Abstract. Characterizing the linear and local linear independence of the functions that span a linear space is a key task if the space is to be used computationally. Given a control net, the spanning functions of one spatial coordinate of a generalized subdivision surface are called nodal functions. They are the limit, under subdivision, of associating the value one with one control net node and zero with all others. No characterization of independence of nodal functions has been published to date, even for the two most popular generalized subdivision algorithms, Catmull–Clark subdivision and Loop’s subdivision. This paper provides a road map for the verification of linear and local linear independence of generalized subdivision functions. It proves the conjectured global independence of the nodal functions of both algorithms, disproves local linear independence (for higher valences), and establishes linear independence on every surface region corresponding to a facet of the control net. Subtle exceptions, even to global independence, underscore the need for a detailed analysis to provide a sound basis for a number of recently developed computational approaches.

Introduction and Summary The primary goal of my research is to develop simple computational models for simulating the behavior of physical systems. I draw on ideas from graphics, applied mathematics, di#erential geometry, and engineering. My research shows that simple approaches need not sacrifice accuracy or performance. Simple approaches lead to more insight, wider adoption by the scientific computing community, and the opportunity to pursue more ambitious applications to mechanical engineering, medicine, biology, and computer graphics. With this philosophy in mind I am focusing on two key aspects of simulation. First, to improve scalability of simulators with respect to large datasets and high geometric complexity, I am developing a framework for adaptive multiresolution simulation. Second, I am developing discrete models: geometric descriptions of physical phenomena distilled into their most fundamental form. Figure 1: Example applications which benefit from our multiresolution s

Numerical integration of ordinary differential equations resulting from the gravitation of nearby celestial small bodies is the subject of this thesis. We present three methods that alleviate the computational burden of evaluating gravitational force near a small body: i) adaptive polynomial interpolation, ii) adaptive polynomial least squares approximation, and iii) acceleration via specialized, commodity hardware. Each method is evaluated on its quantitative accuracy with respect to a reference model, and its observance of qualitative features of gravity. We conclude with a summary of methods available for computing small body gravitation, and recommendations for different scenarios.

The contents of this report are the sole responsibility of the authors. O conteúdo do presente relatório é de única responsabilidade dos autores. Contents Chapter 1. The finite element method 1 1.1. Direct approach 1 1.2. The Galerkin method 3 1.3. The Collocation method 6