An adaptive minimum spanning tree multi-element method for uncertainty quantification of smooth and discontinuous responses

A novel approach for non-intrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalised polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest depends discontinuously on the input parameters. As a remedy we propose an adaptive sampling algorithm based on minimum spanning trees combined with a domain decomposition method based on support vector machines. The minimum spanning tree determines new sample locations based on both the probability density of the input parameters and the gradient in the quantity of interest. The support vector machine efficiently decomposes the random space in multiple elements, avoiding the appearance of Gibbs phenomena near discontinuities. On each element, local approximations are constructed by means of least orthogonal interpolation, in order to produce stable interpolation on the unstructured sample set. The resulting minimum spanning tree multi-element method does not require initial knowledge of the behaviour of the quantity of interest and automatically detects whether discontinuities are present. We present several numerical examples that demonstrate accuracy, efficiency and generality of the method.Comment: 20 pages, 18 figure

An adaptive minimum spanning tree multielement method for uncertainty quantification of smooth and discontinuous responses

A novel approach for nonintrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalized polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest depends discontinuously on the input parameters. As a remedy we propose an adaptive sampling algorithm based on minimum spanning trees combined with a domain d

Using Choquet integrals for kNN approximation and classification

k-nearest neighbors (kNN) is a popular method for function approximation and classification. One drawback of this method is that the nearest neighbors can be all located on one side of the point in question x. An alternative natural neighbors method is expensive for more than three variables. In this paper we propose the use of the discrete Choquet integral for combining the values of the nearest neighbors so that redundant information is canceled out. We design a fuzzy measure based on location of the nearest neighbors, which favors neighbors located all around x. <br /

Adaptive n-Sided Polygonal Finite Element Method for Analysis of Plane Problems

In this work we present an adaptive polygonal nite element method for analysis of two dimensional plane problems. The generation of n sided polygonal nite element mesh is based on generation of a centroidal Vorononi tessellation (CVT). By this method an unstructured tessellation of a scattered point set, that minimally covers the proximal space around each point in the point set can be generated. The method has also been extended to include tessellation for non convex domains. For the numerical integration of Galerkin weak form over polygonal nite element domains we resort to classical Gaussian quadrature applied on triangular sub domains of each polygonal element. An adaptive nite element analysis strategy is proposed and implemented in the present work. A patch recovery type of stress smoothing technique that utilizes polygonal element patches for obtaining smooth stresses has been proposed for obtaining the smoothed nite element stresses. A classical z2 type a - posteriori error estimator that estimates the energy norm of the error from the recovered solution is then implemented. The renement of the polygonal elements is made on an element by element basis through a renement index. Numerical examples of two dimensional plane problems are presented to demonstrate the eciency of the proposed adaptive polygonal nite element method

Optimal Point Placement for Mesh Smoothing

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smoothing problems that do not fit into the generalized linear programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This is the final version, and will appear in a special issue of J. Algorithms for papers from SODA '9