Nonlinear Equations and Optimization

主要研究者将继续他在大型非线性方程组和噪声优化问题的数值解方面的研究计划。在非线性方程领域，PI将继续他的非线性方程的伪瞬态延拓方法的研究计划，以偏微分代数方程（PDAE）的设置和设计和分析完全迭代算法的多层方法，用于参数依赖的紧致不动点问题。在优化领域，首席研究员将把他在抽样方法方面的工作扩展到"隐藏约束“问题，即只有通过目标函数返回一个值的失败才能检测到违反约束的问题，继续他的发展的implicitfiltering与设计的Levenberg-Marquardt形式的算法，并调查的直接方法的性能，无论是从理论上的观点，并作为一种方式来识别隐藏的约束。PI的这方面的研究涉及大学生在一个显着的方式。许多过程和模型，在工程和科学表示为非线性方程。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在手，优化可以用于设计。 在工业设计中出现的优化问题往往是嘈杂的，因为测量或仿真误差。首席研究员将继续他的工作对非线性方程和优化问题的计算解决方案。这项工作将包括算法的研究，需要确保模拟的鲁棒性和可靠性，在分布式内存计算机上实现，这对快速周转很重要，并将该方法应用于与工业，国家实验室和学术界的科学家和工程师的合作。这项工作将用于（a）环境建模，测量和补救，（B）模拟，优化设计，并在航空航天工业控制，并在（c）优化天然气管道网络。

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.