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Sparsity-promoting dynamic mode decomposition

Mihailo R. Jovanović, Peter J. Schmid, Joseph W. Nichols

arXiv:1309.4165v1physics.flu-dynmath.DSmath.OCphysics.data-an

TL;DR

The paper addresses the difficulty of approximating flow snapshots with a compact subset of influential DMD modes. It introduces ℓ1-regularized sparsity-promoting DMD and solves the resulting convex problem with ADMM. Across flow examples, the method identifies compact modal representations while balancing approximation quality and sparsity.

  • Problem

    Selecting a small subset of DMD modes that preserves snapshot-approximation quality is challenging because all modes may contribute to the full representation.

  • Method

    The method adds an ℓ1 penalty on DMD amplitudes to the least-squares reconstruction objective and uses ADMM to compute the globally optimal convex solution.

  • Results

    The method identified relevant structures across three flow examples, including the screech frequency in a supersonic jet and the dominant exponentially growing TS wave in plane Poiseuille flow.

  • Takeaways & Limitations

    Sparsity-promoting DMD selects specific modes and their temporal frequencies and growth or decay rates while providing a user-controlled approximation–sparsity tradeoff.

Abstract

from arXiv · show

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. In our method, 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 $\ell_1$-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.

I. INTRODUCTION

DMD extracts low-order representations of flow dynamics from snapshot data, but selecting a compact subset of influential modes is challenging. The paper develops a sparsity-promoting variant to balance approximation quality against the number of retained modes.

  • Motivation: Reduced mode-based models support computationally tractable studies of flow stability, receptivity, and model-based control.
  • Motivation: Selecting a subset of modes with the strongest influence on the flow remains challenging when all modes collectively approximate the data well.
  • Contribution: The proposed method augments standard DMD with sparsity-promoting regularization to trade off least-squares approximation quality against the number of modes.The regularization penalizes the ℓ1-norm of the DMD amplitude vector.
  • DMD framework: Standard DMD forms a reduced operator from snapshot matrices using an SVD-based POD basis and a least-squares fit between consecutive snapshots.
  • DMD framework: DMD approximates high-dimensional flow dynamics with a low-order representation in the subspace spanned by POD modes.The snapshot matrices are typically tall because measurement points greatly outnumber snapshots.

B. Optimal amplitudes of DMD modes

The standard DMD representation weights dynamic modes by amplitudes to approximate the full snapshot sequence. This section formulates amplitude selection as a convex optimization problem using reduced snapshot and modal quantities.

  • Modal representation: DMD amplitudes weight modes whose temporal evolution is governed by eigenvalues containing frequency and growth or decay information.The temporal evolution is represented through a Vandermonde matrix.
  • Amplitude optimization: The resulting optimization problem is convex and can be formed from the SVD factors of the snapshots together with the DMD eigenvector and Vandermonde matrices.The formulation does not require direct access to the POD modes.
  • Motivation: A superposition of all properly weighted DMD modes optimally approximates the full data sequence, motivating selection of a lower-dimensional representation.

III. SPARSITY-PROMOTING DYNAMIC MODE DECOMPOSITION

The sparsity-promoting DMD formulation selects a mode subset by balancing approximation error against the number of active amplitudes. An ℓ1 relaxation makes the problem convex and enables efficient global optimization.

  • III. SPARSITY-PROMOTING DYNAMIC MODE DECOMPOSITION: The method seeks a hierarchical description of snapshot data by selecting the subset of DMD modes that most influences approximation quality.
  • III. SPARSITY-PROMOTING DYNAMIC MODE DECOMPOSITION: A positive regularization parameter γ controls the tradeoff between least-squares approximation quality and the number of nonzero DMD amplitudes.Larger γ places stronger emphasis on sparsity.
  • III. SPARSITY-PROMOTING DYNAMIC MODE DECOMPOSITION: Replacing the cardinality penalty with an ℓ1-norm avoids the combinatorial search associated with directly minimizing the number of nonzero amplitudes.
  • III. SPARSITY-PROMOTING DYNAMIC MODE DECOMPOSITION: The relaxed sparsity-promoting DMD problem is convex, and ADMM is developed to compute its global solution efficiently for large-scale problems.
  • III. SPARSITY-PROMOTING DYNAMIC MODE DECOMPOSITION: After the sparsity structure is selected, the algorithm fixes zero amplitudes and solves a constrained convex problem for the retained nonzero amplitudes.

A. Alternating direction method of multipliers

The method reformulates sparsity-promoting DMD with auxiliary variables and an augmented Lagrangian so ADMM can exploit the separate structures of its objective terms.

