A majority of methods from dynamical system analysis, especially those in applied settings, rely on Poincaré's geometric picture that focuses on "dynamics of states." While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of "big data" measurements. This overview article presents an alternative framework for dynamical systems, based on the "dynamics of observables" picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics. The first goal of this paper is to make it clear how methods that appeared in different papers and contexts all relate to each other through spectral properties of the Koopman operator. The second goal is to present these methods in a concise manner in an effort to make the framework accessible to researchers who would like to apply them, but also, expand and improve them. Finally, we aim to provide a road map through the literature where each of the topics was described in detail. We describe three main concepts: Koopman mode analysis, Koopman eigenquotients, and continuous indicators of ergodicity. For each concept, we provide a summary of theoretical concepts required to define and study them, numerical methods that have been developed for their analysis, and, when possible, applications that made use of them. The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice. Therefore, the paper highlights its strengths in applied and numerical contexts. Additionally, we point out areas where an additional research push is needed before the approach is adopted as an off-the-shelf framework for analysis and design.

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The importance of using dedicated high-end computing resources to enable high spatial resolution in global climate models and advance knowledge of the climate system has been evaluated in an international collaboration called Project Athena. Inspired by the World Modeling Summit of 2008 and made possible by the availability of dedicated high-end computing resources provided by the National Science Foundation from October 2009 through March 2010, Project Athena demonstrated the sensitivity of climate simulations to spatial resolution and to the representation of subgrid-scale processes with horizontal resolutions up to 10 times higher than contemporary climate models. While many aspects of the mean climate were found to be reassuringly similar, beyond a suggested minimum resolution, the magnitudes and structure of regional effects can differ substantially. Project Athena served as a pilot project to demonstrate that an effective international collaboration can be formed to efficiently exploit dedicated supercomputing resources. The outcomes to date suggest that, in addition to substantial and dedicated computing resources, future climate modeling and prediction require a substantial research effort to efficiently explore the fidelity of climate models when explicitly resolving important atmospheric and oceanic processes.

The Dynamic Mode Decomposition (DMD) is a tool of trade in computational data driven analysis of fluid flows. More generally, it is a computational device for Koopman spectral analysis of nonlinear dynamical systems, with a plethora of applications in applied sciences and engineering. Its exceptional performance triggered developments of several modifications that make the DMD an attractive method in data driven framework. This work offers further improvements of the DMD to make it more reliable, and to enhance its functionality. In particular, data driven formula for the residuals allows selection of the Ritz pairs, thus providing more precise spectral information of the underlying Koopman operator, and the well-known technique of refining the Ritz vectors is adapted to data driven scenarios. Further, the DMD is formulated in a more general setting of weighted inner product spaces, and the consequences for numerical computation are discussed in detail. Numerical experiments are used to illustrate the advantages of the proposed method, designated as DDMD RRR (Refined Rayleigh Ritz Data Driven Modal Decomposition). AMS subject classifications: 15A12, 15A23, 65F35, 65L05, 65M20, 65M22, 93A15, 93A30, 93B18, 93B40, 93B60, 93C05, 93C10, 93C15, 93C20, 93C57 Drmač, Mezić, Mohr / Enhanced DMD / AIMdyn Technical Reports 201708.004v1 2 aimdyn:201708.004v1 AIMDYN INC. Drmač, Mezić, Mohr / Enhanced DMD / AIMdyn Technical Reports 201708.004v1 3to assess the quality of each particular Ritz pair. Further, we show how to apply the well known Ritz vector refinement technique to the DMD data driven setting, and we discuss the importance of data scaling. All these modifications are integrated in §3.5 where we propose a new version of the DMD, designated as DDMD RRR (Refined Rauleigh-Ritz Data Driven Modal Decomposition). In §4 we provide numerical examples that show the benefits of our modifications, and we discuss the fine details of software implementation. In §5 we use the Exact DMD [12] to show that our modifications apply to other versions of DMD. In §6, we provide a compressed form of the new DDMD RRR, designed to improve the computational efficiency in case of extremely large dimensions. A matrixroot-free modification of the Forward-Backward DMD [13] is presented in §7. The column scaling used in the new DDMD implementation in §3.5 is just a particular case of a more general weighting scheme that we address in §8. Using the concept of the generalized SVD introduced by Van Loan [14], we define weighted DDMD with the Hilbert space structures in the spaces of the snapshots and the observables (spatial and temporal) given by two positive definite matrices. aimdyn:201708.004v1 AIMDYN INC. aimdyn:201708.004v1 AIMDYN INC. aimdyn:201708.004v1 AIMDYN INC.

This paper presents a novel learning framework to construct Koopman eigenfunctions for unknown, nonlinear dynamics using data gathered from experiments. The learning framework can extract spectral information from the full nonlinear dynamics by learning the eigenvalues and eigenfunctions of the associated Koopman operator. We then exploit the learned Koopman eigenfunctions to learn a lifted linear statespace model. To the best of our knowledge, our method is the first to utilize Koopman eigenfunctions as lifting functions for EDMD-based methods. We demonstrate the performance of the framework in state prediction and closed loop trajectory tracking of a simulated cart pole system. Our method is able to significantly improve the controller performance while relying on linear control methods to do nonlinear control.