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October  2021, 14(5): 767-817. doi: 10.3934/krm.2021024

Pencil-beam approximation of fractional Fokker-Planck

Guillaume Bal 1, and  Benjamin Palacios 2,,

1. 

Departments of Statistics and Mathematics, University of Chicago, 5747 S. Ellis Avenue, Jones 120B, Chicago, IL 60637, USA
2. 

Departments of Statistics, University of Chicago, 5747 S. Ellis Avenue, Jones 316, Chicago, IL 60637, USA

* Corresponding author: Benjamin Palacios

Received  December 2020 Revised  May 2021 Published  October 2021 Early access  June 2021

Fund Project: This research was partially supported by the Office of Naval Research, Grant N00014-17-1-2096 and by the National Science Foundation, Grant DMS-1908736

We consider the modeling of light beams propagating in highly forward-peaked turbulent media by fractional Fokker-Planck equations and their approximations by fractional Fermi pencil beam models. We obtain an error estimate in a 1-Wasserstein distance for the latter model showing that beam spreading is well captured by the Fermi pencil-beam approximation in the small diffusion limit.

Keywords: Radiative transfer, Fokker–Planck, fractional Laplacian, singular kernel, error estimates, Wasserstein distance.
Mathematics Subject Classification: 35B40, 35H10. 35Q84, 35R11.
Citation: Guillaume Bal, Benjamin Palacios. Pencil-beam approximation of fractional Fokker-Planck. Kinetic & Related Models, 2021, 14 (5) : 767-817. doi: 10.3934/krm.2021024
References:
[1]

R. Alexandre, Fractional order kinetic equations and hypoellipticity, Anal. Appl. (Singap.), 10 (2012), 237-247.  doi: 10.1142/S021953051250011X.  Google Scholar

[2]

R. Alonso and W. Sun, The radiative transfer equation in the forward-peaked regime, Comm. Math. Phys., 338 (2015), 1233-1286.  doi: 10.1007/s00220-015-2395-8.  Google Scholar

[3]

S. Armstrong and J. C. Mourrat, Variational methods for the kinetic Fokker-Planck equation, Preprint, arXiv: 1902.04037, 2019. Google Scholar

[4]

G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001. doi: 10.1088/0266-5611/25/5/053001.  Google Scholar

[5]

G. Bal and A. Jollivet, Generalized stability estimates in inverse transport theory, Inverse Probl. Imaging, 12 (2018), 59-90.  doi: 10.3934/ipi.2018003.  Google Scholar

[6]

G. Bal, T. Komorowski and L. Ryzhik, Kinetic limits for waves in a random medium, Kinet. Relat. Models, 3 (2010), 529-644.  doi: 10.3934/krm.2010.3.529.  Google Scholar

[7]

G. Bal and B. Palacios, Pencil-beam approximation of stationary Fokker–Planck, SIAM J. Math. Anal., 52 (2020), 3487-3519.  doi: 10.1137/19M1295775.  Google Scholar

[8]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles due premier ordre a coefficients réels; théorèmes d'approximation application a l'équation de transport, Ann. Sci. École Norm. Sup., 3 (1970), 185-233.  doi: 10.24033/asens.1190.  Google Scholar

[9]

C. Börgers and E. W. Larsen, Asymptotic derivation of the Fermi pencil-beam approximation, Nuclear Science and Engineering, 123 (1996), 343-357.   Google Scholar

[10]

C. Börgers and E. W. Larsen, On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport, Medical Physics, 23 (1996), 1749-1759.   Google Scholar

[11]

F. Bouchut, Hypoelliptic regularity in kinetic equations, J. Math. Pures Appl., 81 (2002), 1135-1159.  doi: 10.1016/S0021-7824(02)01264-3.  Google Scholar

[12]

J. Brunken and K. Smetana, Stable and efficient Petrov–Galerkin methods for a kinetic Fokker-Planck equation, preprint, arXiv: 2010.15784, 2020. Google Scholar

[13]

E. Cueva, M. Courdurier, A. Osses, V. Castañeda, B. Palacios and S. Härtel, Mathematical modeling for 2D light-sheet fluorescence microscopy image reconstruction, Inverse Problems, 36 (2020), 075005, 27 pp. doi: 10.1088/1361-6420/ab80d8.  Google Scholar

[14]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology: Volume 6 Evolution Problems II, Springer Science & Business Media, 2012. Google Scholar

[15]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[16]

L. C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Soc., 1998.  Google Scholar

[17]

C. Gomez, O. Pinaud and L. Ryzhik, Hypoelliptic estimates in radiative transfer, Comm. Partial Differential Equations, 41 (2016), 150-184.  doi: 10.1080/03605302.2015.1096287.  Google Scholar

