数据驱动的模态分解：分析与增强
Data driven modal decompositions: analysis and enhancements
AI总结：
本文改进了动态模态分解（DMD），通过数据驱动公式选择Ritz对，提升Koopman算子的谱信息精度，并在加权内积空间中扩展了DMD的应用，通过数值实验验证了改进方法DDMD_RRR的优势。
摘要：
动态模态分解（DMD）是计算数据驱动分析流体流动的工具，更广泛地用于非线性动力系统Koopman谱分析。其卓越性能促使开发了多种改进方法，使DMD成为数据驱动框架中的有吸引力方法。本文进一步改进DMD以提高其可靠性并增强其功能。特别地，数据驱动公式允许选择Ritz对，从而提供更精确的Koopman算子谱信息，且已知的Ritz向量细化技术被应用于数据驱动场景。此外，DMD被公式化为更一般的加权内积空间设置，并详细讨论了数值计算的后果。数值实验用于说明所提方法DDMD_RRR的优势。
英文摘要：
The Dynamic Mode Decomposition (DMD) is a tool of trade in computational data driven analysis of fluid flows. More generally, it is a computational device for Koopman spectral analysis of nonlinear dynamical systems, with a plethora of applications in applied sciences and engineering. Its exceptional performance triggered developments of several modifications that make the DMD an attractive method in data driven framework. This work offers further improvements of the DMD to make it more reliable, and to enhance its functionality. In particular, data driven formula for the residuals allows selection of the Ritz pairs, thus providing more precise spectral information of the underlying Koopman operator, and the well-known technique of refining the Ritz vectors is adapted to data driven scenarios. Further, the DMD is formulated in a more general setting of weighted inner product spaces, and the consequences for numerical computation are discussed in detail. Numerical experiments are used to illustrate the advantages of the proposed method, designated as DDMD_RRR (Refined Rayleigh Ritz Data Driven Modal Decomposition).
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