# American Institute of Mathematical Sciences

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

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

 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

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.
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. Google Scholar [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. Google Scholar [3] Y. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis,", Kluwer Academic Publishers, (1995). Google Scholar [4] P. S. Brantley and E. W. Larsen, The simplified $P_3$ approximation,, Nucl. Sci. Eng., 134 (2000). Google Scholar [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). Google Scholar [6] T. A. Brunner, "Forms of Approximate Radiation Transport,", Technical Report SAND2002-1778, (2002), 2002. Google Scholar [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. Google Scholar [8] S. Chandrasekhar, On the radiative equilibrium of a stellar atmosphere,, Astrophys. J., 99 (1944), 180. doi: 10.1086/144606. Google Scholar [9] _____, "Radiative Transfer,", Dover Publications, (1960). Google Scholar [10] A. J. Chorin, Conditional expectations and renormalization,, Multiscale Model. Simul., 1 (2003), 105. doi: 10.1137/S1540345902405556. Google Scholar [11] A. J. Chorin and O. H. Hald, "Stochastic Tools in Mathematics and Science,", Surveys and Tutorials in the Applied Mathematical Sciences, 1 (2006). Google Scholar [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. Google Scholar [13] _____, Non-Markovian optimal prediction, Monte Carlo adn Probablilistic Methods for Patial Differential Equations (Monte Carlo, 2000),, Monte Carlo Meth. Appl., 7 (2001), 99. Google Scholar [14] _____, Optimal prediction with memory,, Physica D, 166 (2002), 239. doi: 10.1016/S0167-2789(02)00446-3. Google Scholar [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. Google Scholar [16] _____, Unresolved computation and optimal predictions,, Comm. Pure Appl. Math., 52 (1998), 1231. Google Scholar [17] _____, "On the Prediction of Large-Scale Dynamics using Unresolved Computations,", Nonlinear Partial Differential Equations (Evanston, 238 (1999), 53. Google Scholar [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. Google Scholar [19] B. Davison, "Neutron Transport Theory,", Clarendon Press, (1958). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] E. M. Gelbard, "Applications of Spherical Harmonics Method to Reactor Problems,", Tech. Report WAPD-BT-20, (1960). Google Scholar [26] _____, "Simplified Spherical Harmonics Equations and their Use in Shielding Problems,", Tech. Report WAPD-T-1182, (1961). Google Scholar [27] _____, "Applications of the Simplified Spherical Harmonics Equations in Spherical Geometry,", Tech. Report WAPD-TM-294, (1962). Google Scholar [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. Google Scholar [29] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite Dimensional Calculus,", Mathematics and its Applications, 253 (1993). Google Scholar [30] S. Karlin and L. S. Shapley, Geometry of moment spaces,, Mem. Amer. Math. Soc., (1953). Google Scholar [31] D. S. Kershaw, "Flux Limiting Nature's Own Way,", Tech. Report UCRL-78378, (1976). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [37] _____, "Transition Regime Models for Radiative Transport,", Presentation at IPAM: Grand challenge problems in computational astrophysics workshop on transfer phenomena, (2005). Google Scholar [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. Google Scholar [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. Google Scholar [40] M. F. Modest, "Radiative Heat Transfer,", second ed., (1993). Google Scholar [41] H. Mori, Transport, collective motion and Brownian motion,, Prog. Theor. Phys., 33 (1965), 423. doi: 10.1143/PTP.33.423. Google Scholar [42] I. Müller and T. Ruggeri, "Rational Extended Thermodynamics,", second ed., (1993). Google Scholar [43] W. H. Reed, Spherical Harmonic solutions of the neutron transport equation from discrete ordinates code,, Nucl. Sci. Eng., 49 (1972), 10. Google Scholar [44] M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations,, Multiscale Model. Simul., 9 (2011), 1. Google Scholar [45] B. Seibold, Optimal prediction in molecular dynamics,, Monte Carlo Methods Appl., 10 (2004), 25. doi: 10.1515/156939604323091199. Google Scholar [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. Google Scholar [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. Google Scholar [48] B. Su, Variable Eddington factors and flux limiters in radiative transfer,, Nucl. Sci. Eng., 137 (2001), 281. Google Scholar [49] D. I. Tomasevic and E. W. Larsen, The simplified $P_2$ approximation,, Nucl. Sci. Eng., 122 (1996), 309. Google Scholar [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. Google Scholar [51] R. Zwanzig, Problems in nonlinear transport theory,, in, (1980), 198. doi: 10.1007/BFb0025619. Google Scholar

