×

zbMATH — the first resource for mathematics

Construction of polygonal interpolants: a maximum entropy approach. (English) Zbl 1073.65505
Summary: 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 \(\varphi_i\), 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 \(\varphi_i\) are computed by maximizing the uncertainty \(H(\varphi_1,\varphi_2, \dots,\varphi_n)=-\sum^n_{i=1}\varphi_i\log\varphi_i\), 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 \(\varphi_i\) 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.

MSC:
65D05 Numerical interpolation
41A05 Interpolation in approximation theory
94A17 Measures of information, entropy
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Shannon, The Bell Systems Technical Journal 27 pp 379– (1948) · Zbl 1154.94303 · doi:10.1002/j.1538-7305.1948.tb01338.x
[2] Jaynes, Physical Review 106 pp 620– (1957)
[3] Jaynes, Physical Review 108 pp 171– (1957)
[4] Jaynes, Proceedings of the IEEE 70 pp 939– (1982)
[5] . (ed.). E.T. Jaynes, Paper on Probability, Statistics and Statistical Physics. Kluwer Academic Publishers: Dordrecht, The Netherlands, 1989.
[6] . Mathematical Foundations of Information Theory. Dover: New York, NY, 1957. · Zbl 0088.10404
[7] . Maximum-Entropy Models in Science and Engineering (1st (revised) edn). Wiley: New Delhi, India, 1993.
[8] Beltzer, International Journal of Solids and Structures 33 pp 3549– (1996)
[9] Sukumar, International Journal for Numerical Methods in Engineering 61 pp 2045– (2004)
[10] Tabarraei, Finite Elements in Analysis and Design (2005)
[11] . Concentration of distributions at entropy maxima. In E.T. Jaynes, Paper on Probability, Statistics and Statistical Physics, (ed.). Kluwer Academic Publishers: Dordrecht, The Netherlands, 1989; 317-336.
[12] Penrose, Proceedings of the Cambridge Philosophical Society 51 pp 406– (1955)
[13] , , . An algorithm for determining the Lagrange parameters in the maximal entropy formalism. In The Maximum Entropy Formalism, , (eds). MIT Press: Cambridge, MA, 1978; 206-209.
[14] Agmon, Journal of Computational Physics 30 pp 250– (1979)
[15] Alhassid, Chemical Physics Letters 53 pp 22– (1978)
[16] , , , . Numerical Recipes in Fortran: The Art of Scientific Computing (2nd edn). Cambridge University Press: New York, NY, 1992. · Zbl 0778.65002
[17] , , . Entropy Optimization and Mathematical Programming. Kluwer Academic Publishers: Dordrecht, The Netherlands, 1997. · Zbl 0933.90051 · doi:10.1007/978-1-4615-6131-6
[18] . Interpolation and Approximation. Dover: New York, NY, 1975.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.