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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An anisotropic interaction model with collision avoidance
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
CMAP, École Polytechnique, CNRS UMR7641, Institut Polytechnique de Paris, Palaiseau, France |
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.
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:
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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. |
| [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. |
| [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. |
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G. Birindelli, Modèle Entropique Pour le Calcul de dose en Radiothérapie Externe et Curiethérapie, PhD thesis, Université de Bordeaux, 2018. Google Scholar |
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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. |
| [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. |
| [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. |
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J. M. Borwein and A. S. Lewis,
Partially finite convex programming: Part Ⅱ, Math. Program., 57 (1992), 49-83.
doi: 10.1007/BF01581073. |
| [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. |
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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 |
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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.
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S. Chandrasekhar, Radiative Transfer, Dover publications, 1960. |
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R. E. Curto and L. A. Fialkow,
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|
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R. E. Curto and L. A. Fialkow,
A duality proof of Tchakaloff's theorem, J. Math. Anal. Appl., 269 (2002), 519-532.
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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. |
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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. |
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L. Fialkow,
The truncated K-moment problem: A survey, Theta Ser. Adv. Math., 18 (2016), 25-51.
|
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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. |
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M. Frank, Partial Moment Models for Radiative Transfer, PhD thesis, T.U. Kaiserslautern, 2005. |
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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. |
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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. |
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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. |
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H. Hamburger,
Über eine erweiterung des stieltjesschen momentenproblems, Math. Ann., 82 (1920), 120-164.
doi: 10.1007/BF01457982. |
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C. Hauck and R. McClarren,
Positive $P_N$ closures, SIAM J. Sci. Comput., 32 (2010), 2603-2626.
doi: 10.1137/090764918. |
| [28] |
C. D. Hauck,
High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.
|
| [29] |
F. Hausdorff,
Summationmethoden und momentfolgen, Math. Z., 9 (1921), 74-109.
doi: 10.1007/BF01378337. |
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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. |
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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. |
| [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. |
| [33] |
M. Junk, Maximum entropy for reduced moment problems, Math. Mod. Meth. Appl. S., 10 (2000), 1001–1025.
doi: 10.1142/S0218202500000513. |
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D. Kershaw, Flux Limiting Nature's Own Way, Technical report, Lawrence Livermore Laboratory, 1976. Google Scholar |
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M. G. Krein and A. A. Nudel'man, The Markov Moment Problem and Extremals Problems, AMS Trans. Math. Monographs : Vol. 50, 1977. |
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K. Kuepper, Models, Numerical Methods, and Uncertainty Quantification for Radiation Therapy, PhD thesis, RWTH Aachen University, 2016. Google Scholar |
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C. D. Levermore,
Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
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R. Li and W. Li, 3D $B_2$model for radiative transfer equation, Int. J. Numer. Anal. Model., 17 (2020), 118–150. |
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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 |
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J. McDonald and M. Torrilhon,
Affordable robust moment closures for cfd based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.
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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. |
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Maximum entropy Eddington factors, J. Quant. Spectros. Radiat. Transfer, 20 (1978), 541-545.
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G. N. Minerbo,
Maximum entropy reconstruction from cone-beam projection data, Comput. Biol. Med., 9 (1979), 29-37.
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P. Monreal, Higher order minimum entropy approximations in radiative transfer, arXiv: 0812.3063, 2008, pages 1–18. Google Scholar |
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T. Pichard, Mathematical Modelling for dose Deposition in Photontherapy, PhD thesis, Université de Bordeaux & RWTH Aachen University, 2016. Google Scholar |
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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. |
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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. |
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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.
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
J. M. Borwein and A. S. Lewis,
Partially finite convex programming: Part Ⅱ, Math. Program., 57 (1992), 49-83.
doi: 10.1007/BF01581073. |
| [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. |
| [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. |
| [15] |
S. Chandrasekhar, Radiative Transfer, Dover publications, 1960. |
| [16] |
R. E. Curto and L. A. Fialkow,
Recursiveness, positivity, and truncated moment problem, Houston j. Math., 17 (1991), 603-635.
|
| [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. |
| [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. |
| [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. |
| [20] |
L. Fialkow,
The truncated K-moment problem: A survey, Theta Ser. Adv. Math., 18 (2016), 25-51.
|
| [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. |
| [22] |
M. Frank, Partial Moment Models for Radiative Transfer, PhD thesis, T.U. Kaiserslautern, 2005. |
| [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. |
| [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. |
| [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. |
| [26] |
H. Hamburger,
Über eine erweiterung des stieltjesschen momentenproblems, Math. Ann., 82 (1920), 120-164.
doi: 10.1007/BF01457982. |
| [27] |
C. Hauck and R. McClarren,
Positive $P_N$ closures, SIAM J. Sci. Comput., 32 (2010), 2603-2626.
doi: 10.1137/090764918. |
| [28] |
C. D. Hauck,
High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.
|
| [29] |
F. Hausdorff,
Summationmethoden und momentfolgen, Math. Z., 9 (1921), 74-109.
doi: 10.1007/BF01378337. |
| [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. |
| [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. |
| [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. |
| [33] |
M. Junk, Maximum entropy for reduced moment problems, Math. Mod. Meth. Appl. S., 10 (2000), 1001–1025.
doi: 10.1142/S0218202500000513. |
| [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. |
| [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. |
| [39] |
A. S. Lewis,
Consistency of moment systems, Can. J. Math., 47 (1995), 995-1006.
doi: 10.4153/CJM-1995-052-2. |
| [40] |
R. Li and W. Li, 3D $B_2$model for radiative transfer equation, Int. J. Numer. Anal. Model., 17 (2020), 118–150. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [54] |
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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
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
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
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
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
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
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
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
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
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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