Optimal prediction for radiative transfer: A new perspective on moment closure

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September 2011, 4(3): 717-733. doi: 10.3934/krm.2011.4.717

Optimal prediction for radiative transfer: A new perspective on moment closure

Martin Frank ^1, and Benjamin Seibold ^2,

1.	RWTH Aachen University, Department of Mathematics, Schinkelstrasse 2 52062 Aachen, Germany
2.	Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122, United States

Received June 2011 Published August 2011

Abstract
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Moment methods are classical approaches that approximate the mesoscopic radiative transfer equation by a system of macroscopic moment equations. An expansion in the angular variables transforms the original equation into a system of infinitely many moments. The truncation of this infinite system is the moment closure problem. Many types of closures have been presented in the literature. In this note, we demonstrate that optimal prediction, an approach originally developed to approximate the mean solution of systems of nonlinear ordinary differential equations, can be used to derive moment closures. To that end, the formalism is generalized to systems of partial differential equations. Using Gaussian measures, existing linear closures can be re-derived, such as $P_N$, diffusion, and diffusion correction closures. This provides a new perspective on several approximations done in the process and gives rise to ideas for modifications to existing closures.

Keywords: diffusion approximation., method of moments, Radiative transfer, optimal prediction, measure.

Mathematics Subject Classification: 85A25, 78M05, 82Cx.

Citation: Martin Frank, Benjamin Seibold. Optimal prediction for radiative transfer: A new perspective on moment closure. Kinetic & Related Models, 2011, 4 (3) : 717-733. doi: 10.3934/krm.2011.4.717

References:

