NDLI: Convex duality and entropy-based moment closures: Characterizing degenerate densities

Content Provider	IEEE Xplore Digital Library
Author	Hauck, C.D. Levermore, C.D. Tits, A.L.
Copyright Year	2008
Description	Author affiliation: Dept. of Electr. & Comput. Eng., Univ. of Maryland, College Park, MD, USA (Tits, A.L.) \|\| Comput. Phys. Group, Los Alamos Nat. Lab., Los Alamos, NM, USA (Hauck, C.D.) \|\| Dept. of Math., Univ. of Maryland, College Park, MD, USA (Levermore, C.D.)
Abstract	A common method for constructing a function from a finite set of moments is to solve a constrained minimization problem. The idea is to find, among all functions with the given moments, that function which minimizes a physically motivated, strictly convex functional. In the kinetic theory of gases, this functional is the kinetic entropy; the given moments are macroscopic densities; and the solution to the constrained minimization problem is used to formally derive a closed system of partial differential equations which describe how the macroscopic densities evolve in time. Moment equations are useful because they simplify the kinetic, phase-space description of a gas, and with entropy-based closures, they retain many of the fundamental properties of kinetic transport. Unfortunately, in many situations, macroscopic densities can take on values for which the constrained minimization problem does not have a solution. In this paper, we give a geometric description of these so-called degenerate densities in a very general setting. Our key tool is the complementary slackness condition that is derived from a dual formulation of a minimization problem with relaxed constraints. We show that the set of degenerate densities is a union of convex cones defined by the complementary slackness conditions and, under reasonable assumptions, that this set is small in both a topological and a measure-theoretic sense. This result is important for further assessment and implementation of entropy-based moment closures. An expanded version of this work can be found in [Hauck et al., SIAM J. Contr. Optim., Vol. 47, 2008, pp. 1977-2015].
Starting Page	5092
Ending Page	5097
File Size	218931
Page Count	6
File Format	PDF
ISBN	9781424431236
ISSN	01912216
DOI	10.1109/CDC.2008.4739510
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2008-12-09
Publisher Place	Mexico
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Kinetic theory Entropy Boltzmann equation Distribution functions Mathematics Educational institutions Minimization methods Gases Constraint theory Partial differential equations
Content Type	Text
Resource Type	Article

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