Robust principal component analysis for modal decomposition of corrupt fluid flows

Modal analysis techniques are used to identify patterns and develop reduced-order models in a variety of fluid applications. However, experimentally acquired flow fields may be corrupted with incorrect and missing entries, which may degrade modal decomposition. Here we use robust principal component analysis (RPCA) to improve the quality of flow-field data by leveraging global coherent structures to identify and replace spurious data points. RPCA is a robust variant of principal component analysis, also known as proper orthogonal decomposition in fluids, that decomposes a data matrix into the sum of a low-rank matrix containing coherent structures and a sparse matrix of outliers and corrupt entries. We apply RPCA filtering to a range of fluid simulations and experiments of varying complexities and assess the accuracy of low-rank structure recovery. First, we analyze direct numerical simulations of flow past a circular cylinder at Reynolds number 100 with artificial outliers, alongside similar particle image velocimetry (PIV) measurements at Reynolds number 413. Next, we apply RPCA filtering to a turbulent channel flow simulation from the Johns Hopkins Turbulence database, demonstrating that dominant coherent structures are preserved in the low-rank matrix. Finally, we investigate PIV measurements behind a two-bladed cross-flow turbine that exhibits both broadband and coherent phenomena. In all cases, we find that RPCA filtering extracts dominant coherent structures and identifies and fills in incorrect or missing measurements. The performance is particularly striking when flow fields are analyzed using dynamic mode decomposition, which is sensitive to noise and outliers.

Figure 1

Schematic of RPCA filtering applied to corrupt flow-field data. Corrupted snapshots are arranged as column vectors in the matrix $\mathbf{X}$, which is decomposed into the sum of a low-rank matrix $\mathbf{L}$ and a sparse matrix of outliers $\mathbf{S}$.

Figure 2

Flowchart showing how we apply RPCA filtering to a data matrix and analyze the results. Depending on the data set in question, the data matrix may be artificially corrupted prior to RPCA filtering (Sec. 4). The results of principal component analysis and dynamic mode decomposition performed on the data matrix ($\mathbf{X}$) are referred to as PCA and DMD modes, respectively, whereas those same operations performed on the low-rank matrix ($\mathbf{L}$) are referred to as RPCA and RDMD modes.

Figure 3

Left: Example flow-field data. Right: Singular value spectrum for each data set. Mean flow travels from left to right in all cases.

Figure 4

RPCA filtering removes noise and outliers in the flow past a cylinder (black circle), from DNS (left) with $10\%$ of velocity field measurements corrupted with salt-and-pepper noise, and PIV measurements (right). All frames show resultant vorticity fields. As the parameter $\lambda$ is decreased, RPCA filtering is more aggressive, eventually incorrectly identifying coherent flow structures as outliers.

Figure 5

RPCA filtering removes vorticity-biased corruption from simulated flow past a cylinder at Reynolds number 100. Unlike results shown in Fig. 4, corrupt entries are concentrated in regions of high vorticity instead of being uniformly distributed. In the flow on the left, $\eta =1\%$ of the velocity field measurements are corrupted and on the right $\eta =10\%$ of the velocity field measurements are corrupted. All frames show resultant vorticity fields.

Figure 6

Error ($\left|\right|{\mathit{X}}_{\mathrm{uncorrupted}}-{\mathit{L}\left|\right|}_{\mathit{F}}/\left|\right|{\mathit{X}}_{\mathrm{uncorrupted}}{\left|\right|}_{\mathit{F}}$) and relative nuclear norm $\left[{\left|\right|\mathit{L}\left|\right|}_{*}/{\left|\right|\mathit{X}\left|\right|}_{*}=\mathrm{sum}\left({\mathbf{\sigma }}_{\mathit{L}}\right)/\mathrm{sum}\left({\mathbf{\sigma }}_{\mathit{X},\mathrm{uncorrupted}}\right)\right]$ of the low-rank matrix $\mathbf{L}$ compared with the uncorrupted data $\mathbf{X}$ for varying percentages of corruption.

Figure 7

Odd PCA vorticity modes of the cylinder simulations from Fig. 5 with 1% of velocity measurements corrupted with a bias toward regions of high vorticity.

Figure 8

Odd PCA vorticity modes of the cylinder simulations from Fig. 5 with 10% of velocity measurements corrupted with a bias toward regions of high vorticity.

Figure 9

$L_2$ error between the true PCA modes of the clean cylinder simulation data $\mathbf{X}$ and the RPCA and PCA modes for corrupted data with $1\%$ (left) and $10\%$ (right) vorticity-biased corruption.

Figure 10

Odd PCA vorticity modes of the experimental cylinder data for PCA and RPCA at $\lambda =0.1,1,$ and 10, along with their singular values.

Figure 11

Discrete-time DMD eigenvalues for the simulated cylinder data for small and large amounts of corruption and for increasing amounts of training data. In all cases, the RPCA-filtered DMD results dramatically outperform the standard DMD results.

Figure 12

Continuous-time DMD eigenvalues for the simulated cylinder data along with parabolic eigenvalue fits to estimate the error as in Ref. [24]. Here we use five vortex shedding periods with $\eta =1\%$ corrupt values. The RDMD parabolic coefficient is approximately $2\times {10}^4$ times smaller than the DMD coefficient.

Figure 13

Continuous-time DMD eigenvalues for the PIV cylinder data along with parabolic eigenvalue fits to estimate the error as in Ref. [24]. The RDMD parabolic coefficient is approximately two times smaller than the DMD coefficient.

Figure 14

RPCA filtering for turbulent channel flow vorticity fields with various levels of added noise and tuning parameter $\lambda$; $\eta$ represents the percentage of corrupted measurements in the velocity fields. The border colors match the color of the curve at the corresponding $\eta$ in Fig. 15.

Figure 15

TKE spectra for various levels of corruption and RPCA filtering. The TKE profiles provide a summary of the filtering that occurs at various scales. As corruption increases, the filtering remove more high-frequency information.

Figure 16

RPCA filtering of cross-flow turbine wake PIV data. The standard PIV processing pipeline (top row) includes several steps where RPCA filtering can be applied (bottom row). In the cross-correlated streamwise velocity field (top left), $23\%$ of the velocity vectors are missing. Vector validation reduces the missing vectors to $20\%$. Finally, linear interpolation is used to fill in these missing vectors. In all cases, RPCA filtering captures the relevant phase-averaged coherent structures with fewer outliers and missing data, which appear as dark spots in the standard deviation plot.

Figure 17

Continuous-time DMD eigenvalues for the turbine wake PIV data, along with parabolic eigenvalue fits to estimate the error as in Ref. [24]. The RDMD parabolic coefficient is approximately six times smaller than the DMD coefficient.