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Construction of polygonal interpolants: a maximum entropy approach

2004; Wiley; Volume: 61; Issue: 12 Linguagem: Inglês

10.1002/nme.1193

1097-0207

N. Sukumar,

Statistical Mechanics and Entropy

Abstract In this paper, we establish a link between maximizing (information‐theoretic) entropy and the construction of polygonal interpolants. The determination of shape functions on n ‐gons ( n >3) leads to a non‐unique under‐determined system of linear equations. The barycentric co‐ordinates ϕ i , which form a partition of unity, are associated with discrete probability measures, and the linear reproducing conditions are the counterpart of the expectations of a linear function. The ϕ i are computed by maximizing the uncertainty H (ϕ 1 ,ϕ 2 ,…,ϕ n )=−∑ ϕ i logϕ i , subject to the above constraints. The description is expository in nature, and the numerical results via the maximum entropy (MAXENT) formulation are compared to those obtained from a few distinct polygonal interpolants. The maximum entropy formulation leads to a feasible solution for ϕ i in any convex or non‐convex polygon. This study is an instance of the application of the maximum entropy principle, wherein least‐biased inference is made on the basis of incomplete information. Copyright © 2004 John Wiley & Sons, Ltd.