# On Numerical Approximations of the Koopman Operator

## Abstract

**:**

## 1. Introduction

- 1.
- The first, suggested already in [9] is based on long time weighted averages over trajectories, rooted in ergodic theory of measure-preserving dynamical systems. An extension of that work that captures properties of continuous spectrum was presented in [10]. This approach was named Generalized Laplace Analysis (GLA) in [11], where concepts pertaining to dissipative systems were discussed also in terms of weighted averages along trajectories. In that sense, the ideas in this context provide an extension of ergodic theory for capturing transient (off-attractor) properties of systems. For on-attractor evolution, the properties of the method acting on ${L}^{2}$ functions were studied in [4]. The off-attractor case was pursued in [4,12] where Fourier averages (which are Laplace averages for the case when the eigenvalue considered is on the imaginary axis) were used to compute the eigenfunction whose level sets are isochrons, and [13] in which analysis for general eigenvalue distributions was pursued in Hardy-type spaces. This study was continued in [14] to construct dynamics-adapted Hilbert spaces. The advantage of the method is that it does not require the approximation of the operator itself, as it constructs eigenfunctions and eigenmodes directly from the data. In this sense, it is close (and in fact related) to the power method of approximating the spectrum of a matrix from data on iteration of a vector. In fact, the methodology extends the power method to the case when eigenvalues can be of magnitude 1. It requires separate computation to first determine the spectrum of the operator, which is also done without constructing it. This can potentially be hard to do because of issues such as spectral pollution—see remarks at the end of Section 3; also note the general long-standing problems of spectral pollution and computing the full spectrum of Schrödinger operators on a lattice were recently solved in [15]. The recent work [16] enables computation of full spectral measures using the combination of resolvent operator techniques (used for the first time in the Koopman operator context in [17]) and ResDMD—an extension of Dynamic Mode Decomposition (introduced next) technique that incorporates computation of residues from data snapshots (computation of residues was considered earlier in [18]).
- 2.
- The second approach requires construction of an approximate operator acting on a finite-dimensional function subspace i.e., a finite section—the problem that is also of concern in a more general context of approximating infinite dimensional operators [7,19,20]. The best known such method is the Dynamic Mode Decomposition (DMD), invented in [21] and connected to the Koopman operator in [22]. It has a number of extensions (many of which are summarized in [23]), for example, Exact DMD [24]; Bayesian/subspace DMD [25]; Optimized DMD [26,27]; Recursive DMD [28]; Variational DMD [29]; DMD with control [30,31]; sparsity promoting DMD [32]; DMD for noisy systems [33,34,35]. The original DMD algorithm featured state observables. The Extended Dynamic Mode Decomposition [36] recognizes that nonlinear functions of state might be necessary to describe a finite-dimensional invariant subset of the Koopman operator and provides an algorithm for finite-section approximation of the Koopman operator. A study of convergence of such approximations is provided in [37], but the convergence was established only along subsequences, and the rate of convergence was not addressed. Here, we provide the first result on the rate of convergence of the finite section approximation under assumptions on the nature of the underlying dynamics. In addition, spectral convergence along subsequences is proven in [37] under the assumption of the weak limit of eigenfunction approximations not being zero. This condition is hard to verify in practice. Instead, in Section 5.2, we prove a result that obviates the weak convergence assumption using some additional information on the underlying dynamics. It was observed already in [9] that, instead of an arbitrary set of observables forming a basis, one can use observables generated by the dynamics—namely time delays of a single observable filling a Krylov subspace—to study spectral properties of the Koopman operator. In the DMD context, the methods developed in this direction are known under the name Hankel-DMD [38,39]. It is worth noticing that the Hankel matrix approach of [38] is in fact based on the Prony approximation and requests a different sample structure than the Dynamic Mode Decomposition. Computation of residues was considered in [18] to address the problem of spectral pollution, where discretization introduces spurious eigenvalues. As mentioned before, the recent work [16] provides another method to resolve the spectral pollution problem, introducing ResDMD—an extension of Dynamic Mode Decomposition that incorporates computation of residues from data snapshots. The relationship between GLA and finite section methods was studied in [40].
- 3.
- The third approach is based on the kernel integral operator combined with the Krylov subspace methodology [41], enabling approximation of continuous spectrum. While GLA and EDMD techniques have been extended to dissipative systems, the kernel integral operator technique is currently available only for measure-preserving (on-attractor) systems.

## 2. Preliminaries

**Remark 1.**

## 3. Generalized Laplace Analysis

#### GLA for Fields of Observables

**Theorem 1.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 4. The Finite Section Method

#### 4.1. Finite Section and the Dual Basis

**Example**

**1.**

**Example**

**2.**

**Theorem**

**2.**

**Proof.**

**Theorem 3.**

**Proof.**

**Remark**

**4.**

**Remark 5.**

**Remark**

**6.**

#### 4.2. Convergence of the Finite Sample Approximation to the Finite Section

**Theorem**

**4.**

**Proof.**

**Remark**

**7.**

**Corollary**

**1.**

**Proof.**

**Remark**

**8.**

**Remark**

**9.**

#### 4.3. The Error in the Finite Section

**Proposition**

**1.**

**Proof.**

## 5. Krylov Subspace Methods

#### 5.1. Single Observable Krylov Subspace Methods

**Example**

**3.**

#### 5.2. Error in the Companion Matrix Representation

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

**Theorem 5.**

**Proof.**

**Remark**

**10.**

**Theorem 6.**

**Proof.**

**Remark 11.**

#### 5.3. Krylov Sequences from Data

**Corollary**

**2.**

**Proof.**

#### 5.4. Schmid’s Dynamical Mode Decomposition as a Finite Section Method

## 6. Weak Eigenfunctions from Data

**Theorem**

**7.**

**Proof.**

**Definition**

**1.**

**Theorem**

**8.**

**Remark**

**12.**

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Mezić, I.
On Numerical Approximations of the Koopman Operator. *Mathematics* **2022**, *10*, 1180.
https://doi.org/10.3390/math10071180

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Mezić I.
On Numerical Approximations of the Koopman Operator. *Mathematics*. 2022; 10(7):1180.
https://doi.org/10.3390/math10071180

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Mezić, Igor.
2022. "On Numerical Approximations of the Koopman Operator" *Mathematics* 10, no. 7: 1180.
https://doi.org/10.3390/math10071180