Joab R. Winkler

Elimination of spurious modes in finite element analysis

Abstract A common cause of spurious (non-physical) modes that arise in either finite difference or finite element derived eigenvalue problems is identified. It is shown that these modes are the result of an excessively flexible system. The flexibility is removed by constraining the problem. An analogy is drawn between electromagnetic wave propagation and acoustic wave propagation in liquids to show that the spurious modes encountered in both cases are due to fundamentally the same cause. Results are presented to show that constraining the problem yields a significant reduction in the number of spurious modes, a big improvement in the quality of the eigenvector of the physical modes and only a marginal increase in the error in the eigenvalue for the low order modes.

A companion matrix resultant for Bernstein polynomials

Abstract A closed form expression for a companion matrix M of a Bernstein polynomial is obtained, and this is used to derive an expression for a resultant matrix of two Bernstein polynomials. It is shown that M differs from its equivalent form for a power basis polynomial because an upper triangular Hankel matrix does not define a similarity transformation between M and M T . A measure of the numerical condition of a resultant matrix, for polynomials in an arbitrary basis, is reviewed and this is used to compare the stability of two resultant matrices. In particular, computational tests are performed and it is shown that the resultant matrix of two Bernstein polynomials is numerically better conditioned than the resultant matrix that is obtained when a simple parameter substitution is used to transform the polynomials to the power basis.

A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix

A non-linear structure preserving matrix method for the computation of a structured low rank approximation S(f@?,g@?) of the Sylvester resultant matrix S(f,g) of two inexact polynomials f=f(y) and g=g(y) is considered in this paper. It is shown that considerably improved results are obtained when f(y) and g(y) are processed prior to the computation of S(f@?,g@?), and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of S(f@?,g@?), which leads to a linear structure preserving matrix method, or they can be incremented during the computation of S(f@?,g@?), which leads to a non-linear structure preserving matrix method. It is shown that the non-linear method yields a better structured low rank approximation of S(f,g) and that the assignment of f(y) and g(y) is important because S(f@?,g@?) may be a good structured low rank approximation of S(f,g), but S(g@?,f@?) may be a poor structured low rank approximation of S(g,f) because its numerical rank is not defined. Examples that illustrate the differences between the linear and non-linear structure preserving matrix methods, and the importance of the assignment of f(y) and g(y), are shown.

Case study of a wave event in the stable atmospheric boundary layer overlying an Antarctic Ice Shelf using the orthogonal wavelet transform

This paper describes a spectacular gravity wave event which was detected by meteorological instruments deployed in the stably stratified atmospheric boundary layer overlying the Brunt Ice Shelf, Antarctica. Varying levels of turbulence activity were also observed. Wind and temperature records from sonic anemometers deployed on a 32 m mast have been studied using cross-spectral analysis and the orthogonal wavelet transform. Wavelet-based variance characteristics and derived parameters are used to identify self-similar processes, noise and coherent structures within a signal, thus providing useful information on the distribution of energy. Local maxima in the wavelet variance characteristics at lower wavelet levels were linked with a propagating internal wave.

The calculation of the degree of an approximate greatest common divisor of two polynomials

The calculation of the degree of an approximate greatest common divisor (AGCD) of two inexact polynomials f(y) and g(y) is a non-trivial computation because it reduces to the estimation of the rank loss of a resultant matrix R(f,g). This computation is usually performed by placing a threshold on the small singular values of R(f,g), but this method suffers from disadvantages because the numerical rank of R(f,g) may not be defined, or the noise level imposed on the coefficients of f(y) and g(y) may not be known, or it may only be known approximately. This paper addresses this problem by considering two methods for estimating the degree of an AGCD of f(y) and g(y), such that knowledge of the noise level is not required. The first method involves the calculation of the smallest angle between two subspaces that are apparent from the structure of the Sylvester resultant matrix S(f,g), and the second method uses the theory of subresultant matrices, which are derived from S(f,g) by the deletion of some of its rows and columns. The two methods are compared computationally on non-trivial polynomials.

A resultant matrix for scaled Bernstein polynomials

Abstract The established theory of the resultant of two polynomials assumes that they are expressed in the power (monomial) basis, and a basis transformation is therefore necessary if the resultant of two Bernstein polynomials is required. In this paper, a resultant matrix for two scaled Bernstein polynomials (polynomials of degree n whose basis functions are (1−x) n−i x i , i=0,…,n ) is constructed. In particular, a companion matrix M for a scaled Bernstein polynomial r(x) is developed, and this is used to form a resultant matrix s(M) , where s(x) is a scaled Bernstein polynomial.

An improved non-linear method for the computation of a structured low rank approximation of the Sylvester resultant matrix

This paper reports on improvements to recent work on the computation of a structured low rank approximation of the Sylvester resultant matrix S(f,g) of two inexact polynomials f=f(y) and g=g(y). Specifically, it has been shown in previous work that these polynomials must be processed before a structured low rank approximation of S(f,g) is computed. The existing algorithm may still, however, yield a structured low rank approximation of S(f,g), but not a structured low rank approximation of S(g,f), which is unsatisfactory. Moreover, a structured low rank approximation of S(f,g) must be equal to, apart from permutations of its columns, a structured low rank approximation of S(g,f), but the existing algorithm does not guarantee the satisfaction of this condition. This paper addresses these issues by modifying the existing algorithm, such that these deficiencies are overcome. Examples that illustrate these improvements are shown.

The transformation of the companion matrix resultant between the power and Bernstein polynomial bases

Many problems in applied mathematics require the computation of the resultant of two polynomials, and it is nearly always assumed that the polynomials are expressed in the power basis. Recent work has shown that significantly improved numerical answers are obtained if the polynomials are expressed in the Bernstein basis, and thus the transformation of a resultant matrix between these bases is required. In this paper, this transformation is considered for one type of resultant, the companion matrix resultant. It is shown that this change of basis of the resultant matrix is defined by a similarity transformation, and that this transformation is ill-conditioned, even for matrices of low order. It is concluded that the companion matrix resultant should be constructed and computed in the Bernstein basis, such that the power basis is not used.

Automatic Face Recognition Using Stereo Images

Face recognition is an important pattern recognition problem in the study of natural and artificial learning systems. In typical optical image based face recognition systems, the systematic variability that arises from representing the three dimensional (3D) shape of a face by a two dimensional (2D) illumination intensity matrix is treated as a random variable, and it is obtained by collecting examples of faces in different poses with respect to the camera. More sophisticated 3D recognition systems employ specialist equipment (e.g. laser scanners) to measure the shape of the face, and they perform either pattern matching in three dimensions or they use projections from 3D models to match against 2D images. It is shown here that optical images obtained with a pair of stereo cameras may be used to extract depth information in the form of disparity values, and thereby significantly enhance the performance of a face recognition system

Structured matrix methods for CAGD: an application to computing the resultant of polynomials in the Bernstein basis

We devise a fast fraction-free algorithm for the computation of the triangular factorization of Bernstein–Bezoutian matrices with entries over an integral domain. Our approach uses the Bareiss fraction-free variant of Gaussian elimination, suitably modified to take into account the structural properties of Bernstein–Bezoutian matrices. The algorithm can be used to solve problems in algebraic geometry that arise in computer aided geometric design and computer graphics. In particular, an example of the application of this algorithm to the numerical computation of the intersection points of two planar rational Bezier curves is presented. Copyright