American Institute of Mathematical Sciences

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December  2020, 13(6): 1243-1280. doi: 10.3934/krm.2020045

A moment closure based on a projection on the boundary of the realizability domain: 1D case

Teddy Pichard

CMAP, École Polytechnique, CNRS UMR7641, Institut Polytechnique de Paris, Palaiseau, France

Received  December 2019 Revised  August 2020 Published  September 2020

This work aims to develop and test a projection technique for the construction of closing equations of moment systems. One possibility to define such a closure consists in reconstructing an underlying kinetic distribution from a vector of moments, then expressing the closure based on this reconstructed function.

Exploiting the geometry of the realizability domain, i.e. the set of moments of positive distribution function, we decompose any realizable vectors into two parts, one corresponding to the moments of a chosen equilibrium function, and one obtain by a projection onto the boundary of the realizability domain in the direction of equilibrium function. A realizable closure of both of these parts are computed with standard techniques providing a realizable closure for the full system. This technique is tested for the reduction of a radiative transfer equation in slab geometry.

Keywords: Moment closure, truncated moment problem, realizability property, projection technique.
Mathematics Subject Classification: 35L40, 47B35, 35B09.
Citation: Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045
References:
[1]

N. I. Akhiezer, The Classical Moment Problem, Edinburgh: Oliver & Boyd, 1965. Google Scholar

[2]

N. I. Akhiezer and M. G. Krein, Some Questions in the Theory of Moments, AMS Trans. Math. Monographs: Vol. 2, 1962. Google Scholar

[3]

G. Alldredge and F. Schneider, A realizability-preserving discontinuous galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, J. Comput. Phys., 295 (2015), 665-684.  doi: 10.1016/j.jcp.2015.04.034.  Google Scholar

[4]

G. W. Alldredge, Optimization Techniques for Entropy-Based Moment Models of Linear Transport, PhD thesis, University of Maryland, 2012. Google Scholar

[5]

G. W. Alldredge, C. D. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry Ⅱ: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), 361-391.  doi: 10.1137/11084772X.  Google Scholar

[6]

C. Bayer and J. Teichmann, The proof of Tchakaloff's theorem, Proc. Amer. Math. Soc., 134 (2006), 3035–3040. doi: 10.1090/S0002-9939-06-08249-9.  Google Scholar

[7]

G. Birindelli, Modèle Entropique Pour le Calcul de dose en Radiothérapie Externe et Curiethérapie, PhD thesis, Université de Bordeaux, 2018. Google Scholar

[8]

G. Birindelli, J.-L. Feugeas, J. Caron, B. Dubroca, G. Kantor, J. Page, T. Pichard, V.T. Tikhonchuk and P. Nicolaï, High performance modelling of the transport of energetic particles for photon radiotherapy, Phys. Medica, 42 (2017), 305-312.  doi: 10.1016/j.ejmp.2017.06.020.  Google Scholar

[9]

J. M. Borwein and A. S. Lewis, Duality relationships for entropy-like minimization problems, SIAM J. Control Optim., 29 (1991), 325-338.  doi: 10.1137/0329017.  Google Scholar

[10]

J. M. Borwein and A. S. Lewis, Partially finite convex programming, part Ⅰ: Quasi relative interiors and duality theory, Math. Program., 57 (1992), 15-48.  doi: 10.1007/BF01581072.  Google Scholar

[11]

J. M. Borwein and A. S. Lewis, Partially finite convex programming: Part Ⅱ, Math. Program., 57 (1992), 49-83.  doi: 10.1007/BF01581073.  Google Scholar

[12]

T. A. Brunner and J. P. Holloway, One-dimensional Riemann solvers and the maximum entropy closure, J. Quant. Spectros. Radiat. Transfer, 69 (2001), 543-566.  doi: 10.1016/S0022-4073(00)00099-6.  Google Scholar

[13]

J. Caron, Étude et Validation Clinique d'un Modèle aux Moments Entropique Pour le Transport de Particules Énergétiques : Application aux Faisceaux D'électrons Pour la Radiothérapie Externe, PhD thesis, Univ. Bordeaux, 2016. Google Scholar

[14]

J. Caron, J.-L. Feugeas, B. Dubroca, G. Kantor, C. Dejean, G. Birindelli, T. Pichard, P. Nicolaï, E. d'Humières, M. Frank and V. Tikhonchuk, Deterministic model for the transport of energetic particles. Application in the electron radiotherapy, Phys. Medica, 31 (2015), 912-921.  doi: 10.1016/j.ejmp.2015.07.148.  Google Scholar

