Iterative Methods for Nonlinear Equations and Optimization

North Carolina State University, Raleigh NC

Investigators

Abstract

Kelley 0707220 The principal investigator studies numerical methods for the solution of nonlinear equations and optimization problems with a focus on continuation methods, nonsmooth problems, and multilevel methods for integral equations. The main topics of the research are (1) the development and analysis of multilevel continuation algorithms for optimization problems with integral equation constraints and the application of those methods in simulating atomic and molecular fluids, (2) the design and analysis of continuation/quasi-Newton methods for nonsmooth nonlinear least squares, with applications to calibration of models of blood flow, (3) continued study of the conditioning of the linear systems and eigenproblems that arise when pseudo-arclength continuation is used in bifurcation analysis, and (4) multi-model methods in which one physical model, based on differential or integral equations, for example, is used as a preconditioner for a solver which is based on a more accurate, more expensive to evaluate model, such as a molecular dynamics or stochastic simulation. Nonlinear equations and optimization problems are commonly encountered in science and engineering. Motivated by applications in chemistry and medicine, the principal investigator studies equations with multiple solutions, the challenge being to determine which of those solutions represents the real world and which cannot be seen in actual physical systems, and optimization problems for which standard methods and computer codes fail. This work should lead to new approaches useful in many branches of science and engineering.