Maximum Entropy Approximation - Sukumar, N (Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 2005)
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 xii = 1nexpression 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 = 1nexpression i(x)ui, where ii = 1nexpression = > = 0 are known as shape functions, and uh must reproduce affine functions (d = 2): i = 1nexpression i = 1, i = 1nexpression ixi = x, i = 1nexpression 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 = 1nexpression 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 ii = 1nexpression are obtained using a convex minimization algorithm, and shape function plots are presented for different nodal configurations.