[15]

S. Chandrasekhar, Radiative Transfer, Dover publications, 1960.  Google Scholar

[16]

R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problem, Houston j. Math., 17 (1991), 603-635.   Google Scholar

[17]

R. E. Curto and L. A. Fialkow, A duality proof of Tchakaloff's theorem, J. Math. Anal. Appl., 269 (2002), 519-532.  doi: 10.1016/S0022-247X(02)00034-3.  Google Scholar

[18]

R. E. Curto and L. A. Fialkow, An analogue of the Riesz-Haviland theorem for the truncated moment problem, J. Functional Analysis, 255 (2008), 2709-2731.  doi: 10.1016/j.jfa.2008.09.003.  Google Scholar

[19]

B. Dubroca and J.-L. Feugeas, Étude théorique et numérique d'une hiérarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris, 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.  Google Scholar

[20]

L. Fialkow, The truncated K-moment problem: A survey, Theta Ser. Adv. Math., 18 (2016), 25-51.   Google Scholar

[21]

R. O. Fox, A quadrature-based third-order moment method for dilute gas-particle flows, J. Comput. Phys., 227 (2008), 6313-6350.  doi: 10.1016/j.jcp.2008.03.014.  Google Scholar

[22]

M. Frank, Partial Moment Models for Radiative Transfer, PhD thesis, T.U. Kaiserslautern, 2005.  Google Scholar

[23]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, J. Comput. Phys., 218 (2006), 1-18.  doi: 10.1016/j.jcp.2006.01.038.  Google Scholar

[24]

K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci., 68 (1971), 1686-1688.  doi: 10.1073/pnas.68.8.1686.  Google Scholar

[25]

C. K. Garrett and C. D. Hauck, A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport theory and Stat. Phys., 42 (2013), 203-235.  doi: 10.1080/00411450.2014.910226.  Google Scholar

[26]

H. Hamburger, Über eine erweiterung des stieltjesschen momentenproblems, Math. Ann., 82 (1920), 120-164.  doi: 10.1007/BF01457982.  Google Scholar

[27]

C. Hauck and R. McClarren, Positive $P_N$ closures, SIAM J. Sci. Comput., 32 (2010), 2603-2626.  doi: 10.1137/090764918.  Google Scholar

[28]

C. D. Hauck, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.   Google Scholar

[29]

F. Hausdorff, Summationmethoden und momentfolgen, Math. Z., 9 (1921), 74-109.  doi: 10.1007/BF01378337.  Google Scholar

[30]

E. K. Haviland, On the momentum problem for distributions in more than one dimension, Amer. J. Math., 57 (1935), 562-568.  doi: 10.2307/2371187.  Google Scholar

[31]

E. K. Haviland, On the momentum problem for distribution functions in more than one dimension. ii, Amer. J. Math., 58 (1936), 164-168.  doi: 10.2307/2371063.  Google Scholar

[32]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy, Phys. Med. Biol., 51 (2006), 675-693.  doi: 10.1088/0031-9155/51/3/013.  Google Scholar

[33]

M. Junk, Maximum entropy for reduced moment problems, Math. Mod. Meth. Appl. S., 10 (2000), 1001–1025. doi: 10.1142/S0218202500000513.  Google Scholar

[34]

D. Kershaw, Flux Limiting Nature's Own Way, Technical report, Lawrence Livermore Laboratory, 1976. Google Scholar

[35]

M. G. Krein and A. A. Nudel'man, The Markov Moment Problem and Extremals Problems, AMS Trans. Math. Monographs : Vol. 50, 1977.  Google Scholar

[36]

K. Kuepper, Models, Numerical Methods, and Uncertainty Quantification for Radiation Therapy, PhD thesis, RWTH Aachen University, 2016. Google Scholar

[37] J. B. Lasserre, Moment, Positive Polynomials, and their Applications, volume 1., Imperial college press, 2009.   Google Scholar
[38]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.  Google Scholar

[39]

A. S. Lewis, Consistency of moment systems, Can. J. Math., 47 (1995), 995-1006.  doi: 10.4153/CJM-1995-052-2.  Google Scholar

[40]

R. Li and W. Li, 3D $B_2$model for radiative transfer equation, Int. J. Numer. Anal. Model., 17 (2020), 118–150.  Google Scholar

[41]

D. S. Lucas, H. D. Gougar, T. Wareing, G. Failla, J. McGhee, D. A. Barnett and I. Davis, Comparison of the 3-D Deterministic Neutron Transport Code Attila® to Measure Data, MCNP and MCNPX for the Advanced Test Reactor, Technical report, Idaho National Laboratory, 2005. Google Scholar

[42]

J. McDonald and M. Torrilhon, Affordable robust moment closures for cfd based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.  doi: 10.1016/j.jcp.2013.05.046.  Google Scholar

[43]

L. R. Mead and N. Papanicolaou, Maximum entropy in the problem of moments, J. Math. Phys., 25 (1984), 2404-2417.  doi: 10.1063/1.526446.  Google Scholar

[44]

G. N. Minerbo, Maximum entropy Eddington factors, J. Quant. Spectros. Radiat. Transfer, 20 (1978), 541-545.  doi: 10.1016/0022-4073(78)90024-9.  Google Scholar

[45]

G. N. Minerbo, Maximum entropy reconstruction from cone-beam projection data, Comput. Biol. Med., 9 (1979), 29-37.  doi: 10.1016/0010-4825(79)90020-9.  Google Scholar

[46]

P. Monreal, Higher order minimum entropy approximations in radiative transfer, arXiv: 0812.3063, 2008, pages 1–18. Google Scholar

[47]

P. Monreal, Moment Realizability and Kershaw Closures in Radiative Transfer, PhD thesis, RWTH Aachen University, 2012. Google Scholar

[48]

T. E. Oliphant, Guide to NumPy, Trelgol Publishing USA, 2006. Google Scholar

[49]

J. Page, Développement et Validation de L'application de la Force de Lorentz Dans le Modèle aux Moments Entropiques $M_1$. Étude de L'effet du Champ Magnétique sur le Dépôt de Dose en Radiothérapie externe, PhD thesis, Univ. Bordeaux, 2018. Google Scholar

[50]

T. Pichard, Mathematical Modelling for dose Deposition in Photontherapy, PhD thesis, Université de Bordeaux & RWTH Aachen University, 2016. Google Scholar

[51]

T. Pichard, G. W. Alldredge, S. Brull, B. Dubroca and M. Frank, An approximation of the $M_2$ closure: application to radiotherapy dose simulation, J. Sci. Comput., 71 (2017), 71-108.  doi: 10.1007/s10915-016-0292-8.  Google Scholar

[52]

T. Pichard, D. Aregba-Driollet, S. Brull, B. Dubroca and M. Frank, Relaxation schemes for the $M_1$ model with space-dependent flux: Application to radiotherapy dose calculation, Commun. Comput. Phys., 19 (2016), 168-191.  doi: 10.4208/cicp.121114.210415a.  Google Scholar

[53]

T. Pichard, S. Brull and B. Dubroca, A numerical approach for a system of transport equations in the field of radiotherapy, Commun. Comput. Phys., 25 (2019), 1097-1126.  doi: 10.4208/cicp.oa-2017-0245.  Google Scholar

[54]

M. Riesz, Sur le problème des moments, troisième note, Ark. Math. Astr. Fys., 17 (1923), 1-52.   Google Scholar

[55]

J. A. R. Sarr and C. P. T. Groth, A second-order maximum-entropy inspired interpolative closure for radiative heat transfer in gray participating media, J. Quant. Spectros. Radiat. Transfer, accepted for publication, 2020. Google Scholar

[56]

F. Schneider, Moment Models in Radiation Transport Equations, PhD thesis, T.U. Kaiserslautern, 2015. Google Scholar

[57]

F. Schneider, Kershaw closures for linear transport equations in slab geometry i: Model derivation, J. Comput. Phys., 322 (2016), 905-919.  doi: 10.1016/j.jcp.2016.02.080.  Google Scholar

[58]

F. Schneider, J. Kall and G. Alldredge, A realizability-preserving high-order kinetic scheme using weno reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kinetic and Related Models, 9 (2016), 193-215.  doi: 10.3934/krm.2016.9.193.  Google Scholar

[59]

F. Schneider, A. Roth and J. Kall, First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions, Kinetic and Related Models, 10 (2017), 1127-1161.  doi: 10.3934/krm.2017044.  Google Scholar

[60]

