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随时间演化。我们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与大时间窗口内的行为不同。该项目旨在开发工具，以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，系统的当前变化率是由最近过去某个时间的配置决定的，这些模型被称为“延迟系统”。令人信服的例子来自生物学，这种延迟可以对应于种群的妊娠期，疾病的潜伏期或药丸溶解的时间。在许多重要的场景中，高度精确的模型需要数千或数百万个变量。该项目将设计高性能计算算法，以有效地评估此类系统的行为。研究小组将开发一类新的算法，这些算法来源于大规模动力系统模型简化的插值技术，用于解决重要的大规模非线性特征值问题。非线性特征值问题在包括时滞微分方程稳定性分析在内的许多应用中发挥着越来越重要的作用。这样的问题给计算带来了巨大的挑战：即使问题的维度是有限的，特征值的数量往往是无限的。该项目的基于插值的技术有望在复杂平面的广泛区域上精确建模非线性算子，从而产生比现有方法更有效和更健壮的算法。将插值与保持结构的子空间投影相结合，可以近似出比传统方法多得多的特征值。为了理解这些新算法的性能并提高它们的速度，该项目还将解决收敛理论；为了提高效率，该项目将探索不精确的解决方法。改进的非线性特征值问题求解器将为本项目的其他两个主要方面提供信息：确定微分方程中作为参数（如时间延迟）的函数的临界稳定性过渡，以及增强对与非线性特征值问题相关的动力系统的瞬态行为的理解。该项目有望在非线性特征值问题和相关稳定性问题的算法和分析方面取得实质性进展。

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.