[1]	A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors,, J. Math. Phys., 32 (1991), 544. doi: 10.1063/1.529391.
[2]	J. Bell, A. J. Chorin and W. Crutchfield, Stochastic optimal prediction with application to averaged Euler equations,, Proc. 7th Nat. Conf. CFD, (2000), 1.
[3]	Y. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis,", Kluwer Academic Publishers, (1995).
[4]	P. S. Brantley and E. W. Larsen, The simplified $P_3$ approximation,, Nucl. Sci. Eng., 134 (2000).
[5]	P. N. Brown, B. Chang, U. R. Hanebutte and J. A. Rathkopf, "Spherical Harmonic Solutions to the 3d Kobayashi Benchmark Suite,", Technical Report UCRL-VG-135163, (2000).
[6]	T. A. Brunner, "Forms of Approximate Radiation Transport,", Technical Report SAND2002-1778, (2002), 2002.
[7]	T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport,, J. Comput. Phys., 210 (2005), 386. doi: 10.1016/j.jcp.2005.04.011.
[8]	S. Chandrasekhar, On the radiative equilibrium of a stellar atmosphere,, Astrophys. J., 99 (1944), 180. doi: 10.1086/144606.
[9]	_____, "Radiative Transfer,", Dover Publications, (1960).
[10]	A. J. Chorin, Conditional expectations and renormalization,, Multiscale Model. Simul., 1 (2003), 105. doi: 10.1137/S1540345902405556.
[11]	A. J. Chorin and O. H. Hald, "Stochastic Tools in Mathematics and Science,", Surveys and Tutorials in the Applied Mathematical Sciences, 1 (2006).
[12]	A. J. Chorin, O. H. Hald, and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes,, Proc. Natl. Acad. Sci. USA, 97 (2000), 2968. doi: 10.1073/pnas.97.7.2968.
[13]	_____, Non-Markovian optimal prediction, Monte Carlo adn Probablilistic Methods for Patial Differential Equations (Monte Carlo, 2000),, Monte Carlo Meth. Appl., 7 (2001), 99.
[14]	_____, Optimal prediction with memory,, Physica D, 166 (2002), 239. doi: 10.1016/S0167-2789(02)00446-3.
[15]	A. J. Chorin, A. P. Kast and R. Kupferman, Optimal prediction of underresolved dynamics,, Proc. Natl. Acad. Sci. USA, 95 (1998), 4094. doi: 10.1073/pnas.95.8.4094.
[16]	_____, Unresolved computation and optimal predictions,, Comm. Pure Appl. Math., 52 (1998), 1231.
[17]	_____, "On the Prediction of Large-Scale Dynamics using Unresolved Computations,", Nonlinear Partial Differential Equations (Evanston, 238 (1999), 53.
[18]	A. J. Chorin and P. Stinis, Problem reduction, renormalization, and memory,, Comm. Appl. Math. Comp. Sc., 1 (2006), 1. doi: 10.2140/camcos.2006.1.1.
[19]	B. Davison, "Neutron Transport Theory,", Clarendon Press, (1958).
[20]	B. Dubroca and J. L. Feugeas, Theoretical and numerical study of a moment closure hierarchy for the radiative transfer equation,, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915.
[21]	B. Dubroca, M. Frank, A. Klar and G. Thömmes, A Half space moment approximation to the radiative heat transfer equations,, Z. Angew. Math. Mech., 83 (2003), 853. doi: 10.1002/zamm.200310055.
[22]	M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer,, J. Comput. Phys., 218 (2006), 1. doi: 10.1016/j.jcp.2006.01.038.
[23]	M. Frank, A. Klar, E. W. Larsen and S. Yasuda, Time-dependent simplified $P_N$ approximation to the equations of radiative transfer,, J. Comput. Phys., 226 (2007), 2289. doi: 10.1016/j.jcp.2007.07.009.
[24]	M. Frank, A. Klar and R. Pinnau, Optimal control of glass cooling using Simplified $P_n$ theory,, Transp. Theory. Stat. Phys., 39 (2010), 282. doi: 10.1080/00411450.2010.533740.
[25]	E. M. Gelbard, "Applications of Spherical Harmonics Method to Reactor Problems,", Tech. Report WAPD-BT-20, (1960).
[26]	_____, "Simplified Spherical Harmonics Equations and their Use in Shielding Problems,", Tech. Report WAPD-T-1182, (1961).
[27]	_____, "Applications of the Simplified Spherical Harmonics Equations in Spherical Geometry,", Tech. Report WAPD-TM-294, (1962).
[28]	D. Givon, O. Hald and R. Kupferman, Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism,, Israel J. Math., 145 (2005), 221. doi: 10.1007/BF02786691.
[29]	T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite Dimensional Calculus,", Mathematics and its Applications, 253 (1993).
[30]	S. Karlin and L. S. Shapley, Geometry of moment spaces,, Mem. Amer. Math. Soc., (1953).
[31]	D. S. Kershaw, "Flux Limiting Nature's Own Way,", Tech. Report UCRL-78378, (1976).
[32]	D. A. Knoll, W. J. Rider and G. L. Olson, Method for non-equilibrium radiation diffusion,, J. Quant. Spectrosc. Radiat. Transfer, 63 (1999), 15. doi: 10.1016/S0022-4073(98)00132-0.
[33]	E. W. Larsen and J. R. Keller, Asymptotic solution of neutron transport problems for small mean free paths,, J. Math. Phys., 15 (1974), 75. doi: 10.1063/1.1666510.
[34]	E. W. Larsen, J. E. Morel and J. M. McGhee, Asymptotic derivation of the multigroup $P_1$ and simplified $P_N$ equations with anisotropic scattering,, Nucl. Sci. Eng., 123 (1996), 328.
[35]	E. W. Larsen and G. C. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry - I: analysis,, Nucl. Sci. Eng., 109 (1991), 49.
[36]	C. D. Levermore, Relating Eddington factors to flux limiters,, J. Quant. Spectrosc. Radiat. Transfer, 31 (1984), 149. doi: 10.1016/0022-4073(84)90112-2.
[37]	_____, "Transition Regime Models for Radiative Transport,", Presentation at IPAM: Grand challenge problems in computational astrophysics workshop on transfer phenomena, (2005).
[38]	R. G. McClarren, Theoretical aspects of the Simplified $P_n$ equations,, Transp. Theory. Stat. Phys., 39 (2010), 73. doi: 10.1080/00411450.2010.535088.
[39]	R. G. McClarren and C. D. Hauck, Robust and accurate filtered spherical Harmonics expansions for radiative transfer,, J. Comput. Phys., 229 (2010), 5597. doi: 10.1016/j.jcp.2010.03.043.
[40]	M. F. Modest, "Radiative Heat Transfer,", second ed., (1993).
[41]	H. Mori, Transport, collective motion and Brownian motion,, Prog. Theor. Phys., 33 (1965), 423. doi: 10.1143/PTP.33.423.
[42]	I. Müller and T. Ruggeri, "Rational Extended Thermodynamics,", second ed., (1993).
[43]	W. H. Reed, Spherical Harmonic solutions of the neutron transport equation from discrete ordinates code,, Nucl. Sci. Eng., 49 (1972), 10.
[44]	M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations,, Multiscale Model. Simul., 9 (2011), 1.
[45]	B. Seibold, Optimal prediction in molecular dynamics,, Monte Carlo Methods Appl., 10 (2004), 25. doi: 10.1515/156939604323091199.
[46]	B. Seibold and M. Frank, Optimal prediction for moment models: Crescendo diffusion and reordered equations,, Continuum Mech. Thermodyn., 21 (2009), 511. doi: 10.1007/s00161-009-0111-7.
[47]	M. Skibinsky, The range of the $(n+1)$th moment for distributions on $[0,1]$,, J. Appl. Probability, 4 (1967), 543. doi: 10.2307/3212220.
[48]	B. Su, Variable Eddington factors and flux limiters in radiative transfer,, Nucl. Sci. Eng., 137 (2001), 281.
[49]	D. I. Tomasevic and E. W. Larsen, The simplified $P_2$ approximation,, Nucl. Sci. Eng., 122 (1996), 309.
[50]	R. Turpault, M. Frank, B. Dubroca and A. Klar, Multigroup half space moment appproximations to the radiative heat transfer equations,, J. Comput. Phys., 198 (2004), 363. doi: 10.1016/j.jcp.2004.01.011.
[51]	R. Zwanzig, Problems in nonlinear transport theory,, in, (1980), 198. doi: 10.1007/BFb0025619.

