Robust Principal Component Analysis for Modal Decomposition of Corrupt Fluid Flows

Abstract: 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 (PCA), also known as proper orthogonal decomposition (POD) 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 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.

An Overview of Robust Principal Component Analysis for Modal Decomposition of Corrupted Fluid Flows

The paper titled "Robust Principal Component Analysis for Modal Decomposition of Corrupt Fluid Flows" provides a nuanced exploration of how robust principal component analysis (RPCA) can be effectively employed to improve the quality of modal analysis in fluid dynamics, especially in scenarios where data corruption is prevalent. The authors, Isabel Scherl et al., handle the degradation issues inherent in experimentally acquired flow fields and propose a RPCA framework to enhance data integrity by addressing spurious and missing entries.

The principal focus of the research is on utilizing RPCA, a robust variant of the traditional principal component analysis (PCA), to recover coherent structures in flow data that might otherwise be masked by noise and corruption. In this context, RPCA acts by decomposing a given data matrix into a low-rank matrix representative of global coherent flow structures and a sparse matrix that encapsulates outliers or corrupt data.

Summary of Methodological Approach

The researchers present RPCA as an advancement over traditional modal decomposition techniques, such as PCA (also referred to as proper orthogonal decomposition in fluids) and dynamic mode decomposition (DMD), both of which are highlighted for their susceptibility to outliers and noise. They delineate a systematic comparison of RPCA against these conventional methods over a series of simulations and experiments.

The paper demonstrates RPCA’s efficacy through multiple fluid flow examples:
- Flow Past a Cylinder: Both from direct numerical simulation (DNS) and particle image velocimetry (PIV) measurements, RPCA effectively filtered the dominant coherent structures from artificially corrupted data.
- Turbulent Channel Flow: Considered a more complex flow, RPCA was evaluated against high-dimensional turbulent data, showing a reduction in noise while preserving essential coherent structures.
- Cross-Flow Turbine Wake: Addressed the practical challenges of PIV data with substantial missing values and demonstrated RPCA's effectiveness in reconstructing coherent wake phenomena.

Key Numerical Results

RPCA filtering demonstrated remarkable capability in retrieving corrupted data structures, notably with recovery performances even when more than 50% of the data points were artificially corrupted. The analysis employed rigorous quantitative assessments, such as comparing singular value spectra and mode structures of both RPCA and standard PCA, substantiating the superior filtering qualities of RPCA especially in coherency preservation and outlier detection.

Implications and Future Directions

The implications of this work are notably practical and theoretical. On the practical front, RPCA introduces a robust method for PIV data processing pipelines, particularly enhancing the pre-processing steps for improved modal analyses. Theoretically, it illuminates new paths for exploring data-driven models in fluid dynamics through robust statistics, potentially affecting future study areas such as turbulence modeling and flow control.

For prospective developments, the insights gained from this work can be extended to investigate the scalability of RPCA in three-dimensional flow fields and other high-dimensional systems. Furthermore, adapting RPCA for online processing in real-time applications could constitute another significant step forward.

In summary, this research positions RPCA as a pivotal tool for enhancing the fidelity of fluid flow data analysis, accentuating its potential for broader applications in the field of fluid dynamics and beyond.