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Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size

Matthew J. Colbrook

TL;DR

This work addresses the challenge of extracting robust Koopman spectral information from snapshot data when the dictionary size exceeds the number of snapshots ($M\le N$). It introduces a dual, data-driven residual framework (ResDMD) that yields infinite-dimensional residuals from finite data, enabling spectral verification, pseudospectra computation, and dictionary validation without dividing data into training and quadrature sets. The authors extend ResDMD to both exact DMD and kernelized EDMD, providing Galerkin and regression perspectives for kernel methods, and demonstrate the approach on cylinder wake, aerofoil cascades, and shockwave data, highlighting improved reliability and efficient mode compression. Overall, the method offers rigorous error control and practical tools for analyzing complex, high-dimensional dynamical systems with nontrivial observables.

Abstract

Residual Dynamic Mode Decomposition (ResDMD) offers a method for accurately computing the spectral properties of Koopman operators. It achieves this by calculating an infinite-dimensional residual from snapshot data, thus overcoming issues associated with finite truncations of Koopman operators, such as spurious eigenvalues. These spectral properties include spectra and pseudospectra, spectral measures, Koopman mode decompositions, and dictionary verification. In scenarios where there are fewer snapshots than dictionary size, particularly for exact DMD and kernelized EDMD, ResDMD has traditionally been applied by dividing snapshot data into a training set and a quadrature set. Through a novel computational approach of solving a dual least-squares problem, we demonstrate how to eliminate the need for two datasets. We provide an analysis of these new residuals for exact DMD and kernelized EDMD, demonstrating ResDMD's versatility and broad applicability across various dynamical systems, including those modeled by high-dimensional and nonlinear observables. The utility of these new residuals is showcased through three diverse examples: the analysis of cylinder wake, the study of aerofoil cascades, and the compression of transient shockwave experimental data. This approach not only simplifies the application of ResDMD but also extends its potential for deeper insights into the dynamics of complex systems.

Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size

TL;DR

This work addresses the challenge of extracting robust Koopman spectral information from snapshot data when the dictionary size exceeds the number of snapshots (). It introduces a dual, data-driven residual framework (ResDMD) that yields infinite-dimensional residuals from finite data, enabling spectral verification, pseudospectra computation, and dictionary validation without dividing data into training and quadrature sets. The authors extend ResDMD to both exact DMD and kernelized EDMD, providing Galerkin and regression perspectives for kernel methods, and demonstrate the approach on cylinder wake, aerofoil cascades, and shockwave data, highlighting improved reliability and efficient mode compression. Overall, the method offers rigorous error control and practical tools for analyzing complex, high-dimensional dynamical systems with nontrivial observables.

Abstract

Residual Dynamic Mode Decomposition (ResDMD) offers a method for accurately computing the spectral properties of Koopman operators. It achieves this by calculating an infinite-dimensional residual from snapshot data, thus overcoming issues associated with finite truncations of Koopman operators, such as spurious eigenvalues. These spectral properties include spectra and pseudospectra, spectral measures, Koopman mode decompositions, and dictionary verification. In scenarios where there are fewer snapshots than dictionary size, particularly for exact DMD and kernelized EDMD, ResDMD has traditionally been applied by dividing snapshot data into a training set and a quadrature set. Through a novel computational approach of solving a dual least-squares problem, we demonstrate how to eliminate the need for two datasets. We provide an analysis of these new residuals for exact DMD and kernelized EDMD, demonstrating ResDMD's versatility and broad applicability across various dynamical systems, including those modeled by high-dimensional and nonlinear observables. The utility of these new residuals is showcased through three diverse examples: the analysis of cylinder wake, the study of aerofoil cascades, and the compression of transient shockwave experimental data. This approach not only simplifies the application of ResDMD but also extends its potential for deeper insights into the dynamics of complex systems.
Paper Structure (16 sections, 3 theorems, 48 equations, 8 figures, 1 table, 8 algorithms)

This paper contains 16 sections, 3 theorems, 48 equations, 8 figures, 1 table, 8 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. EDMD
  4. ResDMD
  5. ResDMD for exact DMD
  6. Exact DMD
  7. Linear dictionaries and the connection with EDMD
  8. Residuals
  9. ResDMD for kernelized DMD methods
  10. Kernelized EDMD
  11. Kernelized ResDMD: Galerkin perspective
  12. Kernelized ResDMD: Regression perspective
  13. Three examples
  14. Example 1: Cylinder wake
  15. Example 2: Verified Koopman modes of a periodic cascade of aerofoils
  16. ...and 1 more sections

Key Result

Proposition 3.1

Suppose that $M\leq d$, $\mathbf{X}^*\mathbf{U}$ has rank $M$ and that $\widetilde{\mathbf{K}}_{\mathrm{DMD}}$ is the output of alg:DMD_vanilla with input $r=M$. Then for any eigenvalue-eigenvector pair $\lambda$ and $\mathbf{v}\in \mathbb{C}^{M}$ of $\widetilde{\mathbf{K}}_{\mathrm{DMD}}^*$, the re

Figures (8)

  • Figure 1: Residuals computed using \ref{['alg:kernel_ResDMD1']} for the cylinder flow. The top row shows the location of the eigenvalues and the residuals. The bottom row plots the residuals against $\sqrt{1-|\lambda|^2}$ with the black line corresponding to an exact relation.
  • Figure 2: Pseudospectra computed using \ref{['alg:kernel_ResDMD2']} for the cylinder flow. The shaded greyscale corresponds to the value of $\epsilon$, and the kernelized EDMD eigenvalues are shown in red, some of which are spurious. We can detect spurious eigenvalues through pseudospectra.
  • Figure 3: Same as \ref{['fig:cylinder2']}, but now pseudospectra are computed using the naive residual in \ref{['biagbfvuibi']}. The approximation of the pseudospectrum is clearly wrong and we cannot detect spurious eigenvalues (without the additional knowledge that the true eigenvalues should be on the circle for this example).
  • Figure 4: Residuals computed using \ref{['alg:exact_ResDMD1']} for the periodic cascade of aerofoils. The black curve in the left panel is a portion of the unit circle.
  • Figure 5: Verified DMD modes for the periodic cascade of aerofoils. Due to conjugate symmetry, we have only shown modes with $\mathrm{Im}(\lambda)\geq 0$. The filled-in magenta regions show the positions of the aerofoils, and the setup is periodic in the vertical direction.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Proposition 3.1
  • proof
  • Proposition 4.1: Proposition 1 of williams2015kernel
  • Proposition 4.2