Distribution of points on the sphere and spherical designs

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Distribution of points on the sphere and spherical designs

 

Author: An, Congpei
Title: Distribution of points on the sphere and spherical designs
Degree: Ph.D.
Year: 2011
Subject: Sphere.
Spherical data.
Sphere -- Mathematical models.
Sphere -- Models.
Hong Kong Polytechnic University -- Dissertations
Department: Dept. of Applied Mathematics
Pages: xiv, 110 leaves : col. ill. ; 30 cm.
Language: English
InnoPac Record: http://library.polyu.edu.hk/record=b2462537
URI: http://theses.lib.polyu.edu.hk/handle/200/6292
Abstract: This thesis concentrates on distribution of points on the unit sphere and polynomial approximation on the unit sphere by using spherical designs. The study of distribution of points on the sphere has many applications, including climate modeling and global approximation in geophysics and virus modeling in bioengineering, as the earth and cell are approximate spheres. For choosing the point set XN , if as in many applications the point set is given by empirical data, then the only option is to selectively delete points so as to improve the distribution. If, on the other hand, the points may be freely chosen, then we shall see that there is merit in choosing XN to be a "spherical t-design" for some appropriate value of t. A set of N points on the unit sphere is a spherical t-design if the average value of any polynomial of degree at most t over the set XN is equal to the average value of the polynomial over the sphere. The main contributions of this thesis consist of the following two parts. 1. We consider the characterization and computation of spherical t-designs on the unit sphere in 3 dimensional space when N ≥ (t + 1)², the dimension of the space of spherical polynomials of degree at most t. We show how construct well conditioned spherical t-designs with N ≥ (t + 1)² points by maximizing the determinant of a Gram matrix which satisfies undetermined nonlinear equations. Interval methods are then used to prove the existence of a true spherical t-design and to provide a guaranteed interval containing the true spherical t-design. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss desirable properties of the points for both equal weight numerical integration and polynomial interpolation on the sphere, and give examples to illustrate the characterization of these points. 2. We consider polynomial approximation on the unit sphere by a class of regularized discrete least squares methods, with novel choices for the regularization operators and the point sets of the discretization. We allow different kinds of rotationally invariant regularization operators, including the zero operator (in which case the approximation includes interpolation, quasi-interpolation and hyperinterpolation); powers of the negative Laplace-Beltrami operator (which can be suitable when there are data errors); and regularization operators that yield filtered polynomial approximations. As node sets we use spherical t-designs. For t ≥ 2L and an approximation polynomial of degree L it turns out that there is no linear algebra problem to be solved, and the approximation in some cases recovers known polynomial approximation schemes, including interpolation, hyperinterpolation and filtered hyperinterpolation. For t ε [L, 2L), the linear system needs to be solved numerically. We present an upper bound for the condition number and show that well conditioned spherical t-designs provide good condition numbers. Finally, we give numerical examples to illustrate the theoretical results and show that well chosen regularization operators and well conditioned spherical t-designs can provide good polynomial approximation on the sphere, with exact data or contaminated data.

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