[18]

C. Gomez, O. Pinaud and L. Ryzhik, Radiative transfer with long-range interactions: regularity and asymptotics, Multiscale Model. Simul., 15 (2017), 1048-1072.  doi: 10.1137/15M1047076.  Google Scholar

[19]

F. Hanson, I. Bendall, C. Deckard and H. Haidar, Off-axis detection and characterization of laser beams in the maritime atmosphere, Applied Optics, 50 (2011), 3050-3056.  doi: 10.1364/AO.50.003050.  Google Scholar

[20]

L. G. Henyey and J. L. Greenstein, Diffuse radiation in the galaxy, The Astrophysical Journal, 93 (1941), 70-83.  doi: 10.1086/144246.  Google Scholar

[21]

H. J. Hwang, J. Jang and J. Jung, On the kinetic Fokker–Planck equation in a half-space with absorbing barriers, Indiana Univ. Math. J., 64 (2015), 1767-1804.  doi: 10.1512/iumj.2015.64.5679.  Google Scholar

[22]

C. Imbert and L. Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc., 22 (2020), 507-592.  doi: 10.4171/jems/928.  Google Scholar

[23]

C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, Anal. PDE, 14 (2021), 171-204.  doi: 10.2140/apde.2021.14.171.  Google Scholar

[24]

C. Imbert and L. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, arXiv preprint, arXiv: 1909.12729v1, 2019. Google Scholar

[25]

A. D. Kim and J. B. Keller, Light propagation in biological tissue, JOSA A, 20 (2003), 92-98.  doi: 10.1364/JOSAA.20.000092.  Google Scholar

[26]

C. L. Leakeas and E. W. Larsen, Generalized Fokker-Planck approximations of particle transport with highly forward-peaked scattering, Nuclear Science and Engineering, 137 (2001), 236-250.  doi: 10.13182/NSE01-A2189.  Google Scholar

[27]

J. L. Lions, Equations Différentielles Opérationnelles et Problèmes aux Limites, Vol. 1a1, Springer-Verlag, 2013. Google Scholar

[28]

E. Olbrant and M. Frank, Generalized Fokker-Planck theory for electron and photon transport in biological tissues: Application to radiotherapy, Comput. Math. Methods Med., 11 (2010), 313-339.  doi: 10.1080/1748670X.2010.491828.  Google Scholar

[29]

G. C. Pomraning, The Fokker-Planck operator as an asymptotic limit, Math. Models Methods Appl. Sci., 2 (1992), 21-36.  doi: 10.1142/S021820259200003X.  Google Scholar

[30]

N. Roy and F. Reid, Off-axis laser detection model in coastal areas, Optical Engineering, 47 (2008), 086002. doi: 10.1117/1.2969119.  Google Scholar

[31]

L. F. Stokols, Hölder continuity for a family of nonlocal hypoelliptic kinetic equations, SIAM J. Math. Anal., 51 (2019), 4815-4847.  doi: 10.1137/18M1234953.  Google Scholar

show all references

References:
[1]

R. Alexandre, Fractional order kinetic equations and hypoellipticity, Anal. Appl. (Singap.), 10 (2012), 237-247.  doi: 10.1142/S021953051250011X.  Google Scholar

[2]

R. Alonso and W. Sun, The radiative transfer equation in the forward-peaked regime, Comm. Math. Phys., 338 (2015), 1233-1286.  doi: 10.1007/s00220-015-2395-8.  Google Scholar

[3]

S. Armstrong and J. C. Mourrat, Variational methods for the kinetic Fokker-Planck equation, Preprint, arXiv: 1902.04037, 2019. Google Scholar

[4]

G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001. doi: 10.1088/0266-5611/25/5/053001.  Google Scholar

[5]

G. Bal and A. Jollivet, Generalized stability estimates in inverse transport theory, Inverse Probl. Imaging, 12 (2018), 59-90.  doi: 10.3934/ipi.2018003.  Google Scholar

[6]

G. Bal, T. Komorowski and L. Ryzhik, Kinetic limits for waves in a random medium, Kinet. Relat. Models, 3 (2010), 529-644.  doi: 10.3934/krm.2010.3.529.  Google Scholar

[7]

G. Bal and B. Palacios, Pencil-beam approximation of stationary Fokker–Planck, SIAM J. Math. Anal., 52 (2020), 3487-3519.  doi: 10.1137/19M1295775.  Google Scholar

[8]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles due premier ordre a coefficients réels; théorèmes d'approximation application a l'équation de transport, Ann. Sci. École Norm. Sup., 3 (1970), 185-233.  doi: 10.24033/asens.1190.  Google Scholar

