Iterative Methods for Nonlinear Equations and Optimization

Kelley 0707220主要研究了求解非线性方程和优化问题的数值方法，包括连续方法、非光滑问题和积分方程组的多层方法。主要研究内容包括：(1)积分方程组约束最优化问题的多层连续算法的发展和分析及其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连续/拟牛顿方法的设计和分析，以及在血流模型的校准中的应用；(3)继续研究在分支分析中使用伪弧连续法时出现的线性系统的条件和特征问题；以及(4)多模型方法，其中基于微分或积分方程组的一个物理模型被用作求解器的预条件，该方法基于更精确、更昂贵的模型来评估例如分子动力学或随机模拟。非线性方程和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，首席调查员研究具有多个解的方程，挑战是确定这些解中的哪些代表现实世界，哪些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。这项工作应该会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.