Session M27: Focus Session: Modal Analysis Methods for Fluid Flows I

Randomized Dynamic Mode Decomposition

The dynamic mode decomposition (DMD) is an equation-free, data-driven matrix decomposition that is capable of providing accurate reconstructions of spatio-temporal coherent structures arising in dynamical systems. We present randomized algorithms to compute the near-optimal low-rank dynamic mode decomposition for massive datasets. Randomized algorithms are simple, accurate and able to ease the computational challenges arising with `big data'. Moreover, randomized algorithms are amenable to modern parallel and distributed computing. The idea is to derive a smaller matrix from the high-dimensional input data matrix using randomness as a computational strategy. Then, the dynamic modes and eigenvalues are accurately learned from this smaller representation of the data, whereby the approximation quality can be controlled via oversampling and power iterations. Here, we present randomized DMD algorithms that are categorized by how many passes the algorithm takes through the data. Specifically, the single-pass randomized DMD does not require data to be stored for subsequent passes. Thus, it is possible to approximately decompose massive fluid flows (stored out of core memory, or not stored at all) using single-pass algorithms, which is infeasible with traditional DMD algorithms.   

Online dynamic mode decomposition for time-varying systems

Dynamic mode decomposition (DMD) is a popular technique for modal decomposition, flow analysis, and reduced-order modeling. In situations where a system is time varying, one would like to update the system’s description online as time evolves. This work provides an efficient method for computing the DMD matrix in real time, updating the approximation of a system’s dynamics as new data becomes available. The algorithm does not require storage of past data, and computes the exact DMD matrix using rank-1 updates. A weighting factor that places less weight on older data can be incorporated in a straightforward manner, making the method particularly well suited to time-varying systems. The efficiency of the method is compared against several existing DMD algorithms: for problems in which the state dimension is less than about 200, the proposed algorithm is the most efficient for real-time computation, and it can be orders of magnitude more efficient than the standard DMD algorithm. The method is demonstrated on several examples, including a time-varying linear system and a more complex example using data from a wind tunnel experiment.   

Dynamic Mode Decomposition based on Kalman Filter for Parameter Estimation

With the development of computational fluid dynamics, large-scale data can now be obtained. In order to model physical phenomena from such data, it is required to extract features of flow field. Dynamic mode decomposition (DMD) is a method which meets the request. DMD can compute dominant eigenmodes of flow field by approximating system matrix. From this point of view, DMD can be considered as parameter estimation of system matrix. To estimate such parameters, we propose a novel method based on Kalman filter. Our numerical experiments indicated that the proposed method can estimate the parameters more accurately if it is compared with standard DMD methods. With this method, it is also possible to improve the parameter estimation accuracy if characteristics of noise acting on the system is given.   

Dynamic Mode Decomposition based on Bootstrapping Extended Kalman Filter Application to Noisy data

In this study, dynamic mode decomposition (DMD) based on bootstrapping extended Kalman filter is proposed for time-series data. In this framework, state variables ($x$ and $y$) are filtered as well as the parameter estimation ($a_{ij}$) which is conducted in the conventional DMD and the standard Kalman-filter-based DMD. The filtering process of state variables enables us to obtain highly accurate eigenvalue of the system with strong noise. In the presentation, formulation, advantages and disadvantages are discussed.   

Dynamic mode shaping for transient growth suppression

Sub-critical transition to turbulence is often triggered by transient energy growth attributed to the non-normality of the linearized Navier-Stokes operator. Here, we formulate a series of feedback control strategies that aim to reduce and suppress transient energy growth using a dynamic mode shaping perspective. In particular, we present controller synthesis techniques for dynamic mode matching and dynamic mode orthogonalization, which allow for spectral specification of the closed-loop dynamics. We demonstrate the control schemes on a number of illustrative examples and discuss extensions to large-scale and nonlinear systems.   

Data-driven discovery of Koopman eigenfunctions for control

Koopman operator theory has emerged as a principled framework to obtain linear embeddings of nonlinear dynamics, enabling the estimation, prediction and control of strongly nonlinear systems using standard linear techniques. Here, we present a data-driven control architecture that utilizes Koopman eigenfunctions to manipulate nonlinear systems using linear control theory. We approximate these eigenfunctions with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. In particular, we show that lightly damped eigenfunctions may be faithfully extracted using sparse regression. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. We formulate the control problem in these intrinsic eigenfunction coordinates and design nonlinear controllers using standard linear optimal control theory. The architecture is demonstrated on a variety of Hamiltonian systems and the double-gyre model for ocean mixing.   

Machine Learning-based discovery of closures for reduced models of dynamical systems

Despite the successful application of machine learning (ML) in fields such as image processing and speech recognition, only a few attempts has been made toward employing ML to represent the dynamics of complex physical systems. Previous attempts mostly focus on parameter calibration or data-driven augmentation of existing models. In this work we present a ML framework to discover closure terms in reduced models of dynamical systems and provide insights into potential problems associated with data-driven modeling. Based on exact closure models for linear system, we propose a general linear closure framework from viewpoint of optimization. The framework is based on trapezoidal approximation of convolution term. Hyperparameters that need to be determined include temporal length of memory effect, number of sampling points, and dimensions of hidden states. To circumvent the explicit specification of memory effect, a general framework inspired from neural networks is also proposed. We conduct both a priori and posteriori evaluations of the resulting model on a number of non-linear dynamical systems.   

Low-dimensional and Data Fusion Techniques Applied to a Rectangular Supersonic Multi-stream Jet

Low-dimensional models of experimental and simulation data for a complex supersonic jet were fused to reconstruct time-dependent proper orthogonal decomposition (POD) coefficients. The jet consists of a multi-stream rectangular single expansion ramp nozzle, containing a core stream operating at $M_{j,1} = 1.6$, and bypass stream at $M_{j,3} = 1.0$ with an underlying deck. POD was applied to schlieren and PIV data to acquire the spatial basis functions. These eigenfunctions were projected onto their corresponding time-dependent large eddy simulation (LES) fields to reconstruct the temporal POD coefficients. This reconstruction was able to resolve spectral peaks that were previously aliased due to the slower sampling rates of the experiments. Additionally, dynamic mode decomposition (DMD) was applied to the experimental and LES datasets, and the spatio-temporal characteristics were compared to POD.