Unsteady Flow Sensing and Estimation via the Gappy Proper Orthogonal Decomposition

By Karen E. Willcox. 2004

485 citations recorded by the source corpus.

Abstract The proper orthogonal decomposition (POD) has been widely used in fluid dynamic applications for extracting dominant flow features. The “gappy” POD is an extension to this method that allows the consideration of incomplete data sets. In this paper, the gappy POD is extended to handle unsteady flow reconstruction problems, such as those encountered when limited flow measurement data is available. In addition, a systematic approach for effective sensor placement is formulated within the gappy framework using a condition number criterion. This criterion allows for accurate flow reconstruction results and yields sensor configurations that are robust to sensor noise. Two applications are considered. The first aims to reconstruct the unsteady flow field using a small number of surface pressure measurements for a subsonic airfoil undergoing plunging motion. The second considers estimation of POD modal content of a cylinder wake flow for active control purposes. In both cases, using the dominant POD basis vectors and a small number of sensor signals, the gappy approach is found to yield accurate flow reconstruction results.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/03dd36fbf10fc1bba666c6569e6d041322690d0d

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Citations in the corpus

Missing Point Estimation in Models Described by Proper Orthogonal Decomposition (2008, background): “…amount to evaluating the inner products over the entire spatial domain, which may become a formidable task in large-scale or high-dimensional systems, and computationally prohibitive in parameter- and time-varying models where the evaluation of these inner products needs to be performed online.”
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows (2012, background): “For example, for the various forms of empirical interpolation outlined in Section 1, several algorithms for selecting sample indices have been developed around the objective of minimizing the error in the interpolated snapshots [22, 31, 25], the difference between the interpolated snapshots and thei”
A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics (2015, methodology): “We next consider the data interpolation perspective: PBDW reduces to the generalized empirical interpolation method (GEIM) [10–12] forN DM , any given ZN ; PBDW reduces to Gappy-POD [13, 14] for M > N , ZN generated by a POD, and u N;M ́ N;M (no update correction).”
Nonlinear model order reduction based on local reduced‐order bases (2012, unknown): “A second class of hyper reduction techniques are based on the concept of empirical interpolation [20, 21]—also known in its discrete form as the Gappy POD technique [6, 20, 26]—or the related discrete empirical interpolation method [5].”
Multi-scale proper orthogonal decomposition of complex fluid flows (2018, methodology): “…et al. 2014), video analysis for adaptive masking or image pre-processing (Mendez et al. 2018a, 2017), flow sensing (Venturi & Karniadakis 2004; Willcox 2006), flow control (Brunton & Noack 2015; Bergmann & Cordier 2008), reduced order modeling (Deane et al. 1991), experimental validation of…”
POD-DEIM model order reduction for strain softening viscoplasticity (2017, methodology): “To overcome this inefficiency, hyper-reduction methods, such as Missing Point Estimation [10], Gappy Proper Orthogonal Decomposition [11] and Discrete Empirical Interpolation Method (DEIM) [5], have been proposed to approximate the nonlinear contributions.”
Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison (2011, methodology): “These include interpolatory methods such as the empirical interpolation method [44], the discrete empirical interpolation method [45,46], the closely related group finite element approach [49], the gappy POD [47,48] and a novel two-level discretization method [50].”
Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations (2011, methodology): “In particular, approaches have been developed to select points that minimize the error in the reconstructed snapshots [27, 30], the difference between the reconstructed snapshots and their optimal projections [25, 28], and the condition number of the normal-equations matrix used for reconstruction [”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “However, reconstruction error bounds for the globally optimal selection are known for DEIM [123], Q-DEIM [124] and related POD sampling methods [119], [120].”
Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, methodology): “These grid point selection procedures are later improved by incorporating a greedy algorithm from [42].”
Using adaptive proper orthogonal decomposition to solve the reaction--diffusion equation (2007, methodology): “Proper orthogonal decomposition is used in many disciplines of science and engineering, including fluid mechanics [16,47], data reconstruction [5, 49 ], controls [1,25,33], heat transfer [30,31], nondestructive evaluation [17], and biology [26].”
A ‘best points’ interpolation method for efficient approximation of parametrized functions (2008, background): “This example has also been investigated in [21, 26] for a particular Reynolds number of 100.”