Algorithms for Large-Scale Nonlinear Eigenvalue Problems: Interpolation, Stability, Transient Dynamics

动力系统是不断变化的世界的数学模型。要理解这样的模型，必须描述系统将如何在时间上演变。令人感兴趣的是解决方案是否随时间增长，以及系统的短期行为是否不同于较大时间窗口的行为。该项目旨在开发工具，帮助科学家和工程师更好地分析一类具有挑战性的模型，在这些模型中，系统当前的变化率由其在最近过去某个时候的配置决定，这些模型被称为“延迟系统”。令人信服的例子来自生物学，这种延迟可能与种群中的怀孕、疾病的潜伏期或药片溶解的时间相对应。在许多重要场景中，高度精确的模型需要数千或数百万个变量。这个项目将设计高性能的计算算法来有效地评估这类系统的行为。研究团队将开发一类新的算法，这些算法源于大规模动力系统的内插技术，用于求解重要的大规模非线性特征值问题。非线性特征值问题在许多应用中发挥着越来越重要的作用，包括时滞微分方程的稳定性分析。这样的问题给计算带来了很大的挑战：即使问题的维度是有限的，特征值的数量也往往是无限的。该项目的基于内插的技术有望在复杂平面的大范围内精确地建模非线性算子，从而产生比现有方法更高效、更健壮的算法。与传统方法相比，结合内插法和保构子空间投影方法有可能逼近更多数量的特征值。为了了解这些新算法的性能并提高它们的速度，该项目还将讨论收敛理论；为了提高效率，该项目将探索不精确的求解方法。改进的非线性特征值问题的求解器将为这个项目的另外两个主要方面的发展提供信息：确定临界稳定性转变为微分方程中参数(如时滞)的函数，以及增强对与非线性特征值问题相关的动力系统的暂态行为的理解。这个项目有望在非线性特征值问题和相关的稳定性问题的算法和分析方面带来实质性的改进。

Dynamical systems are mathematical models of the changing world. To understand such a model, one must describe how the system will evolve in time. Of interest is whether the solution grows with time, and whether the short-term behavior of the system is different from the behavior over a large-time window. This project aims to develop tools to help scientists and engineers better analyze a challenging class of models in which the system's current rate-of-change is dictated by its configuration at some time in the recent past, models known as "delay systems." Compelling examples come from biology, where this delay could correspond to gestation in a population, the incubation of a disease, or the time for a pill to dissolve. In many important scenarios, highly accurate models require thousands or millions of variables. This project will design high-performance computing algorithms to efficiently assess the behavior of such systems.The research team will develop a new class of algorithms that are derived from interpolation techniques for model reduction of large-scale dynamical systems, for solving important large-scale nonlinear eigenvalue problems. Nonlinear eigenvalue problems play an increasingly important role in many applications, including the stability analysis of delay differential equations. Such problems pose a great challenge to computation: the number of eigenvalues is often infinite, even when the dimension of the problem is finite. This project's interpolation-based techniques hold the promise of accurately modeling nonlinear operators over a broad region of the complex plane, resulting in algorithms that are both more efficient and more robust than existing methods. Combining interpolation with structure-preserving subspace projection has the potential to approximate a much larger number of eigenvalues than possible with traditional methods. To understand the performance of these new algorithms and to improve their speed, the project will also address convergence theory; for greater efficiency the project will explore inexact solution methods. Improved solvers for the nonlinear eigenvalue problem will inform development of the other two main aspects of this project: determination of critical stability transitions as a function of parameters (such as the time delays) in differential equations, and an enhanced understanding of the transient behavior of dynamical systems associated with nonlinear eigenvalue problems. This project is expected to lead to substantial improvements in algorithms and analysis for the nonlinear eigenvalue problem and related stability questions.