The article discusses Residual Dynamic Mode Decomposition (ResDMD) for accurately computing Koopman operators' spectral properties. It addresses issues with finite truncations, such as spurious eigenvalues, by calculating infinite-dimensional residuals from snapshot data. ResDMD is applied in scenarios with fewer snapshots than the dictionary size, showcasing its versatility across various dynamical systems. The paper introduces a novel computational approach to eliminate the need for two datasets, simplifying the application of ResDMD. It provides an analysis of new residuals for exact DMD and kernelized EDMD, demonstrating their broad applicability. The utility of these residuals is exemplified through examples like the analysis of a cylinder wake and compression of transient shockwave data. The article also discusses the connection between ResDMD and Koopman operators, highlighting the importance of determining spectral properties from snapshot data.
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