AF: Small: Interpolatory Methods for Dimension Reduction of Parametric and Nonlinear Dynamical Systems

动力系统是建模和控制各种物理现象的主要工具，如神经系统中的信号传播、电路仿真、天气预报和流体动力学。直接数值模拟是研究这些复杂现象的为数不多的方法之一，在许多工程领域，数值模拟已成为设计过程中必不可少的一部分。然而，随着对模型保真度要求的不断提高，不可避免地导致了动态系统的超大规模和复杂性。基于这类系统的模拟通常会给人力资源和计算资源带来难以管理的负担，因此提供了模型缩减的主要动机——创建更小、更便宜的模型，以接近模拟原始系统的行为。模型简化可以产生易于处理的低维系统，适合于分析、仿真、优化和计算机辅助系统设计。本研究的主要主题是经验数据方法与插值相结合，以克服线性问题的标准投影方法的局限性。经验数据可由物理实验或直接数值模拟提供。插值条件以多种方式进入，大大降低了简化模型的计算复杂度。计算时间可以减少三个数量级，同时保持良好的精度。拟议的研究将努力把这些技术建立在坚实的数学基础上，以确保准确性，并大大扩展应用领域，以建立这些新方法的广泛适用性。拟议的办法与现有的方法有很大的不同。将提出的方法发展到更高的成熟度和适用性水平将是模型简化的重大进展。

Dynamical systems are a principal tool in the modeling and control of physical phenomena as diverse as signal propagation in the neural/nervous system, circuit simulation, weather forecasting, and fluid dynamics. Direct numerical simulation has been one of very few available means for studying the rich complexity of these phenomena, and in many areas of engineering numerical simulation has become essential to the design process. However, the ever increasing demand for improved model fidelity leads inevitably to dynamical systems of extremely large scale and complexity. Simulations based on such systems often impose unmanageable burdens on both human and computational resources, and thus provide the principal motivation for model reduction - creating smaller, cheaper models that closely mimic the behaviors of the original system. Model reduction can thus result in tractable low dimensional systems that are suitable for analysis, simulation, optimization, and computer-aided system design. The primary theme of this research is an empirical data approach combined with interpolation to overcome limitations of standard projection methods for linear problems. The empirical data may be provided by physical experimentation or by direct numerical simulation. Interpolation conditions enter in several ways to greatly decrease the computational complexity of the reduced models. A three orders of magnitude reduction in computation time can be achieved while retaining excellent accuracy. The proposed research will strive to put these techniques on firm mathematical foundations in order to assure accuracy and also to greatly extend the areas of application in order to establish broad applicability of these new approaches.The proposed approaches represent a significant departure from existing methodology. Developing the proposed methods to a greater level of maturity and applicability will be a significant advance in model reduction.