Iterative Methods for Nonlinear Equations

本论文的主要研究内容是：（1）发展和分析用于计算时变偏微分方程，特别是具有非光滑非线性项的偏微分方程稳态解的网格无关方法;（2）关于参数的紧致不动点问题解的延拓快速算法。这项研究将把他的成果应用于化学工程、化学、环境科学、电气工程、核工程和航空工程。该项目的两个阶段都解决了全球化不精确牛顿法的传统实现的弱点，即无法在多个解决方案中进行选择。延拓方法使用额外的信息，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的事实，来指导求解器，并且都提高了求解器的鲁棒性，并增加了找到重要解的可能性。拟议的研究将分析这些方法的鲁棒性，因为计算网格是精细的，如果非线性是不可微的，它们如何执行，以及如何设计用于校正迭代的快速算法，该算法最好地利用问题和连续方法的组合的函数分析性质。因此，求解非线性方程组的算法是许多仿真器和工程设计工具的基本组成部分。研究人员将研究依赖于物理参数的方程的求解方法，例如半导体器件上的电压，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着变化，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，使求解器更加健壮。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。PI和他的学生，通过与几个国家实验室和公司的合作，将这些结果应用于纳米级电子，化学工程和航空航天的问题。

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.