Residual Dynamic Mode Decomposition: Robust and verified Koopmanism

• ResDMD computes spectra and pseudospectra with error control, helping detect spectral pollution and providing rigorous convergence guarantees for general Koopman operators.DMD-type discretizations can produce spurious modes without physical relevance.Source: §1 · p. 3
• ResDMD measures how invariant a candidate observable subspace is using residual upper bounds, which also support ranking approximated eigenvalues and Koopman modes.These residuals are associated with the underlying infinite-dimensional operator, unlike residuals from finite DMD matrices.Source: §1 · p. 3
• ResDMD handles continuous and discrete spectra jointly through smoothed spectral-measure approximations with explicit high-order convergence, without assuming ergodicity.It can be applied to data from short or long trajectories.Source: §1 · p. 4
• For nonlinear, high-dimensional flows, ResDMD can use learned dictionaries such as kEDMD or rank-reduced DMD while retaining convergence theory and a-posteriori verification.The introduction identifies turbulent phenomena as an example of strongly nonlinear systems with large state spaces.Source: §1 · p. 4
• The paper applies ResDMD to numerical and experimental fluid flows and introduces residual-based modal ordering for more accurate compression with a smaller DMD dictionary.It also links spectral measures to turbulent power spectra and illustrates the link with experimental boundary-layer measurements.Source: §1 · p. 4
• The paper validates its methods on four cases: numerical cylinder flow, experimental hot-wire boundary-layer flow, PIV wall-jet flow, and microphone measurements of laser-induced plasma.Only flow or pressure fields are used to extract dynamical information in each case.Source: §1 · p. 4

2. Recalling dynamic mode decomposition

• Given discrete snapshot pairs from a general dynamical system, the section places Koopman analysis in L2(Ω, ω) without requiring ω to be invariant or imposing assumptions on K.Snapshots may come from one long trajectory or multiple shorter trajectories.Source: §2 · pp. 4–5

2.1. Extended dynamic mode decomposition

• EDMD uses snapshot data and a user-chosen dictionary to construct a finite-dimensional approximation of K; data-driven dictionaries can help when the state-space dimension is large.The matrix is obtained by solving a weighted least-squares problem that approximates K on the dictionary's span.Source: §2.1 · pp. 5–6

2.2. Quadrature and Galerkin methods

• As the quadrature converges, EDMD approaches a Galerkin representation of K on the dictionary subspace, but spectral pollution can produce spurious modes and motivates independent accuracy checks.The EDMD eigenvalues approach the spectrum of the projected operator in the large-data limit.Source: §2.2 · pp. 6–7

2.2.1. Convergence of the quadrature rule

• Quadrature convergence can arise from random, high-order, or ergodic sampling; the paper uses random and ergodic sampling, common in experimental data collection.Random sampling typically converges at O(M^-1/2); high-order quadrature can be faster but difficult in high dimensions, while ergodic-sampling rates depend on the problem.Source: §2.2.1 · p. 7

2.2.2. Relationship with DMD

• DMD is a Galerkin approximation of the Koopman operator with linear monomials as basis functions, whereas EDMD permits nonlinear dictionary functions.Source: §2.2.2 · p. 7

2.3. Koopman mode decomposition

• With convergent quadrature, projection of an observable into the dictionary subspace supports Koopman mode decomposition; each mode corresponds to a row of the eigenvector-based reconstruction matrix.Source: §2.3 · pp. 7–8

3. Residual dynamic mode decomposition (ResDMD)

• ResDMD approximates both K and K∗K from measured data, enabling convergence guarantees and a posteriori error checks on computed spectra and Koopman modes.These checks help identify reliable features and rectify spectral pollution in DMD-type methods.Source: §3 · p. 8

3.2.1. Avoiding spurious eigenvalues

• Discarding candidate eigenpairs whose relative residual exceeds a tolerance ε rejects spurious or numerically inaccurate modes, avoiding spectral pollution.This cleanup is a modification of EDMD and is also applicable to Koopman mode decompositions.Source: §3.2.1 · p. 9

3.2.2. Computing the full spectrum and pseudospectra

• Algorithm 1 returns eigenvalues inside the ε-pseudospectrum, but may not approximate it all; Algorithm 2 provides approximations with convergence guarantees as N grows and the grid is refined.Under quadrature convergence, Algorithm 2 converges uniformly on compact subsets to the approximate point pseudospectrum and computes ε-approximate eigenfunctions.Source: §3.2.2 · p. 10

3.3. Choice of dictionary

• For large state dimension d, DMD uses a low-rank approximation, while kEDMD uses kernel inner products to construct a typically nonlinear dictionary.When the kernel can be evaluated in O(d) operations, the approximate operator can be constructed in O(dM^2) operations.Source: §3.3 · p. 11

3.3.1. The problem of vanishing residual estimates √

• When dictionary rank r approaches snapshot count M, residual estimates can vanish or become inaccurate, preventing the large-data convergence results that assume a fixed basis.This can prevent application of the results in Section 3.2.Source: §3.3.1 · p. 11