show all references

References:

[1]	A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors,, J. Math. Phys., 32 (1991), 544. doi: 10.1063/1.529391.
[2]	J. Bell, A. J. Chorin and W. Crutchfield, Stochastic optimal prediction with application to averaged Euler equations,, Proc. 7th Nat. Conf. CFD, (2000), 1.
[3]	Y. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis,", Kluwer Academic Publishers, (1995).
[4]	P. S. Brantley and E. W. Larsen, The simplified $P_3$ approximation,, Nucl. Sci. Eng., 134 (2000).
[5]	P. N. Brown, B. Chang, U. R. Hanebutte and J. A. Rathkopf, "Spherical Harmonic Solutions to the 3d Kobayashi Benchmark Suite,", Technical Report UCRL-VG-135163, (2000).
[6]	T. A. Brunner, "Forms of Approximate Radiation Transport,", Technical Report SAND2002-1778, (2002), 2002.
[7]	T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport,, J. Comput. Phys., 210 (2005), 386. doi: 10.1016/j.jcp.2005.04.011.
[8]	S. Chandrasekhar, On the radiative equilibrium of a stellar atmosphere,, Astrophys. J., 99 (1944), 180. doi: 10.1086/144606.
[9]	_____, "Radiative Transfer,", Dover Publications, (1960).
[10]	A. J. Chorin, Conditional expectations and renormalization,, Multiscale Model. Simul., 1 (2003), 105. doi: 10.1137/S1540345902405556.
[11]	A. J. Chorin and O. H. Hald, "Stochastic Tools in Mathematics and Science,", Surveys and Tutorials in the Applied Mathematical Sciences, 1 (2006).
[12]	A. J. Chorin, O. H. Hald, and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes,, Proc. Natl. Acad. Sci. USA, 97 (2000), 2968. doi: 10.1073/pnas.97.7.2968.
[13]	_____, Non-Markovian optimal prediction, Monte Carlo adn Probablilistic Methods for Patial Differential Equations (Monte Carlo, 2000),, Monte Carlo Meth. Appl., 7 (2001), 99.
[14]	_____, Optimal prediction with memory,, Physica D, 166 (2002), 239. doi: 10.1016/S0167-2789(02)00446-3.
[15]	A. J. Chorin, A. P. Kast and R. Kupferman, Optimal prediction of underresolved dynamics,, Proc. Natl. Acad. Sci. USA, 95 (1998), 4094. doi: 10.1073/pnas.95.8.4094.
[16]	_____, Unresolved computation and optimal predictions,, Comm. Pure Appl. Math., 52 (1998), 1231.
[17]	_____, "On the Prediction of Large-Scale Dynamics using Unresolved Computations,", Nonlinear Partial Differential Equations (Evanston, 238 (1999), 53.
[18]	A. J. Chorin and P. Stinis, Problem reduction, renormalization, and memory,, Comm. Appl. Math. Comp. Sc., 1 (2006), 1. doi: 10.2140/camcos.2006.1.1.
[19]	B. Davison, "Neutron Transport Theory,", Clarendon Press, (1958).
[20]	B. Dubroca and J. L. Feugeas, Theoretical and numerical study of a moment closure hierarchy for the radiative transfer equation,, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915.
[21]	B. Dubroca, M. Frank, A. Klar and G. Thömmes, A Half space moment approximation to the radiative heat transfer equations,, Z. Angew. Math. Mech., 83 (2003), 853. doi: 10.1002/zamm.200310055.
[22]	M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer,, J. Comput. Phys., 218 (2006), 1. doi: 10.1016/j.jcp.2006.01.038.
[23]	M. Frank, A. Klar, E. W. Larsen and S. Yasuda, Time-dependent simplified $P_N$ approximation to the equations of radiative transfer,, J. Comput. Phys., 226 (2007), 2289. doi: 10.1016/j.jcp.2007.07.009.
[24]	M. Frank, A. Klar and R. Pinnau, Optimal control of glass cooling using Simplified $P_n$ theory,, Transp. Theory. Stat. Phys., 39 (2010), 282. doi: 10.1080/00411450.2010.533740.
[25]	E. M. Gelbard, "Applications of Spherical Harmonics Method to Reactor Problems,", Tech. Report WAPD-BT-20, (1960).
[26]	_____, "Simplified Spherical Harmonics Equations and their Use in Shielding Problems,", Tech. Report WAPD-T-1182, (1961).
