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A data driven Koopman-Schur decomposition for computational analysis of nonlinear dynamics

Zlatko Drmač, Igor Mezić

TL;DR

This paper introduces a new theoretical and computational framework for a data driven Koopman mode analysis of nonlinear dynamics that introduces a Koopman-Schur decomposition that is entirely based on unitary transformations.

Abstract

This paper introduces a new theoretical and computational framework for a data driven Koopman mode analysis of nonlinear dynamics. To alleviate the potential problem of ill-conditioned eigenvectors in the existing implementations of the Dynamic Mode Decomposition (DMD) and the Extended Dynamic Mode Decomposition (EDMD), the new method introduces a Koopman-Schur decomposition that is entirely based on unitary transformations. The analysis in terms of the eigenvectors as modes of a Koopman operator compression is replaced with a modal decomposition in terms of a flag of invariant subspaces that correspond to selected eigenvalues. The main computational tool from the numerical linear algebra is the partial ordered Schur decomposition that provides convenient orthonormal bases for these subspaces. In the case of real data, a real Schur form is used and the computation is based on real orthogonal transformations. The new computational scheme is presented in the framework of the Extended DMD and the kernel trick is used.

A data driven Koopman-Schur decomposition for computational analysis of nonlinear dynamics

TL;DR

This paper introduces a new theoretical and computational framework for a data driven Koopman mode analysis of nonlinear dynamics that introduces a Koopman-Schur decomposition that is entirely based on unitary transformations.

Abstract

This paper introduces a new theoretical and computational framework for a data driven Koopman mode analysis of nonlinear dynamics. To alleviate the potential problem of ill-conditioned eigenvectors in the existing implementations of the Dynamic Mode Decomposition (DMD) and the Extended Dynamic Mode Decomposition (EDMD), the new method introduces a Koopman-Schur decomposition that is entirely based on unitary transformations. The analysis in terms of the eigenvectors as modes of a Koopman operator compression is replaced with a modal decomposition in terms of a flag of invariant subspaces that correspond to selected eigenvalues. The main computational tool from the numerical linear algebra is the partial ordered Schur decomposition that provides convenient orthonormal bases for these subspaces. In the case of real data, a real Schur form is used and the computation is based on real orthogonal transformations. The new computational scheme is presented in the framework of the Extended DMD and the kernel trick is used.
Paper Structure (38 sections, 13 theorems, 89 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 38 sections, 13 theorems, 89 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction and preliminaries
  2. The Dynamic Mode Decomposition (DMD).
  3. The Koopman connection and the Extended DMD (EDMD).
  4. A quandary about the eigenvectors.
  5. An alternative proposed in this paper.
  6. Outline of the paper.
  7. Koopman operator and its Schur form
  8. Extended DMD
  9. Numerical data driven compression of $\mathbb U$
  10. Convergence as $m\rightarrow\infty$
  11. The Rayleigh quotient of $U_N$
  12. Why is the SVD truncated?
  13. The EDMD with the kernel trick
  14. Connection with the DMD
  15. Evaluation of the approximate eigenfunctions of $\mathbb U$
  16. ...and 23 more sections

Key Result

Theorem 1.1

(Schur form) Let $A\in \mathbb{C}^{r\times r}$ and let $\lambda_1,\ldots, \lambda_r$ be its eigenvalues, listed in an arbitrary order. Then there exist a unitary $Q$ and an upper triangular $T$ such that $T_{ii}=\lambda_i$, $i=1,\ldots, r$, and $A = Q T Q^* .$

Figures (9)

  • Figure 1: (Example \ref{['EX:cylinder']}) The reconstruction error and the consistency of the computed Schur functions. The kernel function is $k_1(\cdot,\cdot)$. The yellow horizontal line indicates the level of machine precision eps times the state space dimension $n$.
  • Figure 2: (Example \ref{['EX:cylinder']}) The reconstruction error and the consistency of the computed Schur functions. The kernel function is $k_2(\cdot,\cdot)$. The yellow horizontal line indicates the level of machine precision eps times the state space dimension $n$.
  • Figure 3: (Example \ref{['EX:cylinder']}) Left: The eigenvalues (Ritz values) computed in the last active window. Right: The relevant condition numbers. The kernel function is $k_1(\cdot,\cdot)$.
  • Figure 4: (Example \ref{['EX:cylinder']}) The forecast error with the horizon $\tau=40$. Left: Koopman-Schur-SSMD. Middle: Koopman-Schur-ESSMD with the kernel function $k_1(\cdot,\cdot)$. The KS-SSMD and the KS-ESSMD method are equivalent and the difference is only due to the finite precision arithmetic. Right: Koopman-Schur-ESSMD with the kernel function $k_2(\cdot,\cdot)$.
  • Figure 5: (Example \ref{['EX:cylinder']}) Left: The eigenvalues (Ritz values) computed in the last active window. Right: The relevant condition numbers. The kernel function is $k_2(\cdot,\cdot)$.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Theorem 2.1
  • Lemma 2.1
  • Theorem 2.2
  • Remark 3.1
  • Definition 3.1
  • Definition 3.2
  • ...and 20 more