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Heterogeneous discrete kinetic model and its diffusion limit
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 |
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.
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. 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. |
[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 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. |
[16] |
L. C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Soc., 1998. |
[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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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 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. |
[16] |
L. C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Soc., 1998. |
[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. |
[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. |
[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. |
[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. 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. |
[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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