Toward compressed DMD: spectral analysis of fluid flows using sub-Nyquist-rate PIV data

• DMD requires Nyquist-Shannon-compliant data, which may be unavailable in fluid experiments because time-resolved PIV is expensive and standard PIV sampling is around 15 Hz.Source: §1 · p. 1
• The proposed method combines DMD and compressed sensing, using POD projections to encode high-dimensional PIV data as low-dimensional coefficients before recovering modes.Source: §1 · p. 2
• The examples test noisy, nearly frequency-sparse synthetic data with mixed POD structures and cylinder-flow PIV data that are only approximately sparse.Source: §1 · p. 2
• The method correctly identifies the characteristic frequencies and oscillatory modes dominating both numerical and experimental signals from sub-Nyquist-rate samples.Source: §1 · p. 2

2 Numerical method

• The numerical method combines compressed sensing for vector-valued signals with POD dimension reduction and sampling strategies for sub-Nyquist data.Source: §2 · p. 2

2.1 Scalar signals

• For scalar signals, the method recovers sparse or compressible basis coefficients from undersampled measurements by minimizing the ℓ1 norm as a tractable proxy for sparsity.Source: §2.1 · p. 3

2.2 Choice of basis, measurement

• The method assumes temporal sparsity in the Fourier basis and measures the signal at selected instants, using random sampling rather than uniform sub-Nyquist sampling.Source: §2.2 · pp. 3–4

2.3 Vector-valued signals

• For vector-valued data, mixed-norm recovery enforces shared row sparsity across measurements, so the data jointly identify dominant frequencies rather than estimating each signal separately.Source: §2.3 · p. 4

2.4 Efficiency through POD projection

• POD projection reduces the cost of compressed sensing on large PIV fields; recovered frequency coefficients are combined with POD modes to form spatial modes that have no growth or decay.Source: §2.4 · pp. 4–5

2.5 Sampling strategy

• The minimum/maximum separation strategy samples at random intervals bounded by smin and smax, while respecting minimum timing constraints and seeking coverage of signal phases.Source: §2.5 · pp. 5–6

2.6 Summary of method

• The workflow triggers data collection from a sampling schedule, computes POD modes and coefficients, solves the sparse-recovery problem, then lifts the coefficients into spatial modes; OMP can replace optimization for long signals.Source: §2.6 · p. 6

3 Results

• Both examples correctly identify the characteristic frequencies and oscillatory modes dominating their signals using only sub-Nyquist-rate samples.The examples are a constructed numerical dataset and data from a fluid-flow experiment.Source: §3 · p. 6

3.1 Canonical dataset

• The canonical signal is sampled at least eight times slower than the Nyquist-Shannon criterion requires, and traditional techniques cannot reconstruct it from those samples.Its two dominant oscillations are embedded in low-amplitude broadband noise, while the POD modes mix features of the true oscillatory modes.Source: §3.1 · pp. 6–7
• The recovered canonical frequencies have less than 2.5% error each and agree exactly with the time-resolved-data DFT; secondary-frequency power has noticeable error.The remaining frequencies have negligible energy after ℓ1 minimization.Source: Figure 4 · p. 7, §3.1 · p. 7
• The recovered canonical modes mostly resemble the original oscillatory modes, with some aberrations.Compressed sensing combines features mixed across the POD modes to recover those structures.Source: §3.1 · p. 7, Figure 5 · p. 8

3.2 Flow past a cylinder

• For cylinder flow, 12 POD modes capture 75% of dataset energy; sampling 33 of 2000 snapshots uses spacings of 50–70 against a Nyquist spacing of 10.The first two POD modes dominate, and the sparse samples do not resolve the fastest oscillations.Source: §3.2 · p. 9, Figure 12 · p. 10
• OMP identifies the cylinder wake-shedding frequency with less than 2.5% error and energy in good agreement with the time-resolved-data DFT, but does not recover its harmonic peaks.The authors attribute this boundary to the Fourier-sparsity assumption: the DFT also identifies only the fundamental frequency, while the OMP computation's other peaks are spurious.Source: §3.2 · p. 10
• The dominant OMP mode captures the main coherent structures and pairs dominant POD modes with the wake-shedding frequency without requiring prior knowledge of the flow dynamics.The mode does not exactly match the corresponding DMD modes; the authors note the approach may generalize to more complex flows where POD-to-frequency pairing is not obvious.Source: §3.2 · p. 10

4 Conclusions and future work

• The method computes oscillatory spatial modes from sub-Nyquist-rate data, but currently represents temporal oscillations without growth or decay.The authors suggest more general bases or adaptive searches for growth and decay rates as future directions.Source: §4 · p. 11
• Correct frequencies and spatial modes were identified in numerical and experimental demonstrations, also verifying the proposed random-sampling strategy.The computations used ℓ1 minimization for numerical data and OMP for experimental data.Source: §4 · p. 11
• Compressed sensing can address insufficient temporal resolution with slow-sampling sensors and can also help faster sensors analyze faster dynamics.The authors describe this as a broader application beyond the PIV motivation.Source: §4 · p. 11
• POD encodes high-dimensional PIV measurements in a low-dimensional subspace, making them amenable to standard compressed sensing algorithms.Compressed sensing can reduce data storage requirements, while POD addresses computational strain from large measurement vectors.Source: §4 · p. 11
• The authors identify systems limited by sampling rates, storage, or high-dimensional measurements as likely targets, and propose exploring bases beyond Fourier.The paper describes broad applicability as likely and non-Fourier bases as a future extension.Source: §4 · p. 11