9 research outputs found
The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
We propose and study discontinuous Galerkin methods for strongly degenerate
convection-diffusion equations perturbed by a fractional diffusion (L\'evy)
operator. We prove various stability estimates along with convergence results
toward properly defined (entropy) solutions of linear and nonlinear equations.
Finally, the qualitative behavior of solutions of such equations are
illustrated through numerical experiments
On the Smoothness of the Solution to the Two-Dimensional Radiation Transfer Equation
In this paper, we deal with the differential properties of the scalar flux
defined over a two-dimensional bounded convex domain, as a solution to the
integral radiation transfer equation. Estimates for the derivatives of the
scalar flux near the boundary of the domain are given based on Vainikko's
regularity theorem. A numerical example is presented to demonstrate the
implication of the solution smoothness on the convergence behavior of the
diamond difference method
Adaptive Algorithms for Relatively Lipschitz Continuous Convex Optimization Problems
Recently there were proposed some innovative convex optimization concepts,
namely, relative smoothness [1] and relative strong convexity [2,3]. These
approaches have significantly expanded the class of applicability of
gradient-type methods with optimal estimates of the convergence rate, which are
invariant regardless of the dimensionality of the problem. Later Yu. Nesterov
and H. Lu introduced some modifications of the Mirror Descent method for convex
minimization problems with the corresponding analogue of the Lipschitz
condition (so-called relative Lipschitz continuity). By introducing an
artificial inaccuracy to the optimization model, we propose adaptive methods
for minimizing a convex Lipschitz continuous function, as well as for the
corresponding class of variational inequalities. We also consider an adaptive
"universal" method, applicable to convex minimization problems both on the
class of relatively smooth and relatively Lipschitz continuous functionals with
optimal estimates of the convergence rate. The universality of the method makes
it possible to justify the applicability of the obtained theoretical results to
a wider class of convex optimization problems. We also present the results of
numerical experiments
The Dynamics of Inhomogeneous Cosmologies
In this thesis we investigate cosmological models more general than the
isotropic and homogeneous Friedmann-Lemaitre models. We focus on cosmologies
with one spatial degree of freedom, whose matter content consists of a perfect
fluid and the cosmological constant. We formulate the Einstein field equations
as a system of quasilinear first order partial differential equations, using
scale-invariant variables.
The primary goal is to study the dynamics in the two asymptotic regimes, i.e.
near the initial singularity and at late times. We highlight the role of
spatially homogeneous dynamics as the background dynamics, and analyze the
inhomogeneous aspect of the dynamics. We perform a variety of numerical
simulations to support our analysis and to explore new phenomena.Comment: PhD thesis, University of Waterloo, September 2004. 205 page
Research in applied mathematics, numerical analysis, and computer science
Research conducted at the Institute for Computer Applications in Science and Engineering (ICASE) in applied mathematics, numerical analysis, and computer science is summarized and abstracts of published reports are presented. The major categories of the ICASE research program are: (1) numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; (2) control and parameter identification; (3) computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and (4) computer systems and software, especially vector and parallel computers
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal