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

Previous Article
Non--local macroscopic models based on Gaussian closures for the Spizer-Härm regime
KRM Home
This Issue
Next Article
Fast diffusion equations: Matching large time asymptotics by relative entropy methods

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
Related Papers

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 and 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), 544-550. 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, 1-13.
[3]	Y. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis," Kluwer Academic Publishers, Dordrecht, 1995.
[4]	P. S. Brantley and E. W. Larsen, The simplified $P_3$ approximation, Nucl. Sci. Eng., 134 (2000), 1.
[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, Lawrence Livermore National Laboratory, May 2000.
[6]	T. A. Brunner, "Forms of Approximate Radiation Transport," Technical Report SAND2002-1778, Sandia National Laboratories, July 2002.
[7]	T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport, J. Comput. Phys., 210 (2005), 386-399. doi: 10.1016/j.jcp.2005.04.011.
[8]	S. Chandrasekhar, On the radiative equilibrium of a stellar atmosphere, Astrophys. J., 99 (1944), 180-190. doi: 10.1086/144606.
[9]	_____, "Radiative Transfer," Dover Publications, Inc., New York, 1960.
[10]	A. J. Chorin, Conditional expectations and renormalization, Multiscale Model. Simul., 1 (2003), 105-118. 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, Springer, New York, 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), 2968-2973. 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), 99-109.
[14]	_____, Optimal prediction with memory, Physica D, 166 (2002), 239-257. 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), 4094-4098. doi: 10.1073/pnas.95.8.4094.
[16]	_____, Unresolved computation and optimal predictions, Comm. Pure Appl. Math., 52 (1998), 1231-1254.
[17]	_____, "On the Prediction of Large-Scale Dynamics using Unresolved Computations," Nonlinear Partial Differential Equations (Evanston, IL, 1998), Contemp. Math., 238, AMS, Providence, RI, (1999), 53-75.
[18]	A. J. Chorin and P. Stinis, Problem reduction, renormalization, and memory, Comm. Appl. Math. Comp. Sc., 1 (2006), 1-27. doi: 10.2140/camcos.2006.1.1.
[19]	B. Davison, "Neutron Transport Theory," Clarendon Press, Oxford, 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), 915-920.
[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), 853-858. 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), 1-18. 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), 2289-2305. 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), 282-311. doi: 10.1080/00411450.2010.533740.
[25]	E. M. Gelbard, "Applications of Spherical Harmonics Method to Reactor Problems," Tech. Report WAPD-BT-20, Bettis Atomic Power Laboratory, 1960.
[26]	_____, "Simplified Spherical Harmonics Equations and their Use in Shielding Problems," Tech. Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961.
[27]	_____, "Applications of the Simplified Spherical Harmonics Equations in Spherical Geometry," Tech. Report WAPD-TM-294, Bettis Atomic Power Laboratory, 1962.
[28]	D. Givon, O. Hald and R. Kupferman, Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism, Israel J. Math., 145 (2005), 221-241. 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, Kluwer Academic Publishers Group, Dordrecht, 1993.
[30]	S. Karlin and L. S. Shapley, Geometry of moment spaces, Mem. Amer. Math. Soc., (1953), 93 pp.
[31]	D. S. Kershaw, "Flux Limiting Nature's Own Way," Tech. Report UCRL-78378, Lawrence Livermore National Laboratory, 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), 15-29. 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), 75-81. 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), 328-342.
[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), 49-75.
[36]	C. D. Levermore, Relating Eddington factors to flux limiters, J. Quant. Spectrosc. Radiat. Transfer, 31 (1984), 149-160. 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), 73-109. 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), 5597-5614. doi: 10.1016/j.jcp.2010.03.043.
[40]	M. F. Modest, "Radiative Heat Transfer," second ed., Academic Press, 1993.
[41]	H. Mori, Transport, collective motion and Brownian motion, Prog. Theor. Phys., 33 (1965), 423-455. doi: 10.1143/PTP.33.423.
[42]	I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," second ed., Springer, New York, 1993.
[43]	W. H. Reed, Spherical Harmonic solutions of the neutron transport equation from discrete ordinates code, Nucl. Sci. Eng., 49 (1972), 10-19.
[44]	M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations, Multiscale Model. Simul., 9 (2011), 1-28.
[45]	B. Seibold, Optimal prediction in molecular dynamics, Monte Carlo Methods Appl., 10 (2004), 25-50. 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), 511-527. 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), 543-552. doi: 10.2307/3212220.
[48]	B. Su, Variable Eddington factors and flux limiters in radiative transfer, Nucl. Sci. Eng., 137 (2001), 281-297.
[49]	D. I. Tomasevic and E. W. Larsen, The simplified $P_2$ approximation, Nucl. Sci. Eng., 122 (1996), 309-325.
[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), 363-371. doi: 10.1016/j.jcp.2004.01.011.
