Iterative Methods for Nonlinear Equations and Optimization

主要研究非线性方程和优化问题的数值求解方法，重点是延拓方法，非光滑问题和积分方程的多层方法。主要研究方向为：(1)积分方程约束优化问题的多水平延拓算法的开发与分析及其在原子和分子流体模拟中的应用；(2)非光滑非线性最小二乘的延拓/准牛顿方法的设计与分析及其在血流模型标定中的应用；(3)继续研究在分岔分析中使用伪弧长延拓时出现的线性系统和特征问题的条件，以及(4)多模型方法，其中一个物理模型，例如基于微分或积分方程，作为基于更精确，更昂贵的评估模型（如分子动力学或随机模拟）的解算器的前置条件。非线性方程和优化问题是科学和工程中经常遇到的问题。在化学和医学应用的激励下，首席研究员研究具有多个解决方案的方程，挑战在于确定哪些解决方案代表现实世界，哪些不能在实际物理系统中看到，以及标准方法和计算机代码无法解决的优化问题。这项工作将导致在科学和工程的许多分支中有用的新方法。

Kelley0707220 The principal investigator studies numerical methods for thesolution of nonlinear equations and optimization problems with afocus on continuation methods, nonsmooth problems, and multilevelmethods for integral equations. The main topics of the researchare (1) the development and analysis of multilevel continuationalgorithms for optimization problems with integral equationconstraints and the application of those methods in simulatingatomic and molecular fluids, (2) the design and analysis ofcontinuation/quasi-Newton methods for nonsmooth nonlinear leastsquares, with applications to calibration of models of bloodflow, (3) continued study of the conditioning of the linearsystems and eigenproblems that arise when pseudo-arclengthcontinuation is used in bifurcation analysis, and (4) multi-modelmethods in which one physical model, based on differential orintegral equations, for example, is used as a preconditioner fora solver which is based on a more accurate, more expensive toevaluate model, such as a molecular dynamics or stochasticsimulation. Nonlinear equations and optimization problems are commonlyencountered in science and engineering. Motivated byapplications in chemistry and medicine, the principalinvestigator studies equations with multiple solutions, thechallenge being to determine which of those solutions representsthe real world and which cannot be seen in actual physicalsystems, and optimization problems for which standard methods andcomputer codes fail. This work should lead to new approachesuseful in many branches of science and engineering.