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On the Five-Moment Hamburger Maximum Entropy Reconstruction

Summy, D. P. and Pullin, D. I. (2018) On the Five-Moment Hamburger Maximum Entropy Reconstruction. Journal of Statistical Physics, 172 (3). pp. 854-879. ISSN 0022-4715. doi:10.1007/s10955-018-2034-9.

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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 (μ_3, μ_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 ( μ_3, μ_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.

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Additional Information:© Springer Science+Business Media 2018. Received: 25 September 2017 / Accepted: 6 April 2018 / Published online: 14 May 2018. This research was partially supported by the National Science Foundation under Award Number: DMS-1418903.
Group:GALCIT, Graduate Aeronautical Laboratories (Fluid Mechanics)
Funding AgencyGrant Number
Subject Keywords:Maximum entropy closure · Moments · Five-moment reconstruction
Issue or Number:3
Classification Code:Mathematics Subject Classification: 00-01; 99-00
Record Number:CaltechAUTHORS:20180801-140416478
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Official Citation:Summy, D.P. & Pullin, D.I. J Stat Phys (2018) 172: 854.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88470
Deposited By: George Porter
Deposited On:01 Aug 2018 21:11
Last Modified:16 Nov 2021 00:27

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