3.3.2. Using two subsets of the snapshot data

• Using separate snapshot sets for dictionary selection and ResDMD analysis allows convergence as the second set grows and enables residual-based checks of dictionary quality and size.DMD or kEDMD may suffer spectral pollution, but here they select the dictionary rather than compute spectral properties; ResDMD checks it using the second set.Source: §3.3.2 · pp. 12–13

4.1. Spectral measures and Koopman mode decompositions

• For measure-preserving systems, the Koopman operator is an isometry, and spectral measures encode discrete and continuous spectral components that support Koopman mode decompositions.The discrete part is atomic on eigenvalues, while the continuous part includes absolutely continuous and singular continuous components; absent singular continuous spectrum, observables admit a decomposition using eigenfunctions and continuously parameterized modes.Source: §4 · p. 13, §4.1 · pp. 13–14

4.2. Spectral measures and autocorrelations

• The Fourier coefficients of a spectral measure are determined by forward-time autocorrelations, making the measure a signature of the system’s forward-time dynamics.The coefficients at negative and positive indices are equal, so the measure is completely determined by autocorrelations for n ≥ 0.Source: §4.2 · p. 14

4.3. Computing spectral measures

• Autocorrelations can be estimated by random sampling, high-order quadrature, or ergodic sampling; ergodic sampling uses one trajectory and computes averages rapidly with the FFT.Source: §4.3 · pp. 14–15
• ResDMD residuals enable adaptive, rigorous selection of the smoothing parameter and computation from general snapshot data without requiring long trajectories.The approach starts from computed autocorrelations and uses a filter function to form a smooth approximation of the spectral measure.Source: §4.3 · pp. 15–16
• Under suitable smoothness assumptions, the filtered measure approximates the spectral density to O(N_ac^-m log(N_ac)); rescaled values approximate point spectrum, with weak convergence for test functions.The filter is even, compactly supported, and satisfies specified smoothness and derivative conditions; the results are attributed to Colbrook & Townsend (2021, Section 3).Source: §4.3 · p. 15

4.4. Interpretation of Koopman spectral measures as power spectra

• The smoothed measure is a discretized power spectrum, avoiding broadening by windowing in the frequency domain and providing rigorous convergence as N_ac increases.The approximation uses a time window [−T,T]; as N_ac tends to infinity at fixed frequency resolution, the spectral measure is an appropriate limit of the power spectrum.Source: §4.4 · p. 16

5.1. Computational setup

• The Re = 100 cylinder flow was simulated with a two-dimensional lattice-Boltzmann solver, and retaining 12 snapshots per shedding period preserved DMD results.The analysis uses 500 snapshots in its first data set and 1,000 further snapshots in the second; it compares a 200-function linear dictionary with a better-conditioned 400-function non-linear dictionary.Source: §5.1 · p. 17, §5.2 · p. 17

5.2. Results

• The minimal residual τ_N(−0.5) converges as M′′ increases and decreases monotonically with N toward 0.5.The limiting value is ∥(K + 0.5)−1∥−1 = 0.5; Figure 4 shows the monotonicity as M′′ increases.Source: §5.2 · p. 18, Figure 4 · p. 18
• The linear dictionary exhibits severe spectral pollution, while the non-linear dictionary better captures the lattice structure of Koopman eigenvalues wrapping around the unit circle.Some linear-dictionary eigenvalues have relative residuals of approximately 10^-8, but ResDMD detects spurious modes through their residuals.Source: §5.2 · p. 18, Figure 5 · p. 19
• The linear dictionary captures lower spectral content up to about j = 35 but misses strongly non-linear eigenfunctions, whereas the non-linear dictionary better captures higher-order spectral information.The computed spectra and eigenspaces are evaluated against reference eigenvalues and eigenfunctions, and the analysis confirms that Algorithm 1 provides error bounds.Source: §5.2 · p. 18
• Modes 1 and 2 agree across dictionaries, but Mode 20 differs substantially; residual measures support the non-linear dictionary’s higher-order modes as physical rather than spurious.The difference indicates that non-linearity matters in the expansion for higher-order modes.Source: §5.2 · pp. 18–19

6.1. Experimental setup

• Experiments compare a canonical zero-pressure-gradient boundary layer (Reτ = 1400) with a BL3 case where perpendicular flow injection creates an embedded shear layer, using traversed hot-wire measurements.The high-Reynolds-number boundary layer presents a wide range of length scales; the embedded shear layer is expected to have characteristic scales with broadband energy distribution.Source: §6 · pp. 19–20, §6.1 · p. 20

6.2. Results

• Because measurements were collected along a single line, DMD-type methods cannot robustly approximate the Koopman operator, although Algorithm 5 can still compute its spectral measures.The study compares Algorithm 5 spectral measures with Welch PSD estimates from the fine-temporal-resolution flow data.Source: §6.2 · pp. 20–21
• Below 10 Hz, spectral measures are approximately 1 dB above PSD while high-frequency behavior is almost identical; higher-order filters produce sharper low-frequency peaks.The difference is attributed to broadening in conventional PSD processing; rigorous convergence assures a more accurate low-frequency power measure as N_ac → ∞.Source: §6.2 · pp. 21–22
• Algorithm 5’s spectral-measure integration converges at expected rates for first-, second-, fourth-, and infinite-order filters, whereas PSD initially appears second-order then stagnates.The convergence test integrates against φ(θ) = exp(sin(θ)) for data at y/δ₀ = 0.1000; it is a numerical demonstration, not a physical example.Source: §6.2 · pp. 21–22, Figure 10 · p. 22