J. Schneider, Entropic approximation in kinetic theory, ESAIM-Math. Model. Num., 38 (2004), 541-561.  doi: 10.1051/m2an:2004025.  Google Scholar

[61]

T.-J. Stieltjes, Recherches sur les fractions continues, Anns Fac. Sci. Toulouse: Mathématiques, 8 (1894), J1–J122. doi: 10.5802/afst.108.  Google Scholar

[62]

V. Tchakaloff, Formules de cubature mécanique à coefficients non négatifs, Bull. Sci. Math., 81 (1957), 123-134.   Google Scholar

[63]

T. A. Wareing, J. M. McGhee, Y. Archambault and S. Thompson, Acuros XB ® advanced dose calculation for the Eclipse TM treatement planning system, Clinical Perspectives, 2010. Google Scholar

[64]

C. Yuan, F. Laurent and R. O. Fox, An extended quadrature method of moments for population balance equations, J. Aerosol Sci., 51 (2012), 1-23.  doi: 10.1016/j.jaerosci.2012.04.003.  Google Scholar

show all references

References:
[1]

N. I. Akhiezer, The Classical Moment Problem, Edinburgh: Oliver & Boyd, 1965. Google Scholar

[2]

N. I. Akhiezer and M. G. Krein, Some Questions in the Theory of Moments, AMS Trans. Math. Monographs: Vol. 2, 1962. Google Scholar

[3]

G. Alldredge and F. Schneider, A realizability-preserving discontinuous galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, J. Comput. Phys., 295 (2015), 665-684.  doi: 10.1016/j.jcp.2015.04.034.  Google Scholar

[4]

G. W. Alldredge, Optimization Techniques for Entropy-Based Moment Models of Linear Transport, PhD thesis, University of Maryland, 2012. Google Scholar

[5]

G. W. Alldredge, C. D. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry Ⅱ: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), 361-391.  doi: 10.1137/11084772X.  Google Scholar

[6]

C. Bayer and J. Teichmann, The proof of Tchakaloff's theorem, Proc. Amer. Math. Soc., 134 (2006), 3035–3040. doi: 10.1090/S0002-9939-06-08249-9.  Google Scholar

[7]

G. Birindelli, Modèle Entropique Pour le Calcul de dose en Radiothérapie Externe et Curiethérapie, PhD thesis, Université de Bordeaux, 2018. Google Scholar

[8]

G. Birindelli, J.-L. Feugeas, J. Caron, B. Dubroca, G. Kantor, J. Page, T. Pichard, V.T. Tikhonchuk and P. Nicolaï, High performance modelling of the transport of energetic particles for photon radiotherapy, Phys. Medica, 42 (2017), 305-312.  doi: 10.1016/j.ejmp.2017.06.020.  Google Scholar

[9]

J. M. Borwein and A. S. Lewis, Duality relationships for entropy-like minimization problems, SIAM J. Control Optim., 29 (1991), 325-338.  doi: 10.1137/0329017.  Google Scholar

[10]

J. M. Borwein and A. S. Lewis, Partially finite convex programming, part Ⅰ: Quasi relative interiors and duality theory, Math. Program., 57 (1992), 15-48.  doi: 10.1007/BF01581072.  Google Scholar

[11]

J. M. Borwein and A. S. Lewis, Partially finite convex programming: Part Ⅱ, Math. Program., 57 (1992), 49-83.  doi: 10.1007/BF01581073.  Google Scholar

[12]

T. A. Brunner and J. P. Holloway, One-dimensional Riemann solvers and the maximum entropy closure, J. Quant. Spectros. Radiat. Transfer, 69 (2001), 543-566.  doi: 10.1016/S0022-4073(00)00099-6.  Google Scholar

[13]

J. Caron, Étude et Validation Clinique d'un Modèle aux Moments Entropique Pour le Transport de Particules Énergétiques : Application aux Faisceaux D'électrons Pour la Radiothérapie Externe, PhD thesis, Univ. Bordeaux, 2016. Google Scholar

[14]

J. Caron, J.-L. Feugeas, B. Dubroca, G. Kantor, C. Dejean, G. Birindelli, T. Pichard, P. Nicolaï, E. d'Humières, M. Frank and V. Tikhonchuk, Deterministic model for the transport of energetic particles. Application in the electron radiotherapy, Phys. Medica, 31 (2015), 912-921.  doi: 10.1016/j.ejmp.2015.07.148.  Google Scholar

