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Reduced-order modelling based on Koopman operator theory

Diana A. Bistrian, Gabriel Dimitriu, Ionel M. Navon

Abstract

The present study focuses on a subject of significant interest in fluid dynamics: the identification of a model with decreased computational complexity from numerical code output using Koopman operator theory. A reduced-order modelling method that incorporates a novel strategy for identifying the most impactful Koopman modes was used to numerically approximate the Koopman composition operator.

Reduced-order modelling based on Koopman operator theory

Abstract

The present study focuses on a subject of significant interest in fluid dynamics: the identification of a model with decreased computational complexity from numerical code output using Koopman operator theory. A reduced-order modelling method that incorporates a novel strategy for identifying the most impactful Koopman modes was used to numerically approximate the Koopman composition operator.
Paper Structure (5 sections, 2 theorems, 38 equations, 7 figures, 1 table)

This paper contains 5 sections, 2 theorems, 38 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical considerations on the Koopman operator
  3. Reduced-order modelling based on Koopman operator
  4. Reduced-order modelling of Saint-Venant equations model
  5. Conclusions

Key Result

Proposition 1

Linearity of the Koopman operator. Consider the Koopman operator ${{\mathcal{K}}^t}$ and two observables ${\varphi _1},{\varphi _2} \in \Omega$ and the scalar $\alpha \in \mathbb{R}$. Using (koop) it follows that:

Figures (7)

  • Figure 1: The spectrum of Koopman decomposition of height field $h$: a) in the first experiment $\left( {\varepsilon = {{10}^{ - 3}}} \right)$, $21$ leading modes are selected (darker colored dots); b) in the second experiment $\left( {\varepsilon = {{10}^{ - 4}}} \right)$, $67$ leading modes are selected (darker colored dots)
  • Figure 2: The spectrum of Koopman decomposition of streamwise field $u$: a) in the first experiment $\left( {\varepsilon = {{10}^{ - 3}}} \right)$, $116$ leading modes are selected (darker colored dots); b) in the second experiment $\left( {\varepsilon = {{10}^{ - 4}}} \right)$, $199$ leading modes are selected (darker colored dots)
  • Figure 3: The spectrum of Koopman decomposition of spanwise field $v$: a) in the first experiment $\left( {\varepsilon = {{10}^{ - 3}}} \right)$, $151$ leading modes are selected (darker colored dots); b) in the second experiment $\left( {\varepsilon = {{10}^{ - 4}}} \right)$, $212$ leading modes are selected (darker colored dots)
  • Figure 4: Full solution of height field $u$ after $T=50h$, compared to its reduced-order model, in the case of the first experiment, the relative error is of order $\mathcal{O}\left( 10^{-3} \right)$
  • Figure 5: Full solution of height field $u$ after $T=50h$, compared to its reduced-order model, in the case of the second experiment, the relative error is of order $\mathcal{O}\left( 10^{-4} \right)$
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof