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New Robust Streaming DMD with Forecasting

Zlatko Drmač, Ela Đimoti

Abstract

The Dynamic Mode Decomposition (DMD) and the more general Extended DMD (EDMD) are powerful tools for computational analysis of dynamical systems in data-driven scenarios. They are built on the theoretical foundation of the Koopman composition operator and can be considered as numerical methods for data snapshot-based extraction of spectral information of the composition operator associated with the dynamics, spectral analysis of the structure of the dynamics, and for forecasting. In high fidelity numerical simulations, the state space is high dimensional and efficient numerical methods leverage the fact that the actual dynamics evolves on manifolds of much smaller dimension. This motivates computing low rank approximations in a streaming fashion and the DMD matrix is adaptively updated with newly received data. In this way, large number of high dimensional snapshots can be processed very efficiently. Low dimensional representation also requires fast updating for online applications. This paper revisits the pioneering works of Hemati, Williams and Rowley (Physics of Fluids, 2014), and Zhang, Rowley, Deem and Cattafesta (SIAM Journal on Applied Dynamical Systems, 2019) on the streaming DMD and proposes improvements in functionality (using residual bounds, Exact DMD vectors), computational efficiency (more efficient algorithm with smaller memory footprint) and numerical robustness (smaller condition numbers and better forecasting skill).

New Robust Streaming DMD with Forecasting

Abstract

The Dynamic Mode Decomposition (DMD) and the more general Extended DMD (EDMD) are powerful tools for computational analysis of dynamical systems in data-driven scenarios. They are built on the theoretical foundation of the Koopman composition operator and can be considered as numerical methods for data snapshot-based extraction of spectral information of the composition operator associated with the dynamics, spectral analysis of the structure of the dynamics, and for forecasting. In high fidelity numerical simulations, the state space is high dimensional and efficient numerical methods leverage the fact that the actual dynamics evolves on manifolds of much smaller dimension. This motivates computing low rank approximations in a streaming fashion and the DMD matrix is adaptively updated with newly received data. In this way, large number of high dimensional snapshots can be processed very efficiently. Low dimensional representation also requires fast updating for online applications. This paper revisits the pioneering works of Hemati, Williams and Rowley (Physics of Fluids, 2014), and Zhang, Rowley, Deem and Cattafesta (SIAM Journal on Applied Dynamical Systems, 2019) on the streaming DMD and proposes improvements in functionality (using residual bounds, Exact DMD vectors), computational efficiency (more efficient algorithm with smaller memory footprint) and numerical robustness (smaller condition numbers and better forecasting skill).

Paper Structure

This paper contains 39 sections, 10 theorems, 72 equations, 13 figures, 9 algorithms.

Table of Contents

  1. Introduction
  2. DMD - a review
  3. Task $\sharp 1$: Approximate eigenpairs
  4. Task $\sharp 2$: Modal analysis of the data and forecasting
  5. DMD of large dimensional perpetual data streams
  6. Organization of the paper and our contributions
  7. Streaming DMD
  8. Adding new data
  9. Reorthogonalization
  10. Low rank constraints
  11. The compressed streaming DMD algorithm
  12. Modifications of the compressed streaming DMD
  13. Avoiding squared condition number
  14. Theory behind the growth of $\kappa_2(G_x)$
  15. Orthogonal factorization based updates
  16. ...and 24 more sections

Key Result

Theorem 3.1

Let $H$ be $n\times n$ Hermitian, $v\in\mathbb{C}^n$, and let $\lambda_i(\cdot)$ denote the $i$-th smallest eigenvalue of a Hermitian matrix ($\lambda_1(\cdot)\leq\cdots\leq\lambda_n(\cdot)$). Then the eigenvalues of $H$ and $H\pm vv^*$ interlace: Also, by the Weyl's theorem, $\max_i |\lambda_i(H)-\lambda_i(H\pm vv^*)|\leq \|vv^*\|_2=\|v\|_2^2$, and by the monotonicity principle $\lambda_i(H+vv^*

Figures (13)

  • Figure 1: Left panels: Test of the low rank representation (\ref{['eq:RRF-XY']}) and the orthogonality of the bases $Q_x$, $Q_y$. Note how the low rank constraints (§ \ref{['SS=Hemati-low_rank_constraints']}) here set to $30$ (first row) and $100$ (second row)) impact the accuracy of the representation. Middle panels: The condition numbers of the key matrices. Right panels: The DMD eigenvalues computed after receiving the last snapshot. The colors encode the corresponding residuals. In the second row, the eigenvalues from the first row are shown as purple hexagons. (The system is measure preserving and the underlying Koopman operator is unitary. The true eigenvalues are on the unit circle.)
  • Figure 2: Comparison of predicted snapshots using the original algorithm HWR-sDMD (middle) and TQ-sDMD (using the LQ decomposition) (right). Left figure is the original snapshot at $n=31$. The apparent differences in the prediction errors ($\texttt{5.02e-5}$ vs. $\texttt{3.78e-1}$) show that TQ-sDMD has improved robustness to ill-conditioning and finite precision roundoff errors. While HWR-sDMD has difficulties even for one step prediction, TQ-sDMD retains accuracy even for multiple steps, see figure \ref{['fig:pred-many-steps']}.
  • Figure 3: Relative error when predicting one or five steps ahead when updating $A$ by using HWR-sDMD and TQ-sDMD and rounding to single precision before calling DMD, compared to updates by HWR-sDMD with calculation done entirely in double precision. Here HWR-sDMD (64) denotes execution in the 64-bit double precision arithmetic.
  • Figure 4: Visualization of predicted dynamics of the Gray-Scott model, using HWR-sDMD (64) and TQ-sDMD (32) five steps ahead after $n=200$ observed snapshots.
  • Figure 5: Left: When predicting one step ahead, TQ-sDMD in simulated single basis maintains relative error norms around $10^{-3}$. Meanwhile, the errors for HWR-sDMD in both simulated single and fully double precision result in errors between $0.1$ and $1$ with occasional oscillation to larger errors. In HWR-sDMD (32) $144$ error norms were NaN. Error norms for TQ-sDMD in double precision are omitted since they coincide with errors when working in single precision. Right: When predicting 5 steps ahead, all errors are larger, but TQ-sDMD errors stay almost always under $0.1$, while both HWR-sDMD errors stay above $10$ after first few iterations. HWR-sDMD (32) error norms contains $191$NaNs.
  • ...and 8 more figures

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Remark 3
  • Example 2.1
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Remark 4
  • Remark 5
  • ...and 18 more