[27]	_____, "Applications of the Simplified Spherical Harmonics Equations in Spherical Geometry,", Tech. Report WAPD-TM-294, (1962).
[28]	D. Givon, O. Hald and R. Kupferman, Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism,, Israel J. Math., 145 (2005), 221. doi: 10.1007/BF02786691.
[29]	T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite Dimensional Calculus,", Mathematics and its Applications, 253 (1993).
[30]	S. Karlin and L. S. Shapley, Geometry of moment spaces,, Mem. Amer. Math. Soc., (1953).
[31]	D. S. Kershaw, "Flux Limiting Nature's Own Way,", Tech. Report UCRL-78378, (1976).
[32]	D. A. Knoll, W. J. Rider and G. L. Olson, Method for non-equilibrium radiation diffusion,, J. Quant. Spectrosc. Radiat. Transfer, 63 (1999), 15. doi: 10.1016/S0022-4073(98)00132-0.
[33]	E. W. Larsen and J. R. Keller, Asymptotic solution of neutron transport problems for small mean free paths,, J. Math. Phys., 15 (1974), 75. doi: 10.1063/1.1666510.
[34]	E. W. Larsen, J. E. Morel and J. M. McGhee, Asymptotic derivation of the multigroup $P_1$ and simplified $P_N$ equations with anisotropic scattering,, Nucl. Sci. Eng., 123 (1996), 328.
[35]	E. W. Larsen and G. C. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry - I: analysis,, Nucl. Sci. Eng., 109 (1991), 49.
[36]	C. D. Levermore, Relating Eddington factors to flux limiters,, J. Quant. Spectrosc. Radiat. Transfer, 31 (1984), 149. doi: 10.1016/0022-4073(84)90112-2.
[37]	_____, "Transition Regime Models for Radiative Transport,", Presentation at IPAM: Grand challenge problems in computational astrophysics workshop on transfer phenomena, (2005).
[38]	R. G. McClarren, Theoretical aspects of the Simplified $P_n$ equations,, Transp. Theory. Stat. Phys., 39 (2010), 73. doi: 10.1080/00411450.2010.535088.
[39]	R. G. McClarren and C. D. Hauck, Robust and accurate filtered spherical Harmonics expansions for radiative transfer,, J. Comput. Phys., 229 (2010), 5597. doi: 10.1016/j.jcp.2010.03.043.
[40]	M. F. Modest, "Radiative Heat Transfer,", second ed., (1993).
[41]	H. Mori, Transport, collective motion and Brownian motion,, Prog. Theor. Phys., 33 (1965), 423. doi: 10.1143/PTP.33.423.
[42]	I. Müller and T. Ruggeri, "Rational Extended Thermodynamics,", second ed., (1993).
[43]	W. H. Reed, Spherical Harmonic solutions of the neutron transport equation from discrete ordinates code,, Nucl. Sci. Eng., 49 (1972), 10.
[44]	M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations,, Multiscale Model. Simul., 9 (2011), 1.
[45]	B. Seibold, Optimal prediction in molecular dynamics,, Monte Carlo Methods Appl., 10 (2004), 25. doi: 10.1515/156939604323091199.
[46]	B. Seibold and M. Frank, Optimal prediction for moment models: Crescendo diffusion and reordered equations,, Continuum Mech. Thermodyn., 21 (2009), 511. doi: 10.1007/s00161-009-0111-7.
[47]	M. Skibinsky, The range of the $(n+1)$th moment for distributions on $[0,1]$,, J. Appl. Probability, 4 (1967), 543. doi: 10.2307/3212220.
[48]	B. Su, Variable Eddington factors and flux limiters in radiative transfer,, Nucl. Sci. Eng., 137 (2001), 281.
[49]	D. I. Tomasevic and E. W. Larsen, The simplified $P_2$ approximation,, Nucl. Sci. Eng., 122 (1996), 309.
[50]	R. Turpault, M. Frank, B. Dubroca and A. Klar, Multigroup half space moment appproximations to the radiative heat transfer equations,, J. Comput. Phys., 198 (2004), 363. doi: 10.1016/j.jcp.2004.01.011.
[51]	R. Zwanzig, Problems in nonlinear transport theory,, in, (1980), 198. doi: 10.1007/BFb0025619.

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