Model Reduction with Rational Krylov Methods

模拟和计算已经成为许多科学和工业中感兴趣的复杂现象的建模和控制的标准必需任务。例子比比皆是，从大型高速交通工具中的声波传播和噪音抑制，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，潜在地耦合到其他复杂的计算机模型，这可能需要额外的模拟，而这些模拟本身是困难和昂贵的。由此产生的计算负担可能不堪重负，并可能产生无法管理的巨大资源需求。在这种大规模环境下，有效利用计算模型成为模拟的必要组成部分。这是模型缩减的主要动机。通常，原始系统模型的行为非常接近于它是一个更简单的系统--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。然后，新的更简单的系统可以用作原始系统的有效替代品。这里支持的研究重点是基于Krylov的投影方法来完成这一任务。在过去的十年里，这些方法已经成为在大规模环境中进行模型简化的有前途的候选方法，但它们的使用仍然需要尚未被很好理解的即兴元素。我们相信，我们的方法将允许对这些想法进行系统的改进，并导致高效地构建高保真的、在某些情况下最优的、用于大规模系统的降阶模型，并精确估计所保持的模型保真的水平。还将生产用于分析、近似和控制大规模复杂系统模型的工具，预计这些工具将为科学研究基础设施做出贡献。

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.