Sparsity-promoting dynamic mode decomposition

By M. Jovanovi'c, Peter J. Schmid, Joseph W. Nichols. 2012

901 citations recorded by the source corpus.

Dynamic mode decomposition (DMD) represents an effective means for capturing the essential features of numerically or experimentally generated flow fields. In order to achieve a desirable tradeoff between the quality of approximation and the number of modes that are used to approximate the given fields, we develop a sparsity-promoting variant of the standard DMD algorithm. Sparsity is induced by regularizing the least-squares deviation between the matrix of snapshots and the linear combination of DMD modes with an additional term that penalizes the l1-norm of the vector of DMD amplitudes. The globally optimal solution of the resulting regularized convex optimization problem is computed using the alternating direction method of multipliers, an algorithm well-suited for large problems. Several examples of flow fields resulting from numerical simulations and physical experiments are used to illustrate the effectiveness of the developed method.

Subfields: Civil and Structural Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/61c5ce3f7f109150b471bf5b7c0f63aad480e564

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Citations in the corpus

Higher Order Dynamic Mode Decomposition (2017, background): “In fact, the accuracy may be an issue [61, 29] when calculating the DMD expansion (1.”
Model Reduction for Flow Analysis and Control (2017, methodology): “Alternatively, one can seek a sparse representation of the dynamics, in the form of a small number of modes with nonzero amplitude, by penalizing the 1 norm of the vector of mode amplitudes ( Jovanović et al. 2014).”
Generalizing Koopman Theory to Allow for Inputs and Control (2016, positive): “The theory also allows for critical enabling theoretical augmentations that can take advantage of compression and sparsity [22, 7, 16], multiresolution/multiscale phenomenon [27], denoising [9, 18], data fusion [52], extended and kernel DMD [51, 50], and control [36].”
De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets (2015, methodology): “Moreover, based on the two-step analysis developed here, the de-biasing procedure is generalizable to other DMD-like algorithms that appeal to Koopman spectral analysis, such as optimal mode decomposition [21,58], streaming DMD [26], sparsity-promoting DMD [29], non-uniform DMD [23], and optimized D”
Dynamic Mode Decomposition with Control (2014, background): “This connection has already been demonstrated for DMD both in time and space [6, 22, 53].”
Multi-scale proper orthogonal decomposition of complex fluid flows (2018, methodology): “More robust variants of this method are the optimized DMD by Chen et al. (2012) or the sparsity promoting DMD by Jovanović et al. (2014) in which the L2 minimization problem is extended to the entire dataset and not just on the last realization.”
Parallel data-driven decomposition algorithm for large-scale datasets: with application to transitional boundary layers (2016, methodology): “The choice and amplitudes of these modes are determined using the optimization algorithm described in [6], and the resulting amplitude distribution is shown versus the detected frequencies in Fig.”
Linearly-Recurrent Autoencoder Networks for Learning Dynamics (2017, neutral): “Sparsity-promoting DMD [10] is a post-processing method which identi-ﬁes the optimal amplitudes of Koopman modes for reconstructing the snapshot sequence with an l 1 penalty.”
Data-driven discovery of Koopman eigenfunctions for control (2017, methodology): “Current data-driven methods to approximate the Koopman operator include extended DMD (EDMD) [70,71,29] and the variational approach of conformation dynamics (VAC) [49,50].”
Colour of turbulence (2016, methodology): “In general, model reduction techniques (Sirovich 1987; Berkoo et al. 1993; Rowley 2005; Mezić 2005, 2013; Lumley 2007; Rowley et al. 2009; Schmid 2010; Chen et al. 2012; Jovanović et al. 2014) can also be used for this purpose.”
Unsteady effects of strong shock-wave/boundary-layer interaction at high Reynolds number (2017, methodology): “Note that the SPDMD method does not simply chose the DMD modes based on their magnitude, but identifies modes having the strongest influence on the complete snapshot sequence (Jovanović et al. 2014).”
Linear observer synthesis for nonlinear systems using Koopman Operator framework (2016, methodology): “EDMD is a Galerkin weighted residual approach which uses a dictionary of basis functions to approximate KEFs and corresponding KEs.”
Adaptive separation control of a laminar boundary layer using online dynamic mode decomposition (2020, methodology): “Furthermore, the TDMD is combined with the sparsity-promoting DMD (SP-DMD) approach of Jovanović et al. (2014) to enable a systematic study of the tradeoff between the quality of approximation and the number of modes that are used.”