Nonlinear Equations and Bound Costrained Optimization

9700569 Kelley首席研究员将继续他在大型非线性方程组的数值解和有界约束优化问题方面的研究计划。在非线性方程领域，PI将在两个方向上扩展他最近在非线性方程方面的工作：(1)伪瞬态延拓，这是一种计算流体力学和燃烧中某些问题稳态解的流行方法；(2)研究非线性迭代的终止准则，并将其应用于偏微分方程的线解法。PI还将继续研究紧不动点问题的多层方法。在函数空间的有界约束优化领域，将继续研究有界约束抛物控制问题的方法，开始一个关于无限维有界约束问题的预条件的新项目，并重新设计他的隐式滤波代码，以期在汽车工程和环境监测中的应用。新的算法特性将被添加到代码中，代码将被移植到并行计算机上，与新算法相关的理论问题将被解决。这些问题包括在噪声存在下的Nelder-Mead停滞的检测和摆脱及其相关算法和准牛顿方法的收敛性。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。然后将仿真用于制造的设计和优化。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的计算解决和优化问题的工作。这项工作将包括算法研究，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快速周转，以及在工程应用环境中进行测试。通过首席研究员与工业界、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)汽车工程的优化设计。

9700569 Kelley The principal investigator will continue his research program in the numerical solution of large systems of nonlinear equations and bound constrained optimization problems. In the area of nonlinear equations the PI will expand his recent work on nonlinear equations in two directions: (1) pseudo transient continuation, a popular method for computing steady-state solutions of certain problems in fluid mechanics and combustion, and (2) research into termination criteria for nonlinear iterations with applications to the method-of-lines solution of partial differential equations. The PI will also continue his work on multilevel methods for compact fixed point problems. In the area of bound constrained optimization in function spaces, the principal investigator will continue his work on methods for bound constrained parabolic control problems, begin a new project on preconditioners for bound constrained problems in infinite dimension, and redesign his implicit filtering code with a view toward applications in automotive engineering and environmental monitoring. New algorithmic features will be added to the code, the code will be ported to a parallel computer, and theoretical questions related to the new algorithms will be addressed. Such questions include detection of and escape from stagnation in Nelder-Mead and related algorithms and convergence of quasi-Newton methods in the presence of noise. Many processes and models in engineering and science are expressed as nonlinear equations. Numerical simulation of these processes requires the rapid and accurate solution of these equations and a clear understanding of the methods and their limitations. Simulation is then used in design and optimization for manufacturing. 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 p roblems. The work will consist of study of algorithms, needed to ensure robustness and reliability of simulations, implementation on distributed memory computers for rapid turn around, and testing in the context of engineering applications. Through the principal investigator's collaborations with scientists and engineers in industry, national laboratories, and academia, the work will be used in (a) environmental measurement and remediation, (b) simulation, optimal design, and control in the aerospace industry, and (c) optimal design in automotive engineering.