Matthew O. Williams

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” (Mezic in Nonlinear Dynamics 41(1–3): 309–325,xa02005). 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.

Dynamic mode decomposition for large and streaming datasets

We formulate a low-storage method for performing dynamic mode decomposition that can be updated inexpensively as new data become available; this formulation allows dynamical information to be extracted from large datasets and data streams. We present two algorithms: the first is mathematically equivalent to a standard “batch-processed” formulation; the second introduces a compression step that maintains computational efficiency, while enhancing the ability to isolate pertinent dynamical information from noisy measurements. Both algorithms reliably capture dominant fluid dynamic behaviors, as demonstrated on cylinder wake data collected from both direct numerical simulations and particle image velocimetry experiments.

Damped-driven granular chains: An ideal playground for dark breathers and multibreathers

By applying an out-of-phase actuation at the boundaries of a uniform chain of granular particles, we demonstrate experimentally that time-periodic and spatially localized structures with a nonzero background (so-called dark breathers) emerge for a wide range of parameter values and initial conditions. We demonstrate a remarkable control over the number of breathers within the multibreather pattern that can be dialed in by varying the frequency or amplitude of the actuation. The values of the frequency (or amplitude) where the transition between different multibreather states occurs are predicted accurately by the proposed theoretical model, which is numerically shown to support exact dark breather and multibreather solutions. Moreover, we visualize detailed temporal and spatial profiles of breathers and, especially, of multibreathers using a full-field probing technology and enable a systematic favorable comparison among theory, computation, and experiments. A detailed bifurcation analysis reveals that the dark and multibreather families are connected in a snaking pattern, providing a roadmap for the identification of such fundamental states and their bistability in the laboratory.

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.

Improving separation control with noise-robust variants of dynamic mode decomposition

Flow separation can lead to degraded performance in many engineered systems, which has led to sustained interest in developing strategies for suppressing and controlling flow separation. Separation control strategies based on open-loop forcing via synthetic jets have demonstrated a relative degree of success in various studies; however, many of these studies have relied upon trial-and-error “tuning” of a synthetic jet’s operating parameters for satisfactory performance with respect to a particular flow configuration. Subsequent work has focused on improving the general understanding of fluid flow separation from a dynamical systems perspective, with the aim of isolating key mechanisms that can be exploited for more systematic controller designs. Numerical studies have shown that dynamically dominant flow characteristics, identified by the dynamic mode decomposition (DMD), can be used to guide the design of open-loop separation control strategies. While these approaches have proven valuable for dynamical analyses in numerics, standard formulations of DMD have recently been shown to possess systematic errors that can lead to misleading results when the data are corrupted by some degree of measurement noise (e.g., sensor noise in experimental studies). Here, we make use of DMD to synthesize time-resolved particle image velocimetry (TR-PIV) data from a canonical separation experiment in an effort to inform the design of open-loop separation control strategies; to this end, we make use of a noise-aware version of DMD—introduced in Hemati et al. (2015)—to assess the impact of measurement noise on the conclusions drawn for informing open-loop controller design. Additionally, we extend the noise-aware framework to formulate a noise-robust version of the streaming DMD algorithm presented in Hemati et al. (2014). Dynamic characterizations afforded by DMD-based techniques are then used to inform open-loop separation control strategies that are tested in experiments. We find that open-loop forcing at a frequency associated with the dominant DMD mode reduces the mean height of the separation bubble, suggesting that DMD-based techniques may provide a systematic means of designing open-loop control strategies aimed at suppressing flow separation.

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 coherent pairs, 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 solving an optimization problem to obtain 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. We exploit this property by using the basis sets associated with spectral or mesh-free methods, and as a result, our approach has the potential to more accurately approximate these functions given a fixed amount of data. In practice, this could enable better approximations of the coherent pairs in problems with relatively limited quantities of Lagrangian data, which is usually the case with experimental 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.

Semiconductor Diode Laser Mode-Locking by a Waveguide Array

We demonstrate theoretically that robust mode-locking can be achieved in an edge-emitting semiconductor diode laser with a waveguide array architecture. The waveguide array provides an ideal saturable absorption mechanism for initial start-up from noise, as well as pulse shaping and stabilization. One element in the waveguide array is forward-biased to generate gain, while nonlinear coupling to the other elements results in an effective saturable absorber. This new approach can be integrated directly with standard edge-emitting designs.

Nonlinear Resonances and Antiresonances of a Forced Sonic Vacuum

We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude- and frequency-dependent boundary drive. Despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period and observe that this can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force (antiresonances) between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings and Neimark-Sacker bifurcations as well as multiple isolas (e.g., of period-3, -4, or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports a notion of a (purely) nonlinear spectrum in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). We rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.

Equation-free Computations as DDDAS Protocols in the Study of Engineered Granular Crystals☆

Abstract We explore the use of Equation-Free algorithms as Dynamic Data Driven experimental design protocols for the computational as well as laboratory study of the dynamics of engineered granular crystals and their models. The ability to prescribe de- sired initial conditions for computational -and, in this case, also possibly for laboratory- experiments provides an interesting link between traditional, matrix-free numerical analysis and the acceleration of dynamic studies. The framework is further enhanced through combination with data-mining algorithms that process detailed, fine-scale data to uncover underlying im- portant, coarse-grained variables (macroscopic system observables).

Coarse graining, dynamic renormalization and the kinetic theory of shock clustering

We demonstrate the utility of the equation free methodology developed by one of the authors (I.G.K) for the study of scalar conservation laws with disordered initial conditions. The numerical scheme is benchmarked on exact solutions in Burgers turbulence corresponding to Levy process initial data. For these initial data, the kinetics of shock clustering is described by Smoluchowskis coagulation equation with additive kernel. The equation free methodology is used to develop a particle scheme that computes self-similar solutions to the coagulation equation, including those with fat tails.