49 research outputs found
Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B麓ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems
Eigenvalue Methods for Interpolation Bases
This thesis investigates eigenvalue techniques for the location of roots of polynomials expressed in the Lagrange basis. Polynomial approximations to functions arise in almost all areas of computational mathematics, since polynomial expressions can be manipulated in ways that the original function cannot. Polynomials are most often expressed in the monomial basis; however, in many applications polynomials are constructed by interpolating data at a series ofpoints. The roots of such polynomial interpolants can be found by computing the eigenvalues of a generalized companion matrix pair constructed directly from the values of the interpolant. This affords the opportunity to work with polynomials expressed directly in the interpolation basis in which they were posed, avoiding the often ill-conditioned transformation between bases.
Working within this framework, this thesis demonstrates that computing the roots of polynomials via these companion matrices is numerically stable, and the matrices involved can be reduced in such a way as to significantly lower the number of operations required to obtain the roots.
Through examination of these various techniques, this thesis offers insight into the speed, stability, and accuracy of rootfinding algorithms for polynomials expressed in alternative bases
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
Polynomial distribution functions on bounded closed intervals
The thesis explores several topics, related to polynomial distribution functions
and their densities on [0,1]M, including polynomial copula functions and their
densities. The contribution of this work can be subdivided into two areas.
- Studying the characterization of the extreme sets of polynomial densities
and copulas, which is possible due to the Choquet theorem.
- Development of statistical methods that utilize the fact that the density
is polynomial (which may or may not be an extreme density).
With regard to the characterization of the extreme sets, we first establish
that in all dimensions the density of an extreme distribution function is an extreme
density. As a consequence, characterizing extreme distribution functions
is equivalent to characterizing extreme densities, which is easier analytically.
We provide the full constructive characterization of the Choquet-extreme polynomial
densities in the univariate case, prove several necessary and sufficient
conditions for the extremality of densities in arbitrary dimension, provide necessary
conditions for extreme polynomial copulas, and prove characterizing
duality relationships for polynomial copulas. We also introduce a special case
of reflexive polynomial copulas.
Most of the statistical methods we consider are restricted to the univariate
case. We explore ways to construct univariate densities by mixing the extreme
ones, propose non-parametric and ML estimators of polynomial densities. We
introduce a new procedure to calibrate the mixing distribution and propose
an extension of the standard method of moments to pinned density moment
matching. As an application of the multivariate polynomial copulas, we introduce
polynomial coupling and explore its application to convolution of coupled
random variables.
The introduction is followed by a summary of the contributions of this thesis
and the sections, dedicated first to the univariate case, then to the general
multivariate case, and then to polynomial copula densities. Each section first
presents the main results, followed by the literature review
Doctor of Philosophy
dissertationPlatelet aggregation, an important part of the development of blood clots, is a complex process involving both mechanical interaction between platelets and blood, and chemical transport on and o the surfaces of those platelets. Radial Basis Function (RBF) interpolation is a meshfree method for the interpolation of multidimensional scattered data, and therefore well-suited for the development of meshfree numerical methods. This dissertation explores the use of RBF interpolation for the simulation of both the chemistry and mechanics of platelet aggregation. We rst develop a parametric RBF representation for closed platelet surfaces represented by scattered nodes in both two and three dimensions. We compare this new RBF model to Fourier models in terms of computational cost and errors in shape representation. We then augment the Immersed Boundary (IB) method, a method for uid-structure interaction, with our RBF geometric model. We apply the resultant method to a simulation of platelet aggregation, and present comparisons against the traditional IB method. We next consider a two-dimensional problem where platelets are suspended in a stationary fluid, with chemical diusion in the fluid and chemical reaction-diusion on platelet surfaces. To tackle the latter, we propose a new method based on RBF-generated nite dierences (RBF-FD) for solving partial dierential equations (PDEs) on surfaces embedded in 2D domains. To robustly tackle the former, we remove a limitation of the Augmented Forcing method (AFM), a method for solving PDEs on domains containing curved objects, using RBF-based symmetric Hermite interpolation. Next, we extend our RBF-FD method to the numerical solution of PDEs on surfaces embedded in 3D domains, proposing a new method of stabilizing RBF-FD discretizations on surfaces. We perform convergence studies and present applications motivated by biology. We conclude with a summary of the thesis research and present an overview of future research directions, including spectrally-accurate projection methods, an extension of the Regularized Stokeslet method, RBF-FD for variable-coecient diusion, and boundary conditions for RBF-FD