside the medium is absent. By using this law we achieve two important points: (1) the invariance of the average path length guarantees that the expected value is known regardless of the complexity of the medium; (2) the accuracy of a Monte Carlo code can be assessed by simple statistical tests. We will show that we can reach an arbitrary accuracy of the estimated average pathlength as the number of simulated trajectories increases. The method can be applied in complete generality versus the scattering and geometrical properties of the medium, as well as in presence of refractive index mismatches in the optical case. In particular, this verification method is reliable to detect inaccuracies in the treatment of boundaries of finite media. The results presented in this paper, obtained by a standard computer machine, show a verification of our Monte Carlo code up to the sixth decimal digit. We discuss how this method can provide a fundamental tool for the verification of Monte Carlo codes in the geometry of interest, without resorting to simpler geometries and uniform distribution of the scattering properties.

Monte Carlo simulations in anomalous radiative transfer: a tutorial

“…Let's now define four different examples of MC simulations based on Eqs. (10) and (15). In the following sub-sections we will give the geometry of the problem, all the necessary optical quantities, together with the light sources and detectors positions.…”

“…The figure highlights the fact that even if in general the average step is always for the power law [Eq. (10)] and the constant step [Eq. ( 15)] models, for the specific ART condition it is not possible to simply divide the path length s c by to find the m c (.)…”

“…( 3), a fundamental physical property of the system cannot be reproduced, i.e., the invariance property (IP) will not be satisfied (see below). Note that, the IP represents a very powerful test allowing us to check the reliability of any MC code [10].…”

“…Thus, this simple model [Eq. (10)] represents a kind unified summary for the classical and anomalous approach.…”

Light diffusion in layered media: A numerical study in the spatial and time-domains

“…Due to the highly scattering nature of these media, the RTE can be reduced to the diffusion equation, which gives analytical solutions in homogeneous, semi-infinite, or infinite slab geometries 12,13 . The RTE can also be solved by the Monte Carlo method which remains the gold-standard approach to calculate light transport in media with complex geometries 14 but is computationally expensive 15 . Although parallel implementations have significantly improved the speed of Monte Carlo simulations 16,17 , they still broadly remain non-viable as inverse solvers to obtain optical properties from experimental measurements 16 .…”