CAREER: Reduced-order Modeling and Controller Design for Large-scale Dynamical Systems via Rational Krylov Methods

提案ID：0645347PI：Gugercin，塞尔维亚机构：弗吉尼亚理工学院和州立大学标题：Career：通过有理Krylov方法为大型动态系统降阶建模和控制器设计摘要动态系统的直接数值模拟是实现对具有科学价值或工业价值的复杂物理现象进行准确预测或控制的少数可用手段之一。对提高精度的不断增长的需求导致了非常大规模和复杂的动力系统。在如此大规模的环境中进行模拟可能会淹没计算资源；这是模型缩减的主要动机。目标是产生一个更简单的降阶模型，尽可能准确地逼近原始模型。由此得到的简化模型可以用作原始模型的有效替代品，以在更大的仿真中取代它，或者开发适合实时应用的更简单、更快的控制器。基于Krylov的方法已经成为在现实的大规模环境中进行模型简化的有前途的候选方法。该项目致力于开发基于Krylov的最优、稳健和系统的投影方法，以有效地构建高保真和最优降阶模型。通过在基于Krylov的模型简化设置中谨慎地使用不精确解，所提出的研究将把这些最优简化技术扩展到具有数百万个自由度的现实设置，同时保持关于精确有理Krylov子空间的误差度量的相同质量。此外，还将开发系统的方法来设计快速有效的低阶最优控制器，更广泛地说，用于减少互联(耦合)系统。该项目将把新开发的方法应用于来自不同应用领域的几个测试问题。大规模的模拟和计算在研究计算科学和工程的许多领域中的各种复杂物理现象方面发挥着至关重要的作用。例如，电路中的信号传播和干扰、分子动力学、天气预报、大型结构中的波传播和振动抑制、各种介质中的温度控制以及微电子机械系统的行为。大规模计算导致了对国家计算资源的难以管理的巨大需求；因此，越来越需要用更简单、高保真的模型来近似复杂的动力系统，这将使这些模拟更容易、更快地计算。该项目致力于开发能够产生复杂系统的最佳、高保真近似的技术；同时导致更快速但仍然可靠的模拟。在这个项目过程中将开发的数学工具和高质量的软件将有助于科学家和工程师研究大规模、复杂的多物理问题。除了研究部分，该项目还提供了一个全面的教育计划，其中包括对本科生和研究生的指导；研究生课程的开发；以及可用于向学生介绍各种高性能科学计算领域的几个跨学科研究项目。

Proposal ID: 0645347PI: Gugercin, SerkanInstitution: Virginia Polytechnic Institute and State UniversityTitle: CAREER: Reduced-order Modeling and Controller Design for Large-scale Dynamical Systems via Rational Krylov MethodsAbstractDirect numerical simulation of dynamical systems has been one of the few available means for achieving accurate prediction or control of complex physical phenomena that are of scientific interest or industrial value. The ever-increasing need for improved accuracy leads to very large-scale and complex dynamical systems. Simulations in such large-scale settings can overwhelm computational resources; this is the main motivation for model reduction. The goal is to produce a simpler reduced-order model approximating the original one as accurately as possible. The resulting reduced model can then be used as an efficient surrogate to the original, to replace it in a larger simulation or to develop a simpler and faster controller suitable for real time applications. Krylov-based methods have emerged as promising candidates for model reduction in realistic large-scale settings. This project seeks to develop optimal, robust, and systematic Krylov-based projection methods for efficient construction of high fidelity and optimal reduced-order models. By carefully employing inexact solves in a Krylov-based model reduction setting, the proposed research will extend these optimal reduction techniques to realistic settings with millions of degrees of freedom while maintaining the same quality of the error measures with respect to the exact rational Krylov subspaces. In addition, systematic approaches will be developed for the design of fast and effective, low-order optimal controllers, and more generally, for reduction of interconnected (coupled) systems. The project will apply the newly developed methods to several test problems drawn from different application areas.Large-scale simulations and computations play a crucial role in studying a great variety of complex physical phenomena in many areas of computational science and engineering. Examples include signal propagation and interference in electric circuits, molecular dynamics, weather forecasting, wave propagation and vibration suppression in large structures, temperature control in various media, and behavior of micro-electro-mechanical systems. Computing in large-scale settings leads to unmanageably large demands on the nation's computational resources; hence there is a growing need to approximate the complex dynamical systems with simpler, high fidelity models that will make these simulations much easier and faster to compute. This project seeks to develop techniques that will yield optimal, high fidelity approximations of complex systems; leading at once to simulations that are more rapid yet still reliable. The mathematical tools and high-quality software that will be developed through the course of this project will contribute to research efforts of scientist and engineers working with large-scale, complex multi-physics problems. In addition to the research component, this project offers a comprehensive education plan that consists of the supervision of undergraduate and graduate students; development of a graduate course; and several interdisciplinary research projects available for introducing students in a variety of areas of high-performance scientific computing.