残差动态模式分解：鲁棒的和验证的Koopmanism
Residual Dynamic Mode Decomposition: Robust and verified Koopmanism
摘要：
动态模式分解（DMD）通过一系列简单的相干特征来描述复杂的动态过程。DMD经常用于理解湍流的基本特征，并且与Koopman算子密切相关。然而，由于Koopman算子的无限维性质，验证分解，等效于Koopman算子的计算谱特征，仍然是一个重大挑战。挑战包括虚假（非物理）模式，并处理连续光谱，这两个经常发生在湍流。由Colbrook & Townsend 2021引入的残差动态模式分解（ResDMD）通过与完整的无限维Koopman算子相关的残差的数据驱动计算克服了其中的一些挑战。ResDMD在误差控制下计算一般Koopman算子的谱和伪谱，并在显式高阶收敛定理下计算谱测度（包括连续谱）的平滑近似。因此，ResDMD提供了强大的和验证的库普曼主义。我们实现ResDMD和演示其应用程序在各种流体动力学的情况下，在不同的雷诺数，所产生的数值和实验数据。示例包括：圆柱体后的旋涡脱落;在湍流边界层中获得的热线数据;聚焦于壁面射流的粒子图像测速数据;以及激光诱导等离子体的声压信号。我们提出了一些优点的ResDMD，即，能够验证解决非线性，瞬态模式，并减少加宽效应的光谱计算。我们还讨论了如何一个新的模态排序的基础上的残差，使更高的精度与一个较小的字典比传统的模排序。这为在不牺牲准确性的情况下更好地动态压缩大型数据集铺平了道路。
英文摘要：
Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a major challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes, and dealing with continuous spectra, both of which occur regularly in turbulent flows. Residual Dynamic Mode Decomposition (ResDMD), introduced by (Colbrook & Townsend 2021), overcomes some of these challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control, and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in a variety of fluid dynamic situations, at varying Reynolds numbers, arising from both numerical and experimental data. Examples include: vortex shedding behind a cylinder; hot-wire data acquired in a turbulent boundary layer; particle image velocimetry data focusing on a wall-jet flow; and acoustic pressure signals of laser-induced plasma. We present some advantages of ResDMD, namely, the ability to verifiably resolve non-linear, transient modes, and spectral calculation with reduced broadening effects. We also discuss how a new modal ordering based on residuals enables greater accuracy with a smaller dictionary than the traditional modulus ordering. This paves the way for greater dynamic compression of large datasets without sacrificing accuracy.
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