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. Google Scholar [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. Google Scholar [3] Y. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis,", Kluwer Academic Publishers, (1995). Google Scholar [4] P. S. Brantley and E. W. Larsen, The simplified $P_3$ approximation,, Nucl. Sci. Eng., 134 (2000). Google Scholar [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). Google Scholar [6] T. A. Brunner, "Forms of Approximate Radiation Transport,", Technical Report SAND2002-1778, (2002), 2002. Google Scholar [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. Google Scholar [8] S. Chandrasekhar, On the radiative equilibrium of a stellar atmosphere,, Astrophys. J., 99 (1944), 180. doi: 10.1086/144606. Google Scholar [9] _____, "Radiative Transfer,", Dover Publications, (1960). Google Scholar [10] A. J. Chorin, Conditional expectations and renormalization,, Multiscale Model. Simul., 1 (2003), 105. doi: 10.1137/S1540345902405556. Google Scholar [11] A. J. Chorin and O. H. Hald, "Stochastic Tools in Mathematics and Science,", Surveys and Tutorials in the Applied Mathematical Sciences, 1 (2006). Google Scholar [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. Google Scholar [13] _____, Non-Markovian optimal prediction, Monte Carlo adn Probablilistic Methods for Patial Differential Equations (Monte Carlo, 2000),, Monte Carlo Meth. Appl., 7 (2001), 99. Google Scholar [14] _____, Optimal prediction with memory,, Physica D, 166 (2002), 239. doi: 10.1016/S0167-2789(02)00446-3. Google Scholar [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. Google Scholar [16] _____, Unresolved computation and optimal predictions,, Comm. Pure Appl. Math., 52 (1998), 1231. Google Scholar [17] _____, "On the Prediction of Large-Scale Dynamics using Unresolved Computations,", Nonlinear Partial Differential Equations (Evanston, 238 (1999), 53. Google Scholar [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. Google Scholar [19] B. Davison, "Neutron Transport Theory,", Clarendon Press, (1958). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] E. M. Gelbard, "Applications of Spherical Harmonics Method to Reactor Problems,", Tech. Report WAPD-BT-20, (1960). Google Scholar [26] _____, "Simplified Spherical Harmonics Equations and their Use in Shielding Problems,", Tech. Report WAPD-T-1182, (1961). Google Scholar [27] _____, "Applications of the Simplified Spherical Harmonics Equations in Spherical Geometry,", Tech. Report WAPD-TM-294, (1962). Google Scholar [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. Google Scholar [29] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite Dimensional Calculus,", Mathematics and its Applications, 253 (1993). Google Scholar [30] S. Karlin and L. S. Shapley, Geometry of moment spaces,, Mem. Amer. Math. Soc., (1953). Google Scholar [31] D. S. Kershaw, "Flux Limiting Nature's Own Way,", Tech. Report UCRL-78378, (1976). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [37] _____, "Transition Regime Models for Radiative Transport,", Presentation at IPAM: Grand challenge problems in computational astrophysics workshop on transfer phenomena, (2005). Google Scholar [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. Google Scholar [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. Google Scholar [40] M. F. Modest, "Radiative Heat Transfer,", second ed., (1993). Google Scholar [41] H. Mori, Transport, collective motion and Brownian motion,, Prog. Theor. Phys., 33 (1965), 423. doi: 10.1143/PTP.33.423. Google Scholar [42] I. Müller and T. Ruggeri, "Rational Extended Thermodynamics,", second ed., (1993). Google Scholar [43] W. H. Reed, Spherical Harmonic solutions of the neutron transport equation from discrete ordinates code,, Nucl. Sci. Eng., 49 (1972), 10. Google Scholar [44] M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations,, Multiscale Model. Simul., 9 (2011), 1. Google Scholar [45] B. Seibold, Optimal prediction in molecular dynamics,, Monte Carlo Methods Appl., 10 (2004), 25. doi: 10.1515/156939604323091199. Google Scholar [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. Google Scholar [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. Google Scholar [48] B. Su, Variable Eddington factors and flux limiters in radiative transfer,, Nucl. Sci. Eng., 137 (2001), 281. Google Scholar [49] D. I. Tomasevic and E. W. Larsen, The simplified $P_2$ approximation,, Nucl. Sci. Eng., 122 (1996), 309. Google Scholar [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. Google Scholar [51] R. Zwanzig, Problems in nonlinear transport theory,, in, (1980), 198. doi: 10.1007/BFb0025619. Google Scholar
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