7.1. Experimental setup

• The TR-PIV measurements used U_j=50 m/s, Re_jet=63.5×10^3, 0.25-mm spatial resolution, and 12,000 frame pairs per second.The approximately 75 × 45 mm field of view did not capture the roughly 200-mm-thick wall jet, so standard DMD under-predicted shear-layer energies.Source: §7.1 · pp. 22, 24
• Higher-order filters produce a sharper peak at low frequencies.Source: §7.1 · p. 23

7.2. Results

• In the wall-jet example, using Algorithms 3 and 4 with N=2000, including nonlinear functions in the dictionary corresponds to a much better approximation of K’s spectral content near zero.The total-energy forecasts are averaged over 12,000 initial conditions. The estimated error bound for the eigenvalues in Fig. 14 (right) is approximately 10^-10.Source: Figure 14 · p. 24, §7.2 · p. 25
• With ε=0.5, residual measures enable accurate selection of physical transient modes among 656 modes, interpreted as short-lived turbulence including shear-layer travel and breakdown.The modes also show rolling eddies, ejection-like motions, and possible interactions between ejection-type vortices and the shear layer.Source: §7.2 · p. 25, Figure 15 · p. 26
• Streamwise-velocity spectra are consistent across boundary-layer heights, while vertical-velocity spectra and spectral measures show more energy as y/y_m increases.The streamwise consistency is attributed to the dominant strong shear layer in the wall jet.Source: §7.2 · pp. 25, 27, Figure 16 · p. 27

8.1. Experimental setup

• The experiment uses 65 laser-induced plasma realizations, gates each signal to retain the direct propagation path, and resolves frequencies up to 100 kHz.A 1/8-inch B&K 4138 microphone records at 1.25 MS/s; the initial supersonic propagation is confined to about 3–4 mm from the focal point.Source: §8.1 · p. 28

8.2. Results

• The analysis uses time-delay embedding, where d is the initial time interval for future-state prediction, and compares a linear dictionary with a nonlinear dictionary formed by adding the Algorithm 4 dictionary at N=200.The final 5 realizations test the resulting Koopman mode decomposition.Source: §8.2 · pp. 28–29
• The nonlinear dictionary better predicts unseen pressure signals, capturing the negative peak and later high-frequency oscillations missed by the linear dictionary, with around 6% average relative L2 error at d=15.The linear dictionary predicts the positive peak but misses the magnitude and shape of the negative peak.Source: Figure 19 · p. 29, §8.2 · p. 29
• Residual ordering gives better dynamics compression than modulus ordering across tested dictionary sizes and delay values.The comparison uses 40 modes for each ordering in the test-set prediction, with error curves shown for varying dictionary sizes and d.Source: §8.2 · pp. 29–30, Figure 21 · p. 30, Figure 20 · p. 30

9. Discussion and conclusions

• ResDMD computes residuals during modal decomposition, enabling error-controlled spectra and pseudospectra and smoothed spectral measures with explicit high-order convergence theorems.The method is presented as a toolbox for robust and verified Koopmanism.Source: §9 · pp. 30–31
• For cylinder flow, residuals identify spurious modes in a linear dictionary; for turbulent boundary-layer flow, they identify physical transient modes and filter spurious ones.The transient-mode identification supports greater understanding of turbulent structures within the boundary layer.Source: §9 · pp. 30–31
• Relating the Koopman spectral measure to power spectral density reduces low-frequency broadening in processed experimental data and provides convergence guarantees.The conclusions contrast this broadening with that commonplace in conventional power spectral density calculations.Source: §9 · p. 31
• For laser-induced plasma acoustics, ordering modes by residual enables greater data compression than ordering them by modulus.Source: §9 · p. 31
• ResDMD may enable exotic Koopman spectral features (Colbrook & Hansen 2020; Colbrook 2019) and extensions to continuous-time systems, control, and stochastic systems.The authors state that calculating residuals is as computationally efficient as traditional DMD.Source: §9 · p. 31

• A normal matrix has an orthonormal eigenbasis whose projections decompose the space and diagonalize the operator.Source: Appendix A · p. 31
• Because an isometric Koopman operator need not be normal, its spectral information is understood through a unitary extension, whose spectral measures are canonical and extension-independent.The extension is defined on a larger Hilbert space containing L2(Ω, ω) (Nagy et al. 2010, Proposition I.2.3).Source: Appendix A · p. 31
• When the unitary extension has continuous spectrum, a projection-valued measure on the unit circle decomposes it across discrete and continuous spectra, and defines each observable’s scalar spectral measure.For an observable g, μ_g(U) := ⟨E(U)g, g⟩; the corresponding measure ν_g is defined on [−π, π]per by λ = exp(iθ).Source: Appendix A · pp. 31–32