The present thesis consists of two parts. The first part is devoted to the analysis of discretizations of a class of basic electronic density functionals. In the second part we suggest and analyze Quasicontinuum Methods for an atomistic interaction potential that is based on a field. We begin by formulating and analyzing a model for the study of finite clusters of atoms or localized defects in infinite crystals based on a version of the classical Thomas{Fermi{Dirac{von Weizs?acker density functional. We show that the resulting constrained optimization problem has a minimizer and we provide a careful analysis of the solvability of the associated Euler{Lagrange equation. Based on these results, and using tools from saddle-point theory and nonlinear analysis, we then show that a Galerkin discretization has a solution that converges to the correct limit (in the case of Dirichlet as well as periodic boundary conditions). Furthermore, we investigate the issue of optimal convergence rates. Using appropriate dual problems, we can show faster convergence for the energy, the Lagrange multiplier of the underlying minimization problem, and the L2-errors of the solutions. We also look at the dependence of the density functional on the nucleus coordinates and show a convergence result for minimizing nucleus configurations. These results are subsequently generalized to the case of discretizations with numerical integration. Existence and convergence of solutions, as well as optimal convergence rates can be established if quadrature rules of sufficiently high order are applied. In the second part of the thesis we consider an atomistic interaction potential in one dimension given through a minimization problem, which gives rise to a field. The forces on atoms are in this case given by local expressions involving this field. A convenient feature of this model is the existence of a weak formulation for the forces, which provides a natural connection point for the coupling with a continuum model. We suggest Quasicontinuum-like coupling mechanisms that are based on a decomposition of the domain into an atomistic and a continuum region. In the continuum region we use an approximation based on the Cauchy{ Born rule. In the atomistic subdomain a version of the atomistic model with Dirichlet boundary conditions is applied. Special attention has to be paid to the dependence of the atomistic subproblem on the boundary and the boundary conditions. Applying concepts from nonlinear analysis we show existence and convergence of solutions to the Quasicontinuum approximation.

Thesis (M.A.)--Marshall University, 2009. / Title from document title page. Includes abstract. Document formatted into pages: contains xii, 104 p. Includes bibliographical references p. 99-104.

This thesis is about computational theory in the setting of the mathematics of information. The first goal is to demonstrate that many commonly considered problems in optimisation theory cannot be solved with an algorithm if the input data is only known up to an arbitrarily small error (modelling the fact that most real numbers are not expressible to infinite precision with a floating point based computational device). This includes computing the minimisers to basis pursuit, linear programming, lasso and image deblurring as well as finding an optimal neural network given training data. These results are somewhat paradoxical given the success that existing algorithms exhibit when tackling these problems with real world datasets and a substantial portion of this thesis is dedicated to explaining the apparent disparity, particularly in the context of compressed sensing. To do so requires the introduction of a variety of new concepts, including that of a breakdown epsilon, which may have broader applicability to computational problems outside of the ones central to this thesis. We conclude with a discussion on future research directions opened up by this work.