Browsing by Subject "approximation theory"
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Item Approximation and interpolation employing divergence-free radial basis functions with applications(Texas A&M University, 2004-09-30) Lowitzsch, SvenjaApproximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. No such class of radial basis functions that reflects these physical properties was known until 1994, when Narcowich and Ward introduced a family of matrix-valued radial basis functions that are divergence free. They also obtained error bounds and stability estimates for interpolation by means of these functions. These divergence-free functions are very smooth, and have unbounded support. In this thesis we introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. This leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We develop error bounds and stability estimates which hold for a broad class of functions. We conclude with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.Item Piecewise polynomial functions on a planar region: boundary constraints and polyhedral subdivisions(Texas A&M University, 2006-08-16) McDonald, Terry LynnSplines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines of degree at most k forms a vector space Crk() Moreover, a nice way to study Cr k()is to embed n Rd+1, and form the cone b of with the origin. It turns out that the set of splines on b is a graded module Cr b() over the polynomial ring R[x1; : : : ; xd+1], and the dimension of Cr k() is the dimension o This dissertation follows the works of Billera and Rose, as well as Schenck and Stillman, who each approached the study of splines from the viewpoint of homological and commutative algebra. They both defined chain complexes of modules such that Cr(b) appeared as the top homology module. First, we analyze the effects of gluing planar simplicial complexes. Suppose 1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the Mayer-Vietoris sequence to obtain a natural relationship between the spline modules Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2). Next, given a simplicial complex , we study splines which also vanish on the boundary of. The set of all such splines is denoted by Cr(b). In this case, we will discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b). Finally, we consider splines which are defined on a polygonally subdivided region of the plane. By adding only edges to to form a simplicial subdivision , we will be able to find bounds for the dimensions of the vector spaces Cr k() for k 0. In particular, these bounds will be given in terms of the dimensions of the vector spaces Cr k() and geometrical data of both and . This dissertation concludes with some thoughts on future research questions and an appendix describing the Macaulay2 package SplineCode, which allows the study of the Hilbert polynomials of the spline modules.