[51]	R. Zwanzig, Problems in nonlinear transport theory, in "Systems Far from Equilibrium" (Berlin) (ed., L. Garrido), Springer, (1980), 198-221. 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), 544-550. 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, 1-13.
[3]	Y. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis," Kluwer Academic Publishers, Dordrecht, 1995.
[4]	P. S. Brantley and E. W. Larsen, The simplified $P_3$ approximation, Nucl. Sci. Eng., 134 (2000), 1.
[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, Lawrence Livermore National Laboratory, May 2000.
[6]	T. A. Brunner, "Forms of Approximate Radiation Transport," Technical Report SAND2002-1778, Sandia National Laboratories, July 2002.
[7]	T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport, J. Comput. Phys., 210 (2005), 386-399. doi: 10.1016/j.jcp.2005.04.011.
[8]	S. Chandrasekhar, On the radiative equilibrium of a stellar atmosphere, Astrophys. J., 99 (1944), 180-190. doi: 10.1086/144606.
[9]	_____, "Radiative Transfer," Dover Publications, Inc., New York, 1960.
[10]	A. J. Chorin, Conditional expectations and renormalization, Multiscale Model. Simul., 1 (2003), 105-118. 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, Springer, New York, 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), 2968-2973. 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), 99-109.
[14]	_____, Optimal prediction with memory, Physica D, 166 (2002), 239-257. 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), 4094-4098. doi: 10.1073/pnas.95.8.4094.
[16]	_____, Unresolved computation and optimal predictions, Comm. Pure Appl. Math., 52 (1998), 1231-1254.
[17]	_____, "On the Prediction of Large-Scale Dynamics using Unresolved Computations," Nonlinear Partial Differential Equations (Evanston, IL, 1998), Contemp. Math., 238, AMS, Providence, RI, (1999), 53-75.
[18]	A. J. Chorin and P. Stinis, Problem reduction, renormalization, and memory, Comm. Appl. Math. Comp. Sc., 1 (2006), 1-27. doi: 10.2140/camcos.2006.1.1.
[19]	B. Davison, "Neutron Transport Theory," Clarendon Press, Oxford, 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), 915-920.
[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), 853-858. 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), 1-18. 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), 2289-2305. 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), 282-311. doi: 10.1080/00411450.2010.533740.
[25]	E. M. Gelbard, "Applications of Spherical Harmonics Method to Reactor Problems," Tech. Report WAPD-BT-20, Bettis Atomic Power Laboratory, 1960.
[26]	_____, "Simplified Spherical Harmonics Equations and their Use in Shielding Problems," Tech. Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961.
[27]	_____, "Applications of the Simplified Spherical Harmonics Equations in Spherical Geometry," Tech. Report WAPD-TM-294, Bettis Atomic Power Laboratory, 1962.
[28]	D. Givon, O. Hald and R. Kupferman, Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism, Israel J. Math., 145 (2005), 221-241. 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, Kluwer Academic Publishers Group, Dordrecht, 1993.
[30]	S. Karlin and L. S. Shapley, Geometry of moment spaces, Mem. Amer. Math. Soc., (1953), 93 pp.
[31]	D. S. Kershaw, "Flux Limiting Nature's Own Way," Tech. Report UCRL-78378, Lawrence Livermore National Laboratory, 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), 15-29. 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), 75-81. 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), 328-342.
[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), 49-75.
[36]	C. D. Levermore, Relating Eddington factors to flux limiters, J. Quant. Spectrosc. Radiat. Transfer, 31 (1984), 149-160. 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), 73-109. 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), 5597-5614. doi: 10.1016/j.jcp.2010.03.043.
[40]	M. F. Modest, "Radiative Heat Transfer," second ed., Academic Press, 1993.
[41]	H. Mori, Transport, collective motion and Brownian motion, Prog. Theor. Phys., 33 (1965), 423-455. doi: 10.1143/PTP.33.423.
[42]	I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," second ed., Springer, New York, 1993.
[43]	W. H. Reed, Spherical Harmonic solutions of the neutron transport equation from discrete ordinates code, Nucl. Sci. Eng., 49 (1972), 10-19.
[44]	M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations, Multiscale Model. Simul., 9 (2011), 1-28.
[45]	B. Seibold, Optimal prediction in molecular dynamics, Monte Carlo Methods Appl., 10 (2004), 25-50. 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), 511-527. 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), 543-552. doi: 10.2307/3212220.
[48]	B. Su, Variable Eddington factors and flux limiters in radiative transfer, Nucl. Sci. Eng., 137 (2001), 281-297.
[49]	D. I. Tomasevic and E. W. Larsen, The simplified $P_2$ approximation, Nucl. Sci. Eng., 122 (1996), 309-325.
[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), 363-371. doi: 10.1016/j.jcp.2004.01.011.
[51]	R. Zwanzig, Problems in nonlinear transport theory, in "Systems Far from Equilibrium" (Berlin) (ed., L. Garrido), Springer, (1980), 198-221. doi: 10.1007/BFb0025619.

