Maximum Entropy Approximation

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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 {x_i} in R^d, and a point p with coordinate x that is located within the convex hull of the set {x_i}. The convex approximation of a function u(x) is written as: u^h(x) = _i(x)u_i, where {_i} 0 are known as shape functions, and u^h must reproduce affine functions (d = 2): _i = 1, _ix_i = x, _iy_i = 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 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} are obtained using a convex minimization algorithm, and shape function plots are presented for different nodal configurations. ©2005 American Institute of Physics