Model Reduction with Rational Krylov Methods

仿真和计算已经成为科学和工业中感兴趣的许多复杂现象的建模和控制的标准要求任务。 例子比比皆是，从大型高速车辆中的声波传播和噪声抑制到合理药物设计中的分子动力学和蛋白质折叠。 对更高精度的需求导致在计算机模型中包含更多细节，可能与其他复杂的计算机模型耦合，这可能需要额外的模拟，这些模拟本身就很困难且昂贵。 由此产生的计算负担可能是压倒性的，并且可能对资源产生不可管理的大量需求。 计算模型的有效利用成为这种大规模环境中模拟的必要组成部分。 这是模型简化的主要动机。 通常，原始系统模型的行为非常接近于一个更简单的系统，但不幸的是，它不是一个事先明确知道的系统。 模型简化的目标是提取这样一个更简单的系统，同时尽可能地模仿原始的完整系统行为。 新的更简单的系统，然后可以被用来作为一个有效的替代原系统。 这里支持的研究重点是基于Krylov的投影方法来完成这项任务。 在过去的十年里，这些方法已经成为大规模环境中模型简化的有希望的候选者，但它们的使用仍然需要即兴的元素，这些元素尚未得到很好的理解。 我们相信，我们的方法将允许这些想法的系统完善，并导致高保真，在某些情况下，最佳的，大规模系统的降阶模型的有效建设与精确估计的模型保真度的水平，一直保持。 还将开发用于分析、近似和控制大规模复杂系统模型的工具，预计这些工具将有助于科学研究基础设施。

Simulation and computing have become a standard required task for the modeling and control of many complex phenomena that are of interest in science and industry. Examples abound and range from acoustic wave propagation and noise suppression in large high-speed vehicles to molecular dynamics and protein folding in rational drug design. The need for greater accuracy leads to inclusion of greater detail in the computer model, with potential coupling to other complex computer models that may require additional simulations that are themselves difficult and expensive. The resulting computational burden can be overwhelming and can create unmanageably large demands on resources. Efficient utilization of the computational model becomes a necessary component of simulations in such large-scale settings. This is the main motivation for model reduction. Often, the original system model behaves very nearly as if it were a simpler system -- but unfortunately not one that is explicitly known beforehand. The goal of model reduction is to extract such a simpler system while mimicking the original full system behavior as closely as possible. The new simpler system can then be used as an efficient surrogate for the original system. The research supported here focuses on Krylov-based projection methods to accomplish this task. These methods have emerged as promising candidates for model reduction in large-scale settings over the last ten years, yet their use still requires improvised elements that are not yet well understood. We believe that our methods will permit a systematic refinement of these ideas and lead to the efficient construction of high-fidelity, in some cases optimal, reduced-order models for large-scale systems with precise estimates of the level of model fidelity that has been maintained. Tools for the analysis, approximation, and control of large-scale, complex system models will be produced as well that are anticipated to contribute to scientific research infrastructure.