Maximum Entropy Approximation

• Published in: Bayesian Inference and Maximum Entropy Methods in Science and Engineering Vol. 803; pp. 337 - 344
• Main Author: Sukumar, N
• Format: Journal Article
• Language: English
• Published: 01.01.2005
• ISBN: 9780735402928; 0735402922
• ISSN: 0094-243X
• DOI: 10.1063/1.2149812

In this paper, the construction of scattered data approximants is studied using the principle of maximum entropy. For under-determined and ill-posed problems, Jaynes's principle of maximum information-theoretic entropy is a means for least-biased statistical inference when insufficient information is available. Consider a set of distinct nodes {xi}i = 1n{expression} in Rd, and a point p with coordinate x that is located within the convex hull of the set {xi}. The convex approximation of a function u(x) is written as: uh(x) = i = 1n{expression} i(x)ui, where {i}i = 1n{expression} = > = 0 are known as shape functions, and uh must reproduce affine functions (d = 2): i = 1n{expression} i = 1, i = 1n{expression} ixi = x, i = 1n{expression} iyi = y. We view the shape functions as a discrete probability distribution, and the linear constraints as the expectation of a linear function. For n > 3, the problem is under-determined. To obtain a unique solution, we compute i by maximizing the uncertainty H() = - i = 1n{expression} i log i, subject to the above three constraints. In this approach, only the nodal coordinates are used, and neither the nodal connectivity nor any user-defined parameters are required to determine i-the defining characteristics of a mesh-free Galerkin approximant. Numerical results for {i}i = 1n{expression} are obtained using a convex minimization algorithm, and shape function plots are presented for different nodal configurations.