A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions

Identifying Finite-Time Coherent Sets from Limited Quantities of Lagrangian Data

A data-driven procedure for identifying the dominant transport barriers in a time-varying flow from limited quantities of Lagrangian data is presented. Our approach partitions state space into pairs of coherent sets, which are sets of initial conditions chosen to minimize the number of trajectories that "leak" from one set to the other under the influence of a stochastic flow field during a pre-specified interval in time. In practice, this partition is computed by posing an optimization problem, which once solved, yields a pair of functions whose signs determine set membership. From prior experience with synthetic, "data rich" test problems and conceptually related methods based on approximations of the Perron-Frobenius operator, we observe that the functions of interest typically appear to be smooth. As a result, given a fixed amount of data our approach, which can use sets of globally supported basis functions, has the potential to more accurately approximate the desired functions than other functions tailored to use compactly supported indicator functions. This difference enables our approach to produce effective approximations of pairs of coherent sets in problems with relatively limited quantities of Lagrangian data, which is usually the case with real geophysical data. We apply this method to three examples of increasing complexity: the first is the double gyre, the second is the Bickley Jet, and the third is data from numerically simulated drifters in the Sulu Sea.Comment: 14 pages, 7 figure

Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition

Dynamic mode decomposition (DMD) provides a practical means of extracting insightful dynamical information from fluids datasets. Like any data processing technique, DMD's usefulness is limited by its ability to extract real and accurate dynamical features from noise-corrupted data. Here we show analytically that DMD is biased to sensor noise, and quantify how this bias depends on the size and noise level of the data. We present three modifications to DMD that can be used to remove this bias: (i) a direct correction of the identified bias using known noise properties, (ii) combining the results of performing DMD forwards and backwards in time, and (iii) a total least-squares-inspired algorithm. We discuss the relative merits of each algorithm, and demonstrate the performance of these modifications on a range of synthetic, numerical, and experimental datasets. We further compare our modified DMD algorithms with other variants proposed in recent literature

Data Fusion via Intrinsic Dynamic Variables: An Application of Data-Driven Koopman Spectral Analysis

We demonstrate that numerically computed approximations of Koopman eigenfunctions and eigenvalues create a natural framework for data fusion in applications governed by nonlinear evolution laws. This is possible because the eigenvalues of the Koopman operator are invariant to invertible transformations of the system state, so that the values of the Koopman eigenfunctions serve as a set of intrinsic coordinates that can be used to map between different observations (e.g., measurements obtained through different sets of sensors) of the same fundamental behavior. The measurements we wish to merge can also be nonlinear, but must be "rich enough" to allow (an effective approximation of) the state to be reconstructed from a single set of measurements. This approach requires independently obtained sets of data that capture the evolution of the heterogeneous measurements and a single pair of "joint" measurements taken at one instance in time. Computational approximations of eigenfunctions and their corresponding eigenvalues from data are accomplished using Extended Dynamic Mode Decomposition. We illustrate this approach on measurements of spatio-temporal oscillations of the FitzHugh-Nagumo PDE, and show how to fuse point measurements with principal component measurements, after which either set of measurements can be used to estimate the other set.Comment: 8 pages, 6 figure

Mindfulness-Oriented Recovery Enhancement for Chronic Pain and Prescription Opioid Misuse: Results from an Early Stage Randomized Controlled Trial

Objective: Opioid pharmacotherapy is now the leading treatment for chronic pain, a problem that affects nearly one third of the U.S. population. Given the dramatic rise in prescription opioid misuse and opioid-related mortality, novel behavioral interventions are needed. The purpose of this study was to conduct an early-stage randomized controlled trial of Mindfulness-Oriented Recovery Enhancement (MORE), a multimodal intervention designed to simultaneously target mechanisms underpinning chronic pain and opioid misuse. Method: Chronic pain patients (N = 115; mean age = 48 ± 14 years; 68% female) were randomized to 8 weeks of MORE or a support group (SG). Outcomes were measured at pre- and posttreatment, and at 3-month follow-up. The Brief Pain Inventory was used to assess changes in pain severity and interference. Changes in opioid use disorder status were measured by the Current Opioid Misuse Measure. Desire for opioids, stress, nonreactivity, reinterpretation of pain sensations, and reappraisal were also evaluated. Results: MORE participants reported significantly greater reductions in pain severity (p = .038) and interference (p = .003) than SG participants, which were maintained by 3-month follow-up and mediated by increased nonreactivity and reinterpretation of pain sensations. Compared with SG participants, participants in MORE evidenced significantly less stress arousal (p = .034) and desire for opioids (p = .027), and were significantly more likely to no longer meet criteria for opioid use disorder immediately following treatment (p = .05); however, these effects were not sustained at follow-up. Conclusions: Findings demonstrate preliminary feasibility and efficacy of MORE as a treatment for co-occurring prescription opioid misuse and chronic pain