  • ADMM solves the sparsity-promoting optimization problem through α-minimization, β-minimization, and Lagrange multiplier updates.Iterations continue until primal and dual feasibility tolerances are met.
  • The formulation replaces α in the sparsity term with β and imposes α = β through an augmented Lagrangian.This doubles the optimization variables but separates the quadratic approximation objective from the sparsity-promoting function.
  • The augmented Lagrangian includes a positive quadratic penalty on the deviation between α and β, with ρ controlling that penalty.When ρ = 0, it reduces to the standard Lagrangian.
  • The α-update is an unconstrained regularized quadratic program, while the β-update applies a soft-thresholding operator.These updates exploit the respective structures of the quadratic and sparsity-promoting terms.

IV. EXAMPLES

The paper evaluates sparsity-promoting DMD on three snapshot databases spanning numerical linearized flow, a simulated turbulent jet, and experimental particle-image velocimetry data.

  • Three databases test the method across plane Poiseuille flow, a screeching supersonic rectangular jet, and flow through a cylinder bundle.The datasets comprise numerical, LES, and time-resolved particle image velocimetry snapshots.

A. Two-dimensional Poiseuille flow with Re = 10000

For two-dimensional Poiseuille flow, sparsity-promoting DMD retains accurate approximations with substantially fewer modes while selecting modes across the flow’s fast, slow, and unstable dynamics.

  • Re = 10000 and kx = 1 define the two-dimensional Poiseuille-flow case, discretized with M = 150 wall-normal collocation points.The dynamics are governed by the Orr–Sommerfeld equation for wall-normal velocity fluctuations.
  • The largest DMD amplitude corresponds to an unstable eigenvalue generating an exponentially growing Tollmien–Schlichting wave.The snapshot matrix has rank r = 26, and amplitudes are examined against frequency and eigenvalue real part.
  • 18 modes provide nearly identical performance to full-rank DMD, while reducing to 13 modes compromises the least-squares residual by only 1.3 percent.Reducing from 13 to 12 modes adds approximately 3% degradation; reducing from 12 to 6 adds only 2.5% more.
  • Nz = 6 offers a reasonable compromise between least-squares approximation quality and the number of retained modes.The study also reports nearly identical full-rank performance at 18 modes.
  • Sparsity increasingly coarsens selections from both slow and fast eigenvalue branches, ultimately retaining one fast, one slow, and one unstable mode at Nz = 3.These branches represent distinct perturbation dynamics in the channel.

B. A screeching supersonic jet

The screeching supersonic jet combines a narrow-banded feedback tone with broadband turbulent fluctuations, providing a demanding test for sparse DMD mode selection. Sparsity-promoting DMD identifies compact representations that retain dominant frequencies while trading approximation quality against the number of modes.

  • Motivation: The screeching jet contains a loud narrow-banded feedback tone embedded in broadband turbulent flow, motivating sparse extraction of its coherent feedback loop.The method targets the screech mechanism with as few modes as possible.
  • Configuration: The simulated jet used a convergent rectangular nozzle of aspect ratio 4 and an LES mesh of about 45 million control volumes.The nozzle geometry matched an experimental nozzle, and the simulation was validated against experimental measurements.
  • Measurements: Pressure spectra at both probes peak near St ≈0.3, but the shear-layer probe exhibits broader frequency content than the narrow-banded screech location.The upstream-directed acoustic-beam location is relatively narrow-banded, whereas turbulent shear-layer fluctuations span a broad range of frequencies.
  • Mode selection: Large DMD amplitudes can belong to strongly damped modes, so amplitude magnitude alone does not identify modes that best approximate the full snapshot history.Strongly damped modes mainly influence early time evolution, while sparse DMD selects modes according to their influence across the available time history.
  • Sparsity trade-off: Increasing γ produces sparser amplitude vectors but worsens least-squares approximation, spanning solutions from 256 nonzero amplitudes to one.This parameter controls the preference between approximation quality and solution sparsity.
  • Results: With Nz = 3, sparse DMD retains the mean flow and one dominant frequency pair, yielding St = 0.3104 versus the measured St = 0.3.The selected frequency agrees well with measurements from a sequence containing ten times more snapshots; increasing the number of modes improves performance.

C. Flow through a cylinder bundle

The cylinder-bundle study uses sparsity-promoting DMD to identify dominant frequencies and structures while controlling the trade-off between compactness and approximation quality. With three retained modes, the method captures the principal oscillatory behavior and reproduces the jet’s lateral swaying.

  • Experimental setting: Cylinder-bundle flow contains multiple shedding frequencies and flip-flop oscillations of the exiting jet, motivating a simplified two-cylinder experimental configuration.The PIV window measures the flow downstream of the passage, where the jet alternately deflects downward and upward.
  • Mode selection: Standard DMD amplitudes do not reliably indicate mode importance because strongly damped modes can have large amplitudes.The analysis therefore applies sparsity-promoting DMD to distinguish modes contributing across the available time history.
  • Sparse representation: The algorithm concentrates on low-frequency modes and eliminates larger-amplitude structures identified by standard DMD as the retained-mode count decreases.At Nz = 3, the selected frequencies still optimally describe the principal oscillatory components of the full dataset.
  • Sparse representation: 7.99 Hz and St = 0.129 characterize the Nz = 3 representation, whose flow field contains a deflected jet and strong vortical components.The selected frequency agrees well with the frequency identified from point measurements of streamwise velocity.
  • Sparse representation: The three-mode representation reproduces the full sequence’s main lateral jet swaying under the influence of wake vortices from the two cylinders.The sparse approximation removes large-amplitude transient features, while residuals grow progressively as sparsity increases.
  • Approximation trade-off: The performance loss increases with sparsity, and degradation is higher than in the channel-flow and screeching-jet examples.The authors suggest that unstructured measurement noise in the experimental dataset may contribute to this behavior.

V. CONCLUDING REMARKS

The paper concludes that sparsity-promoting DMD reduces full-rank representations by selecting modes that contribute most strongly over the observed time interval. Its convex ADMM formulation and three numerical or experimental demonstrations support this approach, while corrupted or incomplete snapshots remain a scope limitation.

  • Contribution: Sparsity-promoting DMD adds an ℓ1 penalty to least-squares fitting, with a user-specified parameter balancing approximation quality against the number of retained modes.The selected modes include their temporal frequencies and growth/decay rates.
  • Optimization: ADMM computes the globally optimal solution of the convex problem by alternating residual minimization and sparsity enhancement.Soft-thresholding promotes sparsity, after which the retained-mode amplitudes are recomputed optimally.
  • Applications: Across three examples, sparsity-promoting DMD extracts dominant modal contributions, identifies screech and prevailing shedding frequencies, and removes transient or less relevant structures.The examples include plane Poiseuille flow, a supersonic screeching jet, and PIV measurements of flow between two cylinders.
  • Novelty: The method provides an efficient alternative for detecting and extracting a limited subset of modes compared with earlier complex or nearly intractable non-convex formulations.The authors frame this as a paradigm for automated detection of pertinent modal flow features.
  • Limitation: Corrupted or incomplete snapshots can significantly diminish the predictive capability of resulting low-dimensional models without regularization.The authors identify exploration of other regularization penalties suited to fluid-mechanics problems as worthwhile future work.
  • Implications: The demonstrated algorithm is positioned as a tool for quantitative analysis of high-dimensional datasets, interpretation of dynamic modes, and eduction of relevant physical mechanisms.This expectation follows applications to the numerical and experimental snapshot sequences considered in the paper.

SUPPLEMENTARY MATERIAL

Supplementary materials provide additional information about the paper’s examples, together with Matlab source code and problem data.

  • Supplementary material: Additional example information, Matlab source codes, and problem data are available from the authors’ supplementary website.The resource supports reproduction and further examination of the considered cases.

Appendix A: An alternative formulation of (5)

The appendix gives an alternative formulation of the optimization objective and explains the ADMM subproblems through matrix-trace identities, quadratic minimization, and soft thresholding.

  • Appendix A: An alternative formulation of (5): Equation (6) is shown to be equivalent to formulation (A1) through algebraic manipulation and repeated use of matrix-trace properties.The appendix organizes these transformations around properties P1–P3.
  • Appendix A: An alternative formulation of (5): Property P1 states commutativity invariance for the matrix trace.This property is repeatedly used to rearrange trace expressions in the equivalence derivation.
  • Appendix A: An alternative formulation of (5): Property P2 concerns products between a matrix Q and the diagonal matrix Dα = diag{α}.It supplies one of the algebraic steps connecting the alternative formulations.
  • Appendix A: An alternative formulation of (5): Property P3 specifies a trace identity for vectors α and β and matrices A and B of compatible dimensions.Together with P2, it establishes the equivalence between (A1) and (6).
  • Appendix B: α- and β-minimization steps in ADMM: The α-minimization step in ADMM becomes an unconstrained regularized quadratic program after completing squares in the augmented Lagrangian.The definition of J yields the corresponding quadratic programming problem.
  • Appendix B: α- and β-minimization steps in ADMM: The β-minimization step is likewise reduced by completing squares in the augmented Lagrangian and solved using soft thresholding.The appendix characterizes this solution as a standard result for the resulting optimization problem.

Appendix C: An efficient algorithm for solving (9)

Appendix C recasts solving (9) as an equality-constrained quadratic program and derives optimality conditions from variations of the Lagrangian. The sparse amplitude vector is then obtained from these conditions.

  • Solving (9) is formulated as an equality-constrained quadratic programming problem.
  • Variations of the Lagrangian with respect to α and the Lagrange multipliers ν provide optimality conditions.
  • The optimal sparse amplitude vector αsp is determined from the resulting conditions.
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