Overview and construction of meshfree basis functions: from moving least squares to entropy approximants

In this paper, an overview of the construction of meshfree basis functions is presented, with particular emphasis on moving least‐squares approximants, natural neighbour‐based polygonal interpolants, and entropy approximants. The use of information‐theoretic variational principles to derive approxim...

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Bibliographic Details
Published in:International journal for numerical methods in engineering 2007-04, Vol.70 (2), p.181-205
Main Authors: Sukumar, N., Wright, R. W.
Format: Article
Language:English
Subjects:
Computational techniques
convex approximation schemes
Exact sciences and technology
Fundamental areas of phenomenology (including applications)
information theory
JAVA programming
Mathematical methods in physics
natural neighbours
Physics
radial basis functions
relative entropy
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Summary:In this paper, an overview of the construction of meshfree basis functions is presented, with particular emphasis on moving least‐squares approximants, natural neighbour‐based polygonal interpolants, and entropy approximants. The use of information‐theoretic variational principles to derive approximation schemes is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints. The maximization (minimization) of the Shannon–Jaynes entropy functional (relative entropy functional) is used to unify the construction of globally and locally supported convex approximation schemes. A JAVA applet is used to visualize the meshfree basis functions, and comparisons and links between different meshfree approximation schemes are presented. Copyright © 2006 John Wiley & Sons, Ltd.
ISSN:0029-5981
1097-0207
DOI:10.1002/nme.1885