[15]

S. Chandrasekhar, Radiative Transfer, Dover publications, 1960.  Google Scholar

[16]

R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problem, Houston j. Math., 17 (1991), 603-635.   Google Scholar

[17]

R. E. Curto and L. A. Fialkow, A duality proof of Tchakaloff's theorem, J. Math. Anal. Appl., 269 (2002), 519-532.  doi: 10.1016/S0022-247X(02)00034-3.  Google Scholar

[18]

R. E. Curto and L. A. Fialkow, An analogue of the Riesz-Haviland theorem for the truncated moment problem, J. Functional Analysis, 255 (2008), 2709-2731.  doi: 10.1016/j.jfa.2008.09.003.  Google Scholar

[19]

B. Dubroca and J.-L. Feugeas, Étude théorique et numérique d'une hiérarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris, 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.  Google Scholar

[20]

L. Fialkow, The truncated K-moment problem: A survey, Theta Ser. Adv. Math., 18 (2016), 25-51.   Google Scholar

[21]

R. O. Fox, A quadrature-based third-order moment method for dilute gas-particle flows, J. Comput. Phys., 227 (2008), 6313-6350.  doi: 10.1016/j.jcp.2008.03.014.  Google Scholar

[22]

M. Frank, Partial Moment Models for Radiative Transfer, PhD thesis, T.U. Kaiserslautern, 2005.  Google Scholar

[23]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, J. Comput. Phys., 218 (2006), 1-18.  doi: 10.1016/j.jcp.2006.01.038.  Google Scholar

[24]

K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci., 68 (1971), 1686-1688.  doi: 10.1073/pnas.68.8.1686.  Google Scholar

[25]

C. K. Garrett and C. D. Hauck, A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport theory and Stat. Phys., 42 (2013), 203-235.  doi: 10.1080/00411450.2014.910226.  Google Scholar

[26]

H. Hamburger, Über eine erweiterung des stieltjesschen momentenproblems, Math. Ann., 82 (1920), 120-164.  doi: 10.1007/BF01457982.  Google Scholar

[27]

C. Hauck and R. McClarren, Positive $P_N$ closures, SIAM J. Sci. Comput., 32 (2010), 2603-2626.  doi: 10.1137/090764918.  Google Scholar

[28]

C. D. Hauck, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.   Google Scholar

[29]

F. Hausdorff, Summationmethoden und momentfolgen, Math. Z., 9 (1921), 74-109.  doi: 10.1007/BF01378337.  Google Scholar

[30]

E. K. Haviland, On the momentum problem for distributions in more than one dimension, Amer. J. Math., 57 (1935), 562-568.  doi: 10.2307/2371187.  Google Scholar

[31]

E. K. Haviland, On the momentum problem for distribution functions in more than one dimension. ii, Amer. J. Math., 58 (1936), 164-168.  doi: 10.2307/2371063.  Google Scholar

[32]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy, Phys. Med. Biol., 51 (2006), 675-693.  doi: 10.1088/0031-9155/51/3/013.  Google Scholar

[33]

M. Junk, Maximum entropy for reduced moment problems, Math. Mod. Meth. Appl. S., 10 (2000), 1001–1025. doi: 10.1142/S0218202500000513.  Google Scholar

[34]

D. Kershaw, Flux Limiting Nature's Own Way, Technical report, Lawrence Livermore Laboratory, 1976. Google Scholar

[35]

M. G. Krein and A. A. Nudel'man, The Markov Moment Problem and Extremals Problems, AMS Trans. Math. Monographs : Vol. 50, 1977.  Google Scholar

[36]

K. Kuepper, Models, Numerical Methods, and Uncertainty Quantification for Radiation Therapy, PhD thesis, RWTH Aachen University, 2016. Google Scholar

[37] J. B. Lasserre, Moment, Positive Polynomials, and their Applications, volume 1., Imperial college press, 2009.   Google Scholar
[38]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.  Google Scholar

[39]

A. S. Lewis, Consistency of moment systems, Can. J. Math., 47 (1995), 995-1006.  doi: 10.4153/CJM-1995-052-2.  Google Scholar

[40]

R. Li and W. Li, 3D $B_2$model for radiative transfer equation, Int. J. Numer. Anal. Model., 17 (2020), 118–150.  Google Scholar

[41]

