The Fokker-Planck Operator as an Asymptotic Limit

Pomraning, G. C.

doi:10.1142/s021820259200003x

Cited by 122 publications

(80 citation statements)

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“…Note that the choice of S 1 is arbitrary, any circle of given radius would be dealt with in the same fashion. The singularity of kernel tends to favor grazing collisions, and as a consequence this LR regime bears some strong similarities with the classical peakedforward regime [20,23,25], with the forward effect magnified by the singularity.…”

Section: Introductionmentioning

confidence: 89%

Monte Carlo Methods for Radiative Transfer with Singular Kernels

Gomez¹,

Pinaud²

2018

SIAM J. Sci. Comput.

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This work is devoted to Monte Carlo methods for radiative transfer equations with singular kernels, and is motivated by the study of wave propagation in random media with long-range dependence. As opposed to the short-range case where the collision cross section is integrable and leads to a non-zero mean free time, the cross section is not integrable in the long-range situation and yields a vanishing mean free time. For computational efficiency, a particular care is then required in the construction of the stochastic processes used in the Monte Carlo methods. For this, we adapt a method of Asmussen-Rosiński and Cohen-Rosiński based on a small jumps/large jumps decomposition of the generator. We compare this method with another approach based on alpha-stable processes, and show the superiority of the first one. We consider various algorithms for the simulation of the jump distribution, and underline the efficiency of an appropriate stochastic collocation method. Comparisons between integrable and singular kernel solutions are given.

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Section: Introductionmentioning

confidence: 89%

Monte Carlo Methods for Radiative Transfer with Singular Kernels

Gomez¹,

Pinaud²

2018

SIAM J. Sci. Comput.

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“…Note that equation (5.3) can also be derived formally from the radiative transfer equation (3.17). First, one considers that scattering is sharply peaked in the forward scattering direction, so that it is possible to take the Fokker-Planck approximation, that is to say, the right-hand side of (3.17) can be approximated by a diffusion operator in K [24,18]. Second, one considers that the source emission is sharply peaked and that the propagation distance is short enough so that the wave remains in the form of a narrow cone beam.…”

Section: The Paraxial Approximationmentioning

confidence: 99%

Derivation of a one-way radiative transfer equation in random media

Borcea

Garnier

2016

Phys. Rev. E

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Abstract. We derive from first principles a one-way radiative transfer equation for the wave intensity resolved over directions (Wigner transform of the wave field) in random media. It is an initial value problem with excitation from a source which emits waves in a preferred, forward direction. The equation is derived in a regime with small random fluctuations of the wave speed but long distances of propagation with respect to the wavelength, so that cumulative scattering is significant. The correlation length of the medium and the scale of the support of the source are slightly larger than the wavelength, and the waves propagate in a wide cone with opening angle less than 180 o , so that the backward and evanescent waves are negligible. The scattering regime is a bridge between that of radiative transfer, where the waves propagate in all directions and the paraxial regime, where the waves propagate in a narrow angular cone. We connect the one-way radiative transport equation with the equations satisfied by the Wigner transform of the wave field in these regimes.

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“…(11) combining the terms containing tissue optical properties into one term I(r, ω), (12) where (13) It is now clear that one can substitute altered optical coefficients, , and a new phase function f* in (12) without affecting the solution as long as the quantity I remains unchanged. This leads to the similarity relations requirement: (14) Upon expanding the radiance and phase functions in spherical harmonics, using orthogonality and the addition theorem for Legendre polynomials, one arrives at the following system of equations: (15) which can also be written (16) where f n , are the Legendre moments of f HG , f*, respectively. The quantities I n combine the optical properties of the medium and the properties of the scattering phase function.…”

Section: Similarity Theory (St Model)mentioning

confidence: 99%

“…Hence an approximate phase function that preserves only a few of the momentum transfer moments should be a good approximation to the exact phase function. However, the precise behavior of the momentum transfer moments as n increases depends very sensitively on the specific description of the single scattering function f. For example, the Henyey-Greenstein phase function, f HG , is such that even when the average cosine g of the scattering angle μ 0 is close to 1 (but not equal to 1) its momentum transfer coefficients, ξ n do not decrease with increasing n [16,17]. Indeed, the merit of each CH method requires a more careful analysis than intuitively based arguments can provide.…”

Section: Discrete Scattering Angles (Dsa Model)mentioning

confidence: 99%

Condensed history Monte Carlo methods for photon transport problems

Bhan

Spanier

2007

Journal of Computational Physics

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We study methods for accelerating Monte Carlo simulations that retain most of the accuracy of conventional Monte Carlo algorithms. These methods -called Condensed History (CH) methodshave been very successfully used to model the transport of ionizing radiation in turbid systems. Our primary objective is to determine whether or not such methods might apply equally well to the transport of photons in biological tissue. In an attempt to unify the derivations, we invoke results obtained first by Lewis, Goudsmit and Saunderson and later improved by Larsen and Tolar. We outline how two of the most promising of the CH models -one based on satisfying certain similarity relations and the second making use of a scattering phase function that permits only discrete directional changes -can be developed using these approaches. The main idea is to exploit the connection between the space-angle moments of the radiance and the angular moments of the scattering phase function. We compare the results obtained when the two CH models studied are used to simulate an idealized tissue transport problem. The numerical results support our findings based on the theoretical derivations and suggest that CH models should play a useful role in modeling light-tissue interactions.

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The Fokker-Planck Operator as an Asymptotic Limit

Cited by 122 publications

References 0 publications

Monte Carlo Methods for Radiative Transfer with Singular Kernels

Monte Carlo Methods for Radiative Transfer with Singular Kernels

Derivation of a one-way radiative transfer equation in random media

Condensed history Monte Carlo methods for photon transport problems

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