[1]	Graham W. Alldredge, Ruo Li, Weiming Li. Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry. Kinetic and Related Models, 2016, 9 (2) : 237-249. doi: 10.3934/krm.2016.9.237
[2]	Arnaud Debussche, Sylvain De Moor, Julien Vovelle. Diffusion limit for the radiative transfer equation perturbed by a Wiener process. Kinetic and Related Models, 2015, 8 (3) : 467-492. doi: 10.3934/krm.2015.8.467
[3]	Yong-Jung Kim. A generalization of the moment problem to a complex measure space and an approximation technique using backward moments. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 187-207. doi: 10.3934/dcds.2011.30.187
[4]	Martin Frank, Cory D. Hauck, Edgar Olbrant. Perturbed, entropy-based closure for radiative transfer. Kinetic and Related Models, 2013, 6 (3) : 557-587. doi: 10.3934/krm.2013.6.557
[5]	J-F. Clouët, R. Sentis. Milne problem for non-grey radiative transfer. Kinetic and Related Models, 2009, 2 (2) : 345-362. doi: 10.3934/krm.2009.2.345
[6]	Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic and Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
[7]	Grégoire Allaire, Zakaria Habibi. Second order corrector in the homogenization of a conductive-radiative heat transfer problem. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 1-36. doi: 10.3934/dcdsb.2013.18.1
[8]	Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic and Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1
[9]	I-Kun Chen, Daisuke Kawagoe. Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography. Inverse Problems and Imaging, 2019, 13 (2) : 337-351. doi: 10.3934/ipi.2019017
[10]	Chen Li, Fajie Wei, Shenghan Zhou. Prediction method based on optimization theory and its application. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1213-1221. doi: 10.3934/dcdss.2015.8.1213
[11]	Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323
[12]	Jacky Cresson. The transfer lemma for Graff tori and Arnold diffusion time. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 787-800. doi: 10.3934/dcds.2001.7.787
[13]	Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[14]	Andrea Tosin, Paolo Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Networks and Heterogeneous Media, 2011, 6 (3) : 561-596. doi: 10.3934/nhm.2011.6.561
[15]	Jakub Cupera. Diffusion approximation of neuronal models revisited. Mathematical Biosciences & Engineering, 2014, 11 (1) : 11-25. doi: 10.3934/mbe.2014.11.11
[16]	Junying Hu, Xiaofei Qian, Jun Pei, Changchun Tan, Panos M. Pardalos, Xinbao Liu. A novel quality prediction method based on feature selection considering high dimensional product quality data. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021099
[17]	Yanfei You, Suhong Jiang. Improved Lagrangian-PPA based prediction correction method for linearly constrained convex optimization. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021174
[18]	Xiaojiao Tong, Felix F. Wu, Jifeng Su. Quadratic approximation and visualization of online contract-based available transfer capability region of power systems. Journal of Industrial and Management Optimization, 2008, 4 (3) : 553-563. doi: 10.3934/jimo.2008.4.553
[19]	Aldana M. González Montoro, Ricardo Cao, Christel Faes, Geert Molenberghs, Nelson Espinosa, Javier Cudeiro, Jorge Mariño. Cross nearest-spike interval based method to measure synchrony dynamics. Mathematical Biosciences & Engineering, 2014, 11 (1) : 27-48. doi: 10.3934/mbe.2014.11.27
[20]	Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056

2020 Impact Factor: 1.432

Tools

Metrics

Other articles
by authors

on AIMS
Martin Frank Benjamin Seibold
on Google Scholar
Martin Frank Benjamin Seibold

[Back to Top]