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Quadtree is a hierarchical data structure that is well-suited for h-adaptive mesh refinement. Due to the presence of hanging nodes, classical shape functions are non-conforming on quadtree meshes. In this paper, we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes. To this end, the recently proposed construction of polygonal basis functions is adapted to quadtree elements. A fast technique for calculating stiffness matrix on quadtree meshes is introduced. Residual-based error estimators and material force technique are used to estimate the error on quadtree meshes. The performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary-value problems. (C) 2007 Elsevier B.V. All rights reserved.

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 unity 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 (c) 2006 John Wiley & Sons, Ltd.

In this paper, we prove the continuity of maximum-entropy basis functions using variational analysis techniques. The use of information-theoretic variational principles to derive basis functions 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. For a set of distinct nodes {x(i)}(i=1)(n) in R-d, the convex approximation of a function u(x) is u(h)(x) = Sigma(n)(i=1) p(i)(x)u(i), where {p(i)}(i=1)(n) are nonnegative basis functions, and u(h)(x) must reproduce a. ne functions Sigma(n)(i=1) p(i)(x) = 1, Sigma(n)(i=1) p(i)(x) x(i) = x. Given these constraints, we compute p(i)(x) by minimizing the relative entropy functional (Kullback-Leibler distance), D(p parallel to m) = Sigma(n)(i=1) p(i)(x) ln(p(i)(x)/m(i)(x)), where m(i)(x) is a known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epiconvergence.

Quadtree is a hierarchical data structure that is well-suited for h-adaptive mesh refinement. Due to the presence of hanging nodes, classical shape functions are non-conforming on quadtree meshes. In this paper, we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes. To this end, the recently proposed construction of polygonal basis functions is adapted to quadtree elements. A fast technique for calculating stiffness matrix on quadtree meshes is introduced. Residual-based error estimators and material force technique are used to estimate the error on quadtree meshes. The performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary-value problems. (C) 2007 Elsevier B.V. All rights reserved.

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 unity 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 (c) 2006 John Wiley & Sons, Ltd.

In this paper, we prove the continuity of maximum-entropy basis functions using variational analysis techniques. The use of information-theoretic variational principles to derive basis functions 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. For a set of distinct nodes {x(i)}(i=1)(n) in R-d, the convex approximation of a function u(x) is u(h)(x) = Sigma(n)(i=1) p(i)(x)u(i), where {p(i)}(i=1)(n) are nonnegative basis functions, and u(h)(x) must reproduce a. ne functions Sigma(n)(i=1) p(i)(x) = 1, Sigma(n)(i=1) p(i)(x) x(i) = x. Given these constraints, we compute p(i)(x) by minimizing the relative entropy functional (Kullback-Leibler distance), D(p parallel to m) = Sigma(n)(i=1) p(i)(x) ln(p(i)(x)/m(i)(x)), where m(i)(x) is a known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epiconvergence.