Nonlinear Equations and Optimization

DMS 0070641文摘。首席研究员将继续他的研究计划，在非线性方程和噪声优化问题的大型系统的数值解决。在非线性方程领域，PI将继续研究非线性方程的拟瞬态延拓方法到偏微分代数方程（PDAE）的设置，并设计和分析紧不动点问题的参数相关族的全迭代算法的多级方法。在优化领域，首席研究员将把他在采样方法上的工作扩展到“隐藏约束”问题，即约束的违反只能通过目标函数不返回值来检测，通过设计Levenberg-Marquardt形式的算法继续他的隐式滤波的发展，并从理论角度和作为识别隐藏约束的方式来研究DIRECT方法的性能。PI的这方面研究在很大程度上涉及到本科生。工程和科学中的许多过程和模型都用非线性方程表示。这些过程的数值模拟需要快速准确地求解这些方程，并清楚地了解这些方法及其局限性。有了高质量的模拟器，优化可以在设计中使用。工业设计中出现的优化问题往往由于测量或仿真误差而产生噪声。首席研究员将继续他的非线性方程和优化问题的计算解决的工作。这项工作将包括算法的研究，以确保模拟的鲁棒性和可靠性，在分布式存储计算机上实现，这对快速周转很重要，并与工业、国家实验室和学术界的科学家和工程师合作应用这些方法。这项工作将用于(a)环境建模、测量和补救，(b)航空航天工业的模拟、优化设计和控制，以及(c)天然气管道网络的优化。

DMS 0070641ABSTRACT.The principal investigator will continue his research program in the numericalsolution of large systems of nonlinear equations and noisy optimization problems. In the area of nonlinear equations the PI will continue his research program on pseudo-transient continuation methods for nonlinear equations to the partial differential algebraic equation (PDAE) settingand design and analyze multilevel methods for fully iterative algorithms for parameter-dependent families of compact fixed point problems.In the area of optimization, the principal investigator will extend his work on sampling methods to problems with ``hidden constraints'', i.e. constraints whose violation can only be detected by failure of the objective function to return a value, continue his development of the implicitfiltering with the design of a Levenberg-Marquardt form of the algorithm, and investigate the performance of the DIRECT method both from the theoretical viewpoint and as a way to identify hidden constraints. This aspect of the PI's research involves undergraduates in a significant way.Many processes and models in engineering and science are expressed as nonlinearequations. Numerical simulation of these processes requires the rapid and accurate solution of these equations and a clear understanding of the methodsand their limitations. With a high-quality simulator in hand, optimizationcan be used in design. The optimization problems that arise in industrial design are often noisy because of measurement or simulation error. The principal investigator will continue his work on the computational solution of nonlinear equations and optimization problems. The work will consist of study of algorithms, needed to ensure robustness and reliability of simulations, implementation on distributed memory computers,which is important for for rapid turnaround, and applying the methodsin collaborations with scientists and engineers in industry, national laboratories, and academia. The work will be used in (a) environmental modeling, measurement, and remediation, (b) simulation, optimal design, and control in the aerospace industry, and in (c) optimization of gas pipeline networks.