Iterative Methods for Nonlinear Equations

非线性方程的迭代方法

• 批准号: 0404537
• 负责人: Carl Kelley
• 金额: $ 19.3万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-15 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0404537&HistoricalAwards=false
• 关键词: Iterative; Methods; Nonlinear; Equations

The main topics of the proposed research are (1) the development and analysis of mesh-independent methods for computation of steady state solutions of time-dependent partial differential equations, especially those with nonsmooth nonlinearities, and (2) fast algorithms for continuation of the solutions of compact fixed point problems with respect to a parameter. The investigate will apply his results to chemical engineering, chemistry, environmental science, electrical engineering, nuclear engineering, and aeronautical engineering. Both phases of the project address a weakness of a conventional implementation of a globalized inexact Newton method, namely the inability to select among multiple solutions. Continuation methods use additional information, such as a physical parameter in the equation or the fact that the nonlinear equation is the steady state equation for a time-dependent problem, to guide the solver, and both improve the solver's robustness and increase the likelihood that the important solution or solutions will be found. The proposed research will analyze the robustness of these methods as computational grids are refined, how they perform if the nonlinearity is not differentiable, and how one can design fast algorithms for the corrector iteration which best exploit the functional analytic properties of the combination of the problem and the continuation method.Nonlinearity is common across all of science and engineering. Therefore, algorithms for solving nonlinear equations are fundamental components of many simulators and engineering design tools. The investigator will study solution methods for equations that depend on physical parameters, such as the voltage across a semiconductor device, the load on a mechanical structure, or the temperature of a chemical reaction. As such parameters change, the nature of the solution can vary significantly, and the performance of the simulator can be affected as well. The PI's research is directed at improving the performance of the algorithms and making the solvers more robust. His success will have an impact on the performance of simulator and design tools, and thereby accelerate the design cycle. The PI's and his students, through collaborations with several national laboratories and companies, will apply these results to problems in nano-scale electronics, chemical engineering, and aerospace.

本研究的主要课题是：(1)发展和分析与网格无关的计算时变偏微分方程稳态解的方法，特别是那些具有非光滑非线性的方法；(2)关于参数的紧不动点问题解的快速延拓算法。研究结果将应用于化学工程、化学、环境科学、电气工程、核工程、航空工程等领域。项目的两个阶段都解决了全球化不精确牛顿方法的传统实现的弱点，即无法在多个解决方案中进行选择。延拓方法使用附加信息，如方程中的物理参数或非线性方程是时间相关问题的稳态方程这一事实，来指导求解器，既提高了求解器的鲁棒性，又增加了找到重要解的可能性。该研究将分析这些方法在计算网格细化时的鲁棒性，当非线性不可微时它们的表现，以及如何设计快速的校正迭代算法，以最好地利用问题和延拓方法相结合的泛函解析特性。非线性在所有的科学和工程中都很常见。因此，求解非线性方程的算法是许多模拟器和工程设计工具的基本组成部分。研究者将研究依赖于物理参数的方程的求解方法，如半导体器件的电压、机械结构上的负载或化学反应的温度。随着这些参数的变化，解决方案的性质可能会发生显著变化，并且模拟器的性能也会受到影响。PI的研究旨在提高算法的性能，使求解器更加鲁棒。他的成功将对模拟器和设计工具的性能产生影响，从而加快设计周期。通过与几个国家实验室和公司的合作，PI和他的学生将把这些结果应用于纳米级电子、化学工程和航空航天领域的问题。