On Curved Simplicial Elements and Best Quadratic Spline Approximation for Hierarchical Data Representation

We present a method for hierarchical data approximation using curved quadratic simplicial elements for domain decomposition. Scientific data defined over two- or three-dimensional domains typically contain boundaries and discontinuities that are to be preserved and approximated well for data analysis and visualization. Curved simplicial elements make possible a better representation of curved geometry, domain boundaries, and discontinuities than simplicial elements with non-curved edges and faces. We use quadratic basis functions and compute best quadratic simplicial spline approximations that are $C^0$-continuous everywhere except where field discontinuities occur whose locations we assume to be given. We adaptively refine a simplicial approximation by identifying and bisecting simplicial elements with largest errors. It is possible to store multiple approximation levels of increasing quality. Our method can be used for hierarchical data processing and visualization

On Simulated Annealing and the Construction of Linear Spline Approximations for Scattered Data

Abstract. We describe a method to create optimal linear spline approximations to arbitrary functions of one or two variables, given as scattered data without known connectivity. We start with an initial approximation consisting of a fixed number of vertices and improve this approximation by choosing different ver-tices, governed by a simulated annealing algorithm. In the case of one variable, the approximation is defined by line segments; in the case of two variables, the vertices are connected to define a Delaunay triangulation of the selected subset of sites in the plane. In a second version of this algorithm, specifically designed for the bivariate case, we choose vertex sets and also change the triangulation to achieve both optimal vertex placement and optimal triangulation. We then cre-ate a hierarchy of linear spline approximations, each one being a superset of all lower-resolution ones.

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Methods for Estimating The Diagonal of Matrix Functions

Many applications such as path integral evaluation in Lattice Quantum Chromodynamics (LQCD), variance estimation of least square solutions and spline ts, and centrality measures in network analysis, require computing the diagonal of a function of a matrix, Diag(f(A)) where A is sparse matrix, and f is some function. Unfortunately, when A is large, this can be computationally prohibitive. Because of this, many applications resort to Monte Carlo methods. However, Monte Carlo methods tend to converge slowly. One method for dealing with this shortcoming is probing. Probing assumes that nodes that have a large distance between them in the graph of A, have only a small weight connection in f(A). to determine the distances between nodes, probing forms Ak. Coloring the graph of this matrix will group nodes that have a high distance between them together, and thus a small connection in f(A). This enables the construction of certain vectors, called probing vectors, that can capture the diagonals of f(A). One drawback of probing is in many cases it is too expensive to compute and store A^k for the k that adequately determines which nodes have a strong connection in f(A). Additionally, it is unlikely that the set of probing vectors required for A^k is a subset of the probing vectors needed for Ak+1. This means that if more accuracy in the estimation is required, all previously computed work must be discarded. In the case where the underlying problem arises from a discretization of a partial dierential equation (PDE) onto a lattice, we can make use of our knowledge of the geometry of the lattice to quickly create hierarchical colorings for the graph of A^k. A hierarchical coloring is one in which colors for A^{k+1} are created by splitting groups of nodes sharing a color in A^k. The hierarchical property ensures that the probing vectors used to estimate Diag(f(A)) are nested subsets, so if the results are inaccurate the estimate can be improved without discarding the previous work. If we do not have knowledge of the intrinsic geometry of the matrix, we propose two new classes of methods that improve on the results of probing. One method seeks to determine structural properties of the matrix f(A) by obtaining random samples of the columns of f(A). The other method leverages ideas arising from similar problems in graph partitioning, and makes use of the eigenvectors of f(A) to form effective hierarchical colorings. Our methods have thus far seen successful use in computational physics, where they have been applied to compute observables arising in LQCD. We hope that the renements presented in this work will enable interesting applications in many other elds

Lectures on Applied Mathematics Part 2: Numerical Analysis

This book is designed to be a continuation of the textbook, Lectures on Applied Mathematics Part I: Linear Algebra which can also be downloaded at http://rbowen.engr.tamu.edu. This textbook evolved from my teaching an undergraduate Numerical Analysis course to Mechanical Engineering students at Texas A&M University. That course was one of the courses I was allowed to teach after my several years out of the classroom. It tries to utilize rigorous concepts in Linear Algebra in combination with the powerful computational tools of MATLAB to provide undergraduate students practical numerical analysis tools. It makes extensive use of MATLAB's graphics capabilities and, to a limited extent, its ability to animate the solutions of ordinary differential equations. It is not a textbook that tries to be comprehensive as a source of MATLAB information. It does contain a large number of links to MATLAB's extensive online resources. This information has been invaluable to me as this work was developed. The version of MATLAB used in the preparation of this textbook is MATLAB 2019b.Chapter 7: Elements of Numerical Linear Algebra; Chapter 8: Errors that Arise in Numerical Analysis; Chapter 9: Roots of Nonlinear Equations; Chapter 10: Regression; Chapter 11: Interpolation; Chapter 12: Ordinary Differential Equations; Appendix A: Introduction to MATLAB; Appendix B: Animation

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds