标题： 近似平衡截断的质量 显示英文标题

标题： Quality of Approximate Balanced Truncation

摘要： 模型降阶是处理大规模动态系统数值模拟的强大工具，用于研究复杂的物理系统。 对于线性时不变动态系统，模型降阶方法主要有两种类型：基于Krylov子空间的方法和基于平衡截断的方法。 第二种方法在理论上比第一种方法更为严谨，因为原系统与降阶系统之间的近似误差存在一个相当紧的全局误差界。 需要注意的是，该误差界是基于精确可控性和可观性Gram矩阵的可用性建立的。 然而，在数值上，Gram矩阵是不可用的，必须进行数值计算，而对于大规模系统，可行的选择是从Gram矩阵的低秩近似中计算出近似的平衡截断。 因此，严格来说，现有的全局误差界不适用于通过近似Gram矩阵得到的任何降阶系统。 本文的目标是通过近似平衡截断建立降阶系统的全局误差界来解决这个问题。 显示英文摘要

摘要： Model reduction is a powerful tool in dealing with numerical simulation of large scale dynamic systems for studying complex physical systems. Two major types of model reduction methods for linear time-invariant dynamic systems are Krylov subspace-based methods and balanced truncation-based methods. The methods of the second type are much more theoretically sound than the first type in that there is a fairly tight global error bound on the approximation error between the original system and the reduced one. It is noted that the error bound is established based upon the availability of the exact controllability and observability Gramians. However, numerically, the Gramians are not available and have to be numerically calculated, and for a large scale system, a viable option is to compute low-rank approximations of the Gramians from which an approximate balanced truncation is then performed. Hence, rigorously speaking, the existing global error bound is not applicable to any reduced system obtained via approximate Gramians. The goal of this paper is to address this issue by establishing global error bounds for reduced systems via approximate balanced truncation.