D. S. Lucas, H. D. Gougar, T. Wareing, G. Failla, J. McGhee, D. A. Barnett and I. Davis, Comparison of the 3-D Deterministic Neutron Transport Code Attila® to Measure Data, MCNP and MCNPX for the Advanced Test Reactor, Technical report, Idaho National Laboratory, 2005. Google Scholar

[42]

J. McDonald and M. Torrilhon, Affordable robust moment closures for cfd based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.  doi: 10.1016/j.jcp.2013.05.046.  Google Scholar

[43]

L. R. Mead and N. Papanicolaou, Maximum entropy in the problem of moments, J. Math. Phys., 25 (1984), 2404-2417.  doi: 10.1063/1.526446.  Google Scholar

[44]

G. N. Minerbo, Maximum entropy Eddington factors, J. Quant. Spectros. Radiat. Transfer, 20 (1978), 541-545.  doi: 10.1016/0022-4073(78)90024-9.  Google Scholar

[45]

G. N. Minerbo, Maximum entropy reconstruction from cone-beam projection data, Comput. Biol. Med., 9 (1979), 29-37.  doi: 10.1016/0010-4825(79)90020-9.  Google Scholar

[46]

P. Monreal, Higher order minimum entropy approximations in radiative transfer, arXiv: 0812.3063, 2008, pages 1–18. Google Scholar

[47]

P. Monreal, Moment Realizability and Kershaw Closures in Radiative Transfer, PhD thesis, RWTH Aachen University, 2012. Google Scholar

[48]

T. E. Oliphant, Guide to NumPy, Trelgol Publishing USA, 2006. Google Scholar

[49]

J. Page, Développement et Validation de L'application de la Force de Lorentz Dans le Modèle aux Moments Entropiques $M_1$. Étude de L'effet du Champ Magnétique sur le Dépôt de Dose en Radiothérapie externe, PhD thesis, Univ. Bordeaux, 2018. Google Scholar

[50]

T. Pichard, Mathematical Modelling for dose Deposition in Photontherapy, PhD thesis, Université de Bordeaux & RWTH Aachen University, 2016. Google Scholar

[51]

T. Pichard, G. W. Alldredge, S. Brull, B. Dubroca and M. Frank, An approximation of the $M_2$ closure: application to radiotherapy dose simulation, J. Sci. Comput., 71 (2017), 71-108.  doi: 10.1007/s10915-016-0292-8.  Google Scholar

[52]

T. Pichard, D. Aregba-Driollet, S. Brull, B. Dubroca and M. Frank, Relaxation schemes for the $M_1$ model with space-dependent flux: Application to radiotherapy dose calculation, Commun. Comput. Phys., 19 (2016), 168-191.  doi: 10.4208/cicp.121114.210415a.  Google Scholar

[53]

T. Pichard, S. Brull and B. Dubroca, A numerical approach for a system of transport equations in the field of radiotherapy, Commun. Comput. Phys., 25 (2019), 1097-1126.  doi: 10.4208/cicp.oa-2017-0245.  Google Scholar

[54]

M. Riesz, Sur le problème des moments, troisième note, Ark. Math. Astr. Fys., 17 (1923), 1-52.   Google Scholar

[55]

J. A. R. Sarr and C. P. T. Groth, A second-order maximum-entropy inspired interpolative closure for radiative heat transfer in gray participating media, J. Quant. Spectros. Radiat. Transfer, accepted for publication, 2020. Google Scholar

[56]

F. Schneider, Moment Models in Radiation Transport Equations, PhD thesis, T.U. Kaiserslautern, 2015. Google Scholar

[57]

F. Schneider, Kershaw closures for linear transport equations in slab geometry i: Model derivation, J. Comput. Phys., 322 (2016), 905-919.  doi: 10.1016/j.jcp.2016.02.080.  Google Scholar

[58]

F. Schneider, J. Kall and G. Alldredge, A realizability-preserving high-order kinetic scheme using weno reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kinetic and Related Models, 9 (2016), 193-215.  doi: 10.3934/krm.2016.9.193.  Google Scholar

[59]

F. Schneider, A. Roth and J. Kall, First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions, Kinetic and Related Models, 10 (2017), 1127-1161.  doi: 10.3934/krm.2017044.  Google Scholar

[60]

J. Schneider, Entropic approximation in kinetic theory, ESAIM-Math. Model. Num., 38 (2004), 541-561.  doi: 10.1051/m2an:2004025.  Google Scholar

