Journal of Statistical Physics

, Volume 172, Issue 3, pp 854–879 | Cite as

On the Five-Moment Hamburger Maximum Entropy Reconstruction

  • D. P. Summy
  • D. I. PullinEmail author


We consider the Maximum Entropy Reconstruction (MER) as a solution to the five-moment truncated Hamburger moment problem in one dimension. In the case of five monomial moment constraints, the probability density function (PDF) of the MER takes the form of the exponential of a quartic polynomial. This implies a possible bimodal structure in regions of moment space. An analytical model is developed for the MER PDF applicable near a known singular line in a centered, two-component, third- and fourth-order moment (\(\mu _3\), \(\mu _4\)) space, consistent with the general problem of five moments. The model consists of the superposition of a perturbed, centered Gaussian PDF and a small-amplitude packet of PDF-density, called the outlying moment packet (OMP), sitting far from the mean. Asymptotic solutions are obtained which predict the shape of the perturbed Gaussian and both the amplitude and position on the real line of the OMP. The asymptotic solutions show that the presence of the OMP gives rise to an MER solution that is singular along a line in (\(\mu _3\), \(\mu _4\)) space emanating from, but not including, the point representing a standard normal distribution, or thermodynamic equilibrium. We use this analysis of the OMP to develop a numerical regularization of the MER, creating a procedure we call the Hybrid MER (HMER). Compared with the MER, the HMER is a significant improvement in terms of robustness and efficiency while preserving accuracy in its prediction of other important distribution features, such as higher order moments.


Maximum entropy closure Moments Five-moment reconstruction 

Mathematics Subject Classification

00-01 99-00 



This research was partially supported by the National Science Foundation under Award Number: DMS-1418903.


  1. 1.
    Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bobylev, A.: The Chapman–Enskog and Grad methods for solving the Boltzmann equation. Akad Nauk SSSR Dokl 262, 71–75 (1982)ADSMathSciNetGoogle Scholar
  3. 3.
    Levermore, C.D., Morokoff, W.J.: The Gaussian moment closure for gas dynamics. SIAM J. Appl. Math. 59, 72–96 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgault, F., Makarenko, A.A., Williams, S.B., Grocholsky, B., Durrant-Whyte, H.F.: Information based adaptive robotic exploration. In: IEEE/RSJ IEEE International Conference on Intelligent Robots and Systems, vol. 1, pp. 540–545 (2002)Google Scholar
  5. 5.
    Fraser, I.: An application of maximum entropy estimation: the demand for meat in the United Kingdom. Appl. Econ. 32, 45–59 (2000)CrossRefGoogle Scholar
  6. 6.
    Gull, S.F., Daniell, G.J.: Image reconstruction from incomplete and noisy data. Nature V 272, 686–690 (1978)ADSCrossRefGoogle Scholar
  7. 7.
    Bird, G.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford (1994)Google Scholar
  8. 8.
    Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E 52, R5760–R5763 (1995)ADSCrossRefGoogle Scholar
  9. 9.
    Jaynes, E.T., Bretthorst, G.L.: Probability Theory: The Logic of Science, vol. 200. Cambridge University Press, Cambridge (1996)Google Scholar
  10. 10.
    Jaynes, E.: The relation of Bayesian and maximum entropy methods. Maximum-Entropy and Bayesian Methods in Science and Engineering. Springer, Berlin (1988)Google Scholar
  11. 11.
    Junk, M.: Domain of definition of Levermore’s five-moment system. J. Stat. Phys. 93, 1143–1167 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Junk, M.: Maximum entropy for reduced moment problems. Math. Models Methods Appl. Sci. 10, 1001–1025 (2000)MathSciNetzbMATHGoogle Scholar
  13. 13.
    McDonald, J., Torrilhon, M.: Affordable robust moment closures for CFD based on the maximum-entropy hierarchy. J. Comput. Phys. 251, 500–523 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Junk, M., Unterreiter, A.: Maximum entropy moment systems and Galilean invariance. Contin. Mech. Thermodyn. 14, 563–576 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    McDonald, J.G., Groth, C.P.: Numerical solution of maximum-entropy-based hyperbolic moment closures for the prediction of viscous heat-conducting gaseous flows. In: Kuzmin, A. (ed.) Computational Fluid Dynamics 2010: Proceedings of the Sixth International Conference on Computational Fluid Dynamics, ICCFD6, St Petersburg, Russia, 12–16 July 2010, pp. 653–659. Springer, Berlin (2011)Google Scholar
  16. 16.
    Aheizer, N.I., Kemmer, N.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd, Edinburgh (1965)Google Scholar
  17. 17.
    Shohat, J.A., Tamarkin, J.D.: The Problem of Moments. American Mathematical Society, New York (1943)CrossRefzbMATHGoogle Scholar
  18. 18.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  19. 19.
    Shannon, C.E.: A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev. 5, 3–55 (2001)CrossRefGoogle Scholar
  20. 20.
    Mead, L.R., Papanicolaou, N.: Maximum entropy in the problem of moments. J. Math. Phys. 25, 2404–2417 (1984)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Tribus, M.: Chapter Five: the principle of maximum entropy. In: Tribus, M. (ed.) Rational Descriptions, Decisions and Designs. Pergamon Press, New York (1969)Google Scholar
  22. 22.
    Hildebrand, F.B.: Introduction to Numerical Analysis. Courier Corporation, North Chelmsford (1987)zbMATHGoogle Scholar
  23. 23.
    Young, D.M., Gregory, R.T.: A Survey of Numerical Mathematics, vol. 1. Courier Corporation, North Chelmsford (1988)zbMATHGoogle Scholar
  24. 24.
    Junk, M.: Maximum entropy moment problems and extended Euler equations. Transport in Transition Regimes, pp. 189–198. Springer, Berlin (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate Aerospace Laboratories California Institute of TechnologyPasadenaUSA
  2. 2.Thermal Sciences Exponent, Inc.Menlo ParkUSA

Personalised recommendations