[9]

C. Börgers and E. W. Larsen, Asymptotic derivation of the Fermi pencil-beam approximation, Nuclear Science and Engineering, 123 (1996), 343-357.   Google Scholar

[10]

C. Börgers and E. W. Larsen, On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport, Medical Physics, 23 (1996), 1749-1759.   Google Scholar

[11]

F. Bouchut, Hypoelliptic regularity in kinetic equations, J. Math. Pures Appl., 81 (2002), 1135-1159.  doi: 10.1016/S0021-7824(02)01264-3.  Google Scholar

[12]

J. Brunken and K. Smetana, Stable and efficient Petrov–Galerkin methods for a kinetic Fokker-Planck equation, preprint, arXiv: 2010.15784, 2020. Google Scholar

[13]

E. Cueva, M. Courdurier, A. Osses, V. Castañeda, B. Palacios and S. Härtel, Mathematical modeling for 2D light-sheet fluorescence microscopy image reconstruction, Inverse Problems, 36 (2020), 075005, 27 pp. doi: 10.1088/1361-6420/ab80d8.  Google Scholar

[14]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology: Volume 6 Evolution Problems II, Springer Science & Business Media, 2012. Google Scholar

[15]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[16]

L. C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Soc., 1998.  Google Scholar

[17]

C. Gomez, O. Pinaud and L. Ryzhik, Hypoelliptic estimates in radiative transfer, Comm. Partial Differential Equations, 41 (2016), 150-184.  doi: 10.1080/03605302.2015.1096287.  Google Scholar

[18]

C. Gomez, O. Pinaud and L. Ryzhik, Radiative transfer with long-range interactions: regularity and asymptotics, Multiscale Model. Simul., 15 (2017), 1048-1072.  doi: 10.1137/15M1047076.  Google Scholar

[19]

F. Hanson, I. Bendall, C. Deckard and H. Haidar, Off-axis detection and characterization of laser beams in the maritime atmosphere, Applied Optics, 50 (2011), 3050-3056.  doi: 10.1364/AO.50.003050.  Google Scholar

[20]

L. G. Henyey and J. L. Greenstein, Diffuse radiation in the galaxy, The Astrophysical Journal, 93 (1941), 70-83.  doi: 10.1086/144246.  Google Scholar

[21]

H. J. Hwang, J. Jang and J. Jung, On the kinetic Fokker–Planck equation in a half-space with absorbing barriers, Indiana Univ. Math. J., 64 (2015), 1767-1804.  doi: 10.1512/iumj.2015.64.5679.  Google Scholar

[22]

C. Imbert and L. Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc., 22 (2020), 507-592.  doi: 10.4171/jems/928.  Google Scholar

[23]

C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, Anal. PDE, 14 (2021), 171-204.  doi: 10.2140/apde.2021.14.171.  Google Scholar

[24]

C. Imbert and L. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, arXiv preprint, arXiv: 1909.12729v1, 2019. Google Scholar

[25]

A. D. Kim and J. B. Keller, Light propagation in biological tissue, JOSA A, 20 (2003), 92-98.  doi: 10.1364/JOSAA.20.000092.  Google Scholar

[26]

C. L. Leakeas and E. W. Larsen, Generalized Fokker-Planck approximations of particle transport with highly forward-peaked scattering, Nuclear Science and Engineering, 137 (2001), 236-250.  doi: 10.13182/NSE01-A2189.  Google Scholar

[27]

J. L. Lions, Equations Différentielles Opérationnelles et Problèmes aux Limites, Vol. 1a1, Springer-Verlag, 2013. Google Scholar

[28]

E. Olbrant and M. Frank, Generalized Fokker-Planck theory for electron and photon transport in biological tissues: Application to radiotherapy, Comput. Math. Methods Med., 11 (2010), 313-339.  doi: 10.1080/1748670X.2010.491828.  Google Scholar

[29]

G. C. Pomraning, The Fokker-Planck operator as an asymptotic limit, Math. Models Methods Appl. Sci., 2 (1992), 21-36.  doi: 10.1142/S021820259200003X.  Google Scholar

[30]

N. Roy and F. Reid, Off-axis laser detection model in coastal areas, Optical Engineering, 47 (2008), 086002. doi: 10.1117/1.2969119.  Google Scholar

[31]

L. F. Stokols, Hölder continuity for a family of nonlocal hypoelliptic kinetic equations, SIAM J. Math. Anal., 51 (2019), 4815-4847.  doi: 10.1137/18M1234953.  Google Scholar

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