Iterative Methods for Nonlinear Equations and Optimization

Kelley0707220 主要研究非线性方程和优化问题的数值方法，重点是连续方法，非光滑问题和积分方程的多级方法。本论文的主要研究内容是：（1）积分方程约束优化问题的多层延拓算法的发展和分析，以及这些方法在模拟原子和分子流体中的应用;（2）非光滑非线性最小二乘的延拓/拟牛顿方法的设计和分析，以及在血流模型校准中的应用;（3）继续研究拟弧长延拓在分岔分析中所引起的线性系统的条件和特征值问题;（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，被用作求解器的预处理器，该求解器基于更精确、更昂贵的评估模型，例如分子动力学或随机模拟。 非线性方程和最优化问题是科学和工程中经常遇到的问题。 受化学和医学应用的启发，主要研究者研究具有多个解的方程，挑战是确定哪些解代表真实的世界，哪些解在实际的物理系统中看不到，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.