[61]

T.-J. Stieltjes, Recherches sur les fractions continues, Anns Fac. Sci. Toulouse: Mathématiques, 8 (1894), J1–J122. doi: 10.5802/afst.108.  Google Scholar

[62]

V. Tchakaloff, Formules de cubature mécanique à coefficients non négatifs, Bull. Sci. Math., 81 (1957), 123-134.   Google Scholar

[63]

T. A. Wareing, J. M. McGhee, Y. Archambault and S. Thompson, Acuros XB ® advanced dose calculation for the Eclipse TM treatement planning system, Clinical Perspectives, 2010. Google Scholar

[64]

C. Yuan, F. Laurent and R. O. Fox, An extended quadrature method of moments for population balance equations, J. Aerosol Sci., 51 (2012), 1-23.  doi: 10.1016/j.jaerosci.2012.04.003.  Google Scholar

Figure 1.  Schematic representation of a ray starting at a point $ \mathbf{V}\in\mathcal{R}_{\mathbf{b}} $ directed by $ -\mathbf{V}_{eq} $ and crossing $ \partial\mathcal{R}_{\mathbf{b}} $ in $ \mathbf{W}(\bar{x}) $
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Figure 2.  Moments of order 0 (left) and 1 (right) obtained with $ P_7 $, $ K_7 $, $ \Pi_7 $ and reference solution for the simple beam test case
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Figure 3.  Discrete $ l^1 $ (top left), $ l^2 $ (top right) and $ l^\infty $ (bottom) errors on the moment of order 0 compared to a reference solution for the $ P_N $, $ K_N $ and $ \Pi_N $ as a function of $ N $ for the simple beam test case
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Figure 4.  Moments of order 0 (left) and 1 (right) obtained with $ P_N $, $ K_N $, $ \Pi_N $ for $ N = 2 $ (first line), $ N = 3 $ (second line), $ N = 6 $ (third line), $ N = 7 $ (fourth line) and reference solution for the double beam test case
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Figure 5.  Discrete $ l^1 $ (top left), $ l^2 $ (top right) and $ l^\infty $ (bottom) errors on the moment of order 0 compared to a reference solution for the $ P_N $, $ K_N $ and $ \Pi_N $ as a function of $ N $ for the double beam test case
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Figure 6.  Representation of the measures representing the vector $ \mathbf{f}(x = L/2) $ with $ P_N $, $ K_N $ and $ \Pi_N $ models with $ N = 2 $ (top left), $ 3 $ (top right), $ 4 $ (middle left), $ 5 $ (middle right), $ 6 $ (bottom left), $ 7 $ (bottom right) for the double beam test case
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Figure 7.  Moments of order 0 (left) and 1 (right) obtained with $ P_N $, $ K_N $, $ \Pi_N $ for $ N = 2 $ (first line), $ N = 3 $ (second line), $ N = 6 $ (third line), $ N = 7 $ (fourth line) and a reference $ P_{24} $ solution for the point source test case
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Figure 8.  Discrete $ l^1 $ (top left), $ l^2 $ (top right) and $ l^\infty $ (bottom) errors on the moment of order 0 compared to a reference $ P_{24} $ simulation for the $ P_N $, $ K_N $ and $ \Pi_N $ as a function of $ N $ for the point source test case
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Figure 9.  Representation of the measures representing the vector $ \mathbf{f}(x = L/2) $ with $ P_N $, $ K_N $ and $ \Pi_N $ models for $ N = 2 $ (top left), $ 3 $ (top right), $ 4 $ (middle left), $ 5 $ (middle right), $ 6 $ (bottom left), $ 7 $ (bottom right) for the point source test case
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Figure 10.  Moments of order 0 (left) and 1 (right) obtained with $ P_N $, $ K_N $, $ \Pi_N $ for $ N = 2 $ (first line), $ N = 3 $ (second line), $ N = 6 $ (third line), $ N = 7 $ (fourth line) and those of the analytical solution for the Riemann problem
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Figure 11.  Representation of the measures representing the vector $ \mathbf{f}(x = L/2) $ with $ P_N $, $ K_N $ and $ \Pi_N $ models for $ N = 2 $ (top left), $ 3 $ (top right), $ 4 $ (middle left), $ 5 $ (middle right), $ 6 $ (bottom left), $ 7 $ (bottom right) for the Riemann problem
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