Paul Castillo

FEMSTER: An object-oriented class library of high-order discrete differential forms

FEMSTER is a modular finite element class library for solving three-dimensional problems arising in electromagnetism. The library was designed using a modern geometrical approach based on differential forms (or p-forms) and can be used for high-order spatial discretizations of well-known H(div)- and H(curl)-conforming finite element methods. The software consists of a set of abstract interfaces and concrete classes, providing a framework in which the user is able to add new schemes by reusing the existing classes or by incorporating new user-defined data types.

Comparative Study of Semi-Implicit Schemes for Nonlinear Diffusion in Hyperspectral Imagery

Nonlinear diffusion has been successfully employed over the past two decades to enhance images by reducing undesirable intensity variability within the objects in the image, while enhancing the contrast of the boundaries (edges) in scalar and, more recently, in vector-valued images, such as color, multispectral, and hyperspectral imagery. In this paper, we show that nonlinear diffusion can improve the classification accuracy of hyperspectral imagery by reducing the spatial and spectral variability of the image, while preserving the boundaries of the objects. We also show that semi-implicit schemes can speedup significantly the evolution of the nonlinear diffusion equation with respect to traditional explicit schemes

An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method

In this work an a posteriori global error estimate for the Local Discontinuous Galerkin (LDG) applied to a linear second order elliptic problem is analyzed. Using a mixed formulation, an upper bound of the error in the primal variable is derived from explicit computations. Finally, a local adaptive scheme based on explicit error estimators is studied numerically using one dimensional problems.

Fast multiscale regularization and segmentation of hyperspectral imagery via anisotropic diffusion and algebraic multigrid solvers

This paper presents an algorithm that generates a scale-space representation of hyperspectral imagery using Algebraic Multigrid (AMG) solvers. The scale-space representation is obtained by solving with AMG a vector-valued anisotropic diffusion equation, with the hyperspectral image as its initial condition. AMG also provides the necessary structure to obtain a hierarchical segmentation of the image. The scale space representation of the hyperspectral image can be segmented in linear time complexity. Results in the paper show that improved segmentation is achieved. The proposed methodology to solve vector PDEs can be used to extend a number of techniques currently being developed for the fast computation of geometric PDEs and its application for the processing of hyperspectral and multispectral imagery.

Computational Performance of LDG Methods Applied to Time Harmonic Maxwell Equation in Polyhedral Domains

A numerical study of the classical and penalized LDG method applied to vector Helmholtz equation on three dimensional domains is presented. Using a simple numerical flux based on convex combinations classical rates of convergence can be obtained on unstructured meshes while achieving a substantial reduction of the stencil. The superconvergent behaviour of the auxiliary field is investigated on Cartesian meshes. Numerical experiments also suggest convergence of the method for constant approximations on Cartesian meshes. We explore existing scalable preconditioning techniques adapted to the discontinuous Galerkin framework for the low frequency case. Finally the method is tested on examples arising in practical engineering problems with complex valued electric field.

Local Discontinuous Galerkin and Classical Finite Element Methods: Differences and Similarities

In this work a quantitative and qualitative comparison of the Local Discontinuous Galerkin method and classical finite element methods applied to elliptic problems is performed. High order discretizations are considered. The methods are compared with respect to accuracy of the approximation, rates of convergence, asymptotic behavior of the spectral condition number of the stiffness matrix.Copyright

THE WALLMAN COMPACTIFICATION AND INTERMEDIATE SPACES

Abstract In this paper we introduce new characterizations of the Wallman compactification among the T 1 compactifications of a space. We then use one of these characterizations to investigate conditions which would imply that an intermediate Wallman space have a Wallman compactification homeomorphic to the original compactification. The most general of these conditions is that disjoint closed subsets of the intermediate space have disjoint closures in the compactification. This has, as a special case, the consequence that if X ⊆ Y ⊆ X and if Y\X is closed in WX\X, then WY ≅ W X.