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Long-lived flow patterns in the atmosphere such as weather fronts, mid-latitude blockings or tropical cyclones often induce extreme weather conditions. As a consequence, their description, detection, and tracking has received increasing attention in recent years. Similar objectives also arise in diverse fields such as turbulence and combustion research, image analysis, and medical diagnostics under the headlines of "feature tracking", "coherent structure detection" or "image registration" - to name just a few. A host of different approaches to addressing the underlying, often very similar, tasks have been developed and successfully used. Here, several typical examples of such approaches are summarized, further developed and applied to meteorological data sets. Common abstract operational steps form the basis for a unifying framework for the specification of "persistent structures" involving the definition of the physical state of a system, the features of interest, and means of measuring their persistence.

In this article, we show that these well-established spectral algorithms (like PCCA+, Perron Cluster Cluster Analysis) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.

Given a time-dependent stochastic process with trajectories x(t) in a space $\Omega$, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space $M\subset\Omega$, coherent sets $M(t)\subset\Omega$ are defined in space and time. Hence, if we extend the space by the time-variable t, coherent sets are metastable sets in $\Omega\times[0,\infty]$. This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-timediscretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.

We utilize the theory of coherent sets to build Markov state models for non- equilibrium molecular dynamical systems. Unlike for systems in equilibrium, “meta- stable” sets in the non-equilibrium case may move as time evolves. We formalize this concept by relying on the theory of coherent sets, based on this we derive finite-time non-stationary Markov state models, and illustrate the concept and its main differences to equilibrium Markov state modeling on simple, one-dimensional examples.

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.

Optimal data-driven estimation of generalized Markov state models for non-equilibrium dynamics
(2018)

Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review different methods that have been developed over the last decades to compute infinite-dimensional approximations of these infinite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and differences between these approaches. The results will be illustrated using simple stochastic differential equations and molecular dynamics examples.

We used transition path theory (TPT) to infer "reactive" pathways of floating marine debris trajectories. The TPT analysis was applied on a pollution-aware time-homogeneous Markov chain model constructed from trajectories produced by satellite-tracked undrogued buoys from the NOAA Global Drifter Program. The latter involved coping with the openness of the system in physical space, which further required an adaptation of the standard TPT setting. Directly connecting pollution sources along coastlines with garbage patches of varied strengths, the unveiled reactive pollution routes represent alternative targets for ocean cleanup efforts. Among our specific findings we highlight: constraining a highly probable pollution source for the Great Pacific Garbage Patch; characterizing the weakness of the Indian Ocean gyre as a trap for plastic waste; and unveiling a tendency of the subtropical gyres to export garbage toward the coastlines rather than to other gyres in the event of anomalously intense winds.