William F. Mitchell

Applied and Computational Mathematics Division National Institute of Standards and Technology Gaithersburg, MD, USA 20899-8910

Abstract: One aspect of adaptive mesh refinement in the finite element method for solving partial differential equations is the method by which elements are refined. In the early 1980’s the dominant method for refining triangles was the red-green algorithm of Bankand Sherman. The red refinements are the desired refinements, but will result in an incompatible mesh when used alone. The green refinements are used to recover compatibility for stability of the finite element discretization, and are removed before the next adaptive step. Prof. Bob Skeel raised the question as to whether it is possible to perform adaptive refinement of triangles without this complicated patching/unpatching process. As a result, a new triangle refinement method, called newest vertex bisection, was devised as an alternative to red-green refinement in the mid 1980’s. The new approach is simpler and maintains compatibility of the mesh at all times, avoiding the patching/unpatching of the green refinement. In this historical paper we review the development of the newest vertex bisection method for adaptive refinement, and subsequent extensions of the method.

Keywords: finite elements; adaptive mesh refinement; newest vertex bisection
Mathematics Subject Classification: 65-03, 65N30, 65N50
PACS: 02.60.Lj, 02.70.Dh

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Gaussian Scale Mixtures Miguel Martins Felgueiras ^{1 *}, João Paulo Martins^**, Rui Filipe Santos^**

^* CEAUL Lisbon and ESTG, CIGS, Polytechnic Institute of Leiria, Portugal
^** CEAUL Lisbon and ESTG, Polytechnic Institute of Leiria, Portugal

Abstract: In this paper we present a parsimonious approximation of a Gaussian mixture when its components share a common mean value, i.e. a scale mixture. We show that a shifted and scaled Student’s t-distribution can be approximated to this type of mixture, and use the result to develop a hypothesis test for the equality of the components mean value. A simulation study to check the quality of the approximation is also provided, together with an application to logarithmic daily returns.

Keywords: Gaussian scale mixture, Student’s t-distribution, log-returns, simulation.
Mathematics Subject Classification: 60E05; 62P20.

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