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

Abstract

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. Essentially, this is because the moments are not continuous functionals with respect to the L 1 topology. In this paper, we give a geometric description of these so-called degenerate densities in the most general possible 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 and, under reasonable assumptions, that this set is small in both a topological and a measuretheoretic sense. This result is important for further assessment and implementation of entropy-based moment closures.

High-Order Entropy-Based Closures for Linear Transport in Slab Geometry II: A Computational Study of the Optimization Problem

“…In general the answer to the first question is "no." The lack of existence is related to the fact that the constraints in (2.6) are not always continuous in the L 1 norm [29,32,33,57]. However, for the case under consideration, the domain of integration is bounded and the components of m are bounded on that domain.…”

“…Roughly speaking, a vector is realizable if it is the moment of a kinetic distribution. In some applications, there are realizable vectors (on the boundary of the set of realizable moments) for which the defining optimization problem has no solution [29,32,33,57] and so the closure is not well-defined.…”

“…In transport and kinetic theory, entropy-based closures are used to derive moment models which retain fundamental properties of the underlying kinetic equations such as hyperbolicity, entropy dissipation, and positivity. The resulting models have been studied extensively in the areas of extended thermodynamics [18,48], gas dynamics [24,29,32,33,40,42,57], semiconductors [2][3][4][5]26,31,34,41,56], quantum fluids [16,19], radiative transport [9-12, 20-22, 28, 30, 46, 47, 59, 62], and phonon transport in solids [19]. In spite of their attractive mathematical properties and wide application, entropy methods still suffer from several short-comings.…”

High-order entropy-based closures for linear transport in slab geometry

“…For gas dynamics, the formal properties of entropy-based models were elucidated in [26]. However, it is also known [17,20,21,38] that the defining optimization problem in this case is ill-posed. As a result, alternative approaches are currently being pursued which regularize the problem in some suitable fashion; see [15] and references therein.…”

Error estimate and convergence analysis of moment-preserving discrete approximations of continuous distributions

“…For the relation to Bayesian inference, see Van Campenhout and Cover [48] and Csiszár [13]. For the duality theory of entropy maximization, see [7,15,20]. For numerical algorithms for computing maximum entropy densities, see [1,2,5].…”