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.
Highlights
Monte Carlo (MC) methods for transport processes have been crucial for calculating physical measurands used in the description of complex phenomena
All the presented results provide a direct comparison between the calculated values by the MC simulation and the exact theoretical reference of the invariance property (IP) allowing the achievement of arbitrary accuracy only limited by the finite computation time and by the finite precision exploited in the calculations
In this work we have proposed a general method for testing a MC code for transport phenomena
Summary
Monte Carlo (MC) methods for transport processes have been crucial for calculating physical measurands used in the description of complex phenomena. Benchmark solutions of RTE, mostly in semianalytical form, are only available for regularly bounded geometries, e.g. semi-infinite, slab, layers, and mostly for uniform scattering p roperties[45,46,47,48] This verification method has two drawbacks: these solutions cannot be usually expressed in closed form and they are known with limited accuracy and they are likely to be affected by convergence problems of the calculated quantities. For finite geometries and in cases with scattering and/or refractive index mismatch between different regions of the medium the accuracy of the solutions of the diffusion equation is further limited[3]. Known as “Cauchy’s formula”, the invariance property has been studied in different contexts, such as nuclear physics[56,57] and optics[22,52,58], and has been recently experimentally verified for light propagation through scattering media[53] and for bacterial random w alks[19]
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