Pencil-Beam Approximation of Stationary Fokker--Planck

Bal, Guillaume; Palacios, Benjamin

doi:10.1137/19m1295775

Cited by 6 publications

(9 citation statements)

References 36 publications

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“…The source term f (x, θ) is highly concentrated near a single point (x 0 , θ 0 ) ∈ R d × S d−1 . The exponent 2s in condition ii is included here for notational convenience, and agrees with the exponent considered in [7] as we approach the local case s = 1. Without loss of generality, we assume in most of the article that (x 0 , θ 0 ) = (0, N ) with N = (0, .…”

mentioning

confidence: 54%

“…In this setting, the magnitude of the diffusion coefficient in the fractional Laplacian is small compared to the spatial dynamic of particles and characterized by a small parameter . Analogously to what the authors have done in the local case (s = 1) in [7], we provide a higher order approximation to the Fokker-Planck solution by means of pencil-beams, which are solutions to appropriate fractional Fermi pencil-beam equations. We establish error estimates to contrast the accuracy of the approximations for the pencil-beam model and the ballistic (i.e., unscattered) transport solution.…”

mentioning

confidence: 98%

“…In our setting, while solutions preserve positivity, their total mass may vary. As we have done in, e.g., [5,7] for similar reasons, we consider a generalization of the standard 1-Wasserstein distance. We denote by BL 1,κ the set of…”

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confidence: 99%

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Pencil-beam approximation of fractional Fokker-Planck

Bal¹,

Palacios²

2021

KRM

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confidence: 54%

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confidence: 98%

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Pencil-beam approximation of fractional Fokker-Planck

Bal¹,

Palacios²

2021

KRM

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“…This definition is similar to the spaces used for other variants of the kinetic Fokker-Planck equation e.g. in [1,11,5]. We use ideas from [1] to show the following:…”

Section: Function Spacesmentioning

confidence: 99%

Stable and efficient Petrov-Galerkin methods for a kinetic Fokker-Planck equation

Brunken¹,

Smetana²

2020

Preprint

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We propose a stable Petrov-Galerkin discretization of a kinetic Fokker-Planck equation constructed in such a way that uniform inf-sup stability can be inferred directly from the variational formulation. Inspired by well-posedness results for parabolic equations, we derive a lower bound for the dual inf-sup constant of the Fokker-Planck bilinear form by means of stable pairs of trial and test functions. The trial function of such a pair is constructed by applying the kinetic transport operator and the inverse velocity Laplace-Beltrami operator to a given test function. For the Petrov-Galerkin projection we choose an arbitrary discrete test space and then define the discrete trial space using the same application of transport and inverse Laplace-Beltrami operator. As a result, the spaces replicate the stable pairs of the continuous level and we obtain a well-posed numerical method with a discrete inf-sup constant identical to the inf-sup constant of the continuous problem independently of the mesh size. We show how the specific basis functions can be efficiently computed by low-dimensional elliptic problems, and confirm the practicability and performance of the method for a numerical example.

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“…It is based on neglecting back-scattering and is thus valid as long as beams remain sufficiently narrow. The derivation of the Fermi pencil beam and fractional Fermi pencil beam models from Fokker-Planck models was derived recently in [7,8]. We present the models in detail in section 3.…”

Section: Introductionmentioning

confidence: 99%

Fermi pencil beams and off-axis laser detection

Bal¹,

Palacios²

2021

Preprint

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This paper concerns the reconstruction of properties of narrow laser beams propagating in turbulent atmospheres. We consider the setting of off-axis measurements, based on light detection away from the main path of the beam.We first model light propagation in the beam itself by macroscopic approximations of radiative transfer equations that take the form of Fermi pencil beam or fractional Fermi pencil beam equations. Such models are effective in the small mean-free-path, large transport-mean-free path regime. The reconstruction of their constitutive parameters is also greatly simplified compared to the more accurate radiative transfer equations or (fractional) Fokker-Planck models.From off-axis measurements based on wide-angle single scattering off the beam, we propose a framework allowing us to reconstruct the main features of the beam, and in particular its direction of propagation and the location of the emitting source.

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Pencil-Beam Approximation of Stationary Fokker--Planck

Cited by 6 publications

References 36 publications

Pencil-beam approximation of fractional Fokker-Planck

Pencil-beam approximation of fractional Fokker-Planck

Stable and efficient Petrov-Galerkin methods for a kinetic Fokker-Planck equation

Fermi pencil beams and off-axis laser detection

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