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

Pencil-beam approximation of fractional Fokker-Planck

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.

Citation: Guillaume Bal, Benjamin Palacios. Pencil-beam approximation of fractional Fokker-Planck. Kinetic and 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.

[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.

[3]

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

[4]

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

[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.

[6]

G. BalT. 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.

[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.

[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.

[9]

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

[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. 

[11]

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

[12]

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

[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.

[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.

[15]

E. Di NezzaG. 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.

[16]

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

[17]

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

[18]

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

[19]

F. HansonI. BendallC. 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.

[20]

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

[21]

H. J. HwangJ. 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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[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.

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[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.

[6]

G. BalT. 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.

[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.

[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.

[9]

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

[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. 

[11]

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

[12]

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

[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.

[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.

[15]

E. Di NezzaG. 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.

[16]

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

[17]

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

[18]

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

[19]

F. HansonI. BendallC. 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.

[20]

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

[21]

H. J. HwangJ. 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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[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.

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