Abstract
Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems.
Highlights
Received: 30 June 2021Accepted: 17 August 2021Published: 27 August 2021Suppose that we have an autonomous dynamical system F1 (x(t)) ẋ(t) = F(x(t)) ≡Publisher’s Note: MDPI stays neutral, x ( t ) ∈ Rn, (1) Fn (x(t))with regard to jurisdictional claims in published maps and institutional affiliations
We give detailed description of a finite dimensional compression of the Koopman operator (Section 2.1), and we review the basic properties of the matrix logarithm (Section 2.3)
For the reader’s convenience, we summarize the key properties of the matrix logarithm and refer to [24] for proofs and more details
Summary
With regard to jurisdictional claims in published maps and institutional affiliations. It can happen that with more data the potentially better approximation does not materialize in the computation This is an undesirable chasm between analytical properties and numerical finite precision realization of the method. Even if computed accurately, the matrix UN may be so ill-conditioned as to preclude stable computation of the logarithm Both issues are analyzed in detail, and we propose a new numerical algorithm that implements the method. This is standard, well known material and it is included for the reader’s convenience and to introduce necessary notation.
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