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Randomized Methods for Kernelized DMD

Peter Oehme

TL;DR

The paper tackles the scalability of kernelized DMD (KDMD) for large datasets by introducing randomized, pivot-based approximations of the kernel Gram matrix. It centers on RPCholesky as a robust alternative to oASIS, enabling a reduced $R\times R$ kernel eigensystem and a data-driven residual for ranking DMD modes. Through experiments on cylinder flow, the Duffing oscillator, and Sea Surface Temperature data, the authors demonstrate improved numerical stability and consistent mode ordering with reduced computational cost, illustrating the method’s practical applicability to high-dimensional dynamical systems. The work offers a scalable framework for obtaining accurate dynamic modes in KDMD when data size is a major constraint, and it provides benchmarks and code for broader adoption.

Abstract

Dynamic Mode Decomposition (DMD) is a data-driven method related to Koopman operator theory that extracts information about dominant dynamics from data snapshots. In this paper we examine techniques to accelerate the application of DMD to large-scale data sets with an eye on randomized techniques. Randomized techniques exploit low-rank matrix approximations at a much smaller computational cost, therefore permitting the use of increased data set sizes. In particular, we propose the application of the RPCholesky algorithm in the setting of kernelized DMD (KDMD). This algorithm relies on adaptive randomized sampling to approximate positive semidefinite kernel matrices and provides better stability guarantees than previously implemented randomized methods for KDMD. Differences between existing competitive randomized techniques and our proposed implementation are discussed with a focus on numerical stability and tradeoff between exploration and exploitation of information obtained from data. The efficacy of this new combination of algorithms is demonstrated on well-established benchmark problems from DMD literature increasing in problem dimension.

Randomized Methods for Kernelized DMD

TL;DR

The paper tackles the scalability of kernelized DMD (KDMD) for large datasets by introducing randomized, pivot-based approximations of the kernel Gram matrix. It centers on RPCholesky as a robust alternative to oASIS, enabling a reduced kernel eigensystem and a data-driven residual for ranking DMD modes. Through experiments on cylinder flow, the Duffing oscillator, and Sea Surface Temperature data, the authors demonstrate improved numerical stability and consistent mode ordering with reduced computational cost, illustrating the method’s practical applicability to high-dimensional dynamical systems. The work offers a scalable framework for obtaining accurate dynamic modes in KDMD when data size is a major constraint, and it provides benchmarks and code for broader adoption.

Abstract

Dynamic Mode Decomposition (DMD) is a data-driven method related to Koopman operator theory that extracts information about dominant dynamics from data snapshots. In this paper we examine techniques to accelerate the application of DMD to large-scale data sets with an eye on randomized techniques. Randomized techniques exploit low-rank matrix approximations at a much smaller computational cost, therefore permitting the use of increased data set sizes. In particular, we propose the application of the RPCholesky algorithm in the setting of kernelized DMD (KDMD). This algorithm relies on adaptive randomized sampling to approximate positive semidefinite kernel matrices and provides better stability guarantees than previously implemented randomized methods for KDMD. Differences between existing competitive randomized techniques and our proposed implementation are discussed with a focus on numerical stability and tradeoff between exploration and exploitation of information obtained from data. The efficacy of this new combination of algorithms is demonstrated on well-established benchmark problems from DMD literature increasing in problem dimension.
Paper Structure (15 sections, 1 theorem, 25 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 15 sections, 1 theorem, 25 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Fundamentals of DMD
  3. SVD-Based DMD
  4. EDMD
  5. KDMD
  6. Randomized sampling for KDMD
  7. Random Fourier Features (RFF)
  8. Diagonal Pivoting and oASIS
  9. Randomly Pivoted Partial Cholesky Factorization
  10. Computational Complexity of Randomized KDMD
  11. Numerical Experiments
  12. Flow Around a Cylinder
  13. Unforced Duffing Equation
  14. Sea Surface Temperature Dataset
  15. Conclusion

Key Result

Lemma 1

Let $U \Sigma V^*$ be the SVD of $\Psi_X$. Then, $\lambda \neq 0$ and $\eta$ are a right eigenpair of $\hat{K}$ from eq:kdmd-K if and only if $\lambda$ and $\hat{\eta} = V \eta$ are a right eigenpair of $K$ from eq:edmd-K. In an analogous way it follows that $\lambda \neq 0$ and $\eta$ are a left ei

Figures (5)

  • Figure 1: The real parts of the first three DMD modes for fluid flow around cylinder. The modes were increasingly sorted according to their respective residuals $r(i)$. For the randomized methods RFF and RPCholesky we used $S = 50$ random samples. For RFF, KDMD with RPCholesky, and KDMD with greedily pivoted partial Cholesky the signs are flipped, but the same phenomenon occurs to the corresponding eigenfunctions.
  • Figure 2: Residuals of the DMD modes for the fluid flow around a cylinder. All randomized methods use $S = 50$ random samples.
  • Figure 3: Reconstruction errors for the synthetic Duffing oscillator example. The shape parameters of the Gaussian distribution generating the RFF samples and the Gaussian kernel used in the KDMD algorithms are chosen such that $\sigma = 1/\sqrt{2}$
  • Figure 4: Elapsed computational times for the synthetic Duffing oscillator data set.
  • Figure 5: The first 6 DMD modes for the sea surface temperature data set. Here, both techniques are randomized and use $S = 100$ as the sample size.

Theorems & Definitions (1)

  • Lemma 1: Williams2015