Identification of large-scale atmospheric structures under different stability conditions using Dynamic Mode Decomposition

The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition

The objective of this study is to develop a reduced order model (ROM) that accurately predicts the key features of the wake generated by a wind turbine, based on the results of a computational fluid dynamics (CFD) simulation with the actuator line model (ALM). The simulation of a three-bladed up-wind wind turbine was performed using Reynolds-averaged Navier-Stokes (RANS) equations and the finite volume method. The two-equation turbulence model K-ωSST was used to approximate the Reynolds stresses and the Actuator Line Method (ALM) was used to simulate the effect of the wind turbine rotor on the flow field, representing the rotor blades as a distribution of forces acting on the fluid. Data from the RANS simulation was collected to be used by the Dynamic Mode Decomposition (DMD) algorithm. Velocity field was defined as the object of study for the construction of the reduced order model (ROM). From the implementation of the DMD algorithm, the eigenvalues and energy content were obtained for the field of interest. From these results, the main modes and their spectrum were calculated. The velocity field reconstruction was performed using the DMD algorithm, and the RMSE was computed relative to the simulation and experimental data, yielding a good approximation. Due to the turbulence model used, very stable results were obtained in the reduced order model, showing great capability to predict the fields of interest in time.

Abstract. We introduce the optimized dynamic mode decomposition (DMD) algorithm for constructing an adaptive and computationally efficient reduced-order model and forecasting tool for global atmospheric chemistry dynamics. By exploiting a low-dimensional set of global spatio-temporal modes, interpretable characterizations of the underlying spatial and temporal scales can be computed. Forecasting is also achieved with a linear model that uses a linear superposition of the dominant spatio-temporal features. The DMD method is demonstrated on 3 months of global chemistry dynamics data, showing its significant performance in terms of computational speed and interpretability. We show that the presented decomposition method successfully extracts and forecasts chemical patterns for leading chemical indicators, including nitric oxide, ozone, nitrogen dioxide, hydroxyl radical, isoprene, and carbon monoxide. Moreover, the DMD algorithm allows for rapid reconstruction of the underlying linear model, which can then easily accommodate non-stationary data and changes in the dynamics.

Analysis of aggregated load consumption forecasting in short, medium and long term horizons using Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of temporal, or spatio-temporal, data. A variety of regression techniques have been developed for producing the linear model approximation whose solutions are exponentials in time. For spatio-temporal data, DMD provides low-rank and interpretable models in the form of dominant modal structures along with their exponential/oscillatory behaviour in time. The majority of DMD algorithms, however, are prone to bias errors from noisy measurements of the dynamics, leading to poor model fits and unstable forecasting capabilities. The optimized DMD algorithm minimizes the model bias with a variable projection optimization, thus leading to stabilized forecasting capabilities. Here, the optimized DMD algorithm is improved by using statistical bagging methods whereby a single set of snapshots is used to produce an ensemble of optimized DMD models. The outputs of these models are averaged to produce a bagging, optimized dynamic mode decomposition (BOP-DMD). BOP-DMD improves performance by stabilizing and cross-validating the DMD model by ensembling; it also robustifies the model and provides both spatial and temporal uncertainty quantification (UQ). Thus, unlike currently available DMD algorithms, BOP-DMD provides a stable and robust model for probabilistic, or Bayesian, forecasting with comprehensive UQ metrics. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Trading strategy based on dynamic mode decomposition: Tested in Chinese stock market

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 DMD 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. A variant of the method may also be applied to online computation of “windowed DMD,” in which only the most recent data are used. 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. In particular, we show that the method is effective at capturing the dynamics of surface pressure measurements in the flow over a flat plate with an unsteady separation bubble.

In practical engineering applications, the multivariate signal contains more fault feature information than the single-channel signal. How to realize synchronous extraction of fault features from the multivariate signal is of great significance in fault diagnosis of rotary machinery. Dynamic mode decomposition (DMD) has attracted much attention due to its excellent dynamic feature extraction ability. However, DMD lacks mode aliasing property when dealing with the multivariate signal, which may lead to the loss of critical fault feature information. Cater to this problem, this article proposed a multivariate DMD (MDMD) algorithm that is the multivariate extension of DMD. First, snapshot tensors are defined to convert multivariate signals to tensor format. Then, the MDMD algorithm is proposed by introducing tensor operations into the original DMD algorithm, where a tensor low tubal rank component extraction framework is constructed to enable simultaneous extraction of bearing fault features from the multivariate signal, to enhance the robustness and effectiveness of the algorithm. Finally, both numerical simulations and experiments verify that the proposed MDMD has higher fault diagnosis accuracy than other multivariate signal-processing methods.

Operating convergent–divergent nozzles at low pressure ratios can lead to flow separation. Certain nozzle contours are known to produce a discrete acoustical tone under such conditions that are believed to be generated by a phenomenon known as transonic resonance, which involves a standing pressure wave situated between the separation shock and the nozzle exit plane. This paper reports the findings of a dynamic mode decomposition analysis of a perturbed axisymmetric unsteady Reynolds-averaged Navier–Stokes simulation of a nozzle flow known to exhibit this phenomenon. Two cases of different pressure ratios were studied in depth. The results show that the two cases differ in the shape of the standing pressure wave. The lower pressure ratio produces a standing pressure wave, whereas the higher pressure ratio produces a wave. In both cases, dynamic mode decomposition modes that match the standing pressure wave shape were found to be the least damped and the most energetic modes of the modes produced by the dynamic mode decomposition algorithm. The frequency of the mode for the lower pressure ratio matched the experimentally observed transonic tone frequency extraordinarily well, whereas the higher pressure ratio case was in fair agreement. The dynamic mode decomposition algorithm captured the transonic frequency for five additional pressure ratios.

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

Analysis of characteristic wake flow modes on a generic transonic backward-facing step configuration

A note on the use of Dynamic Mode Decomposition in mechanics

A new dynamic mode decomposition (DMD) method is introduced for simultaneous system identification and denoising in conjunction with the adoption of an extended Kalman filter algorithm. The present paper explains the extended-Kalman-filter-based DMD (EKFDMD) algorithm which is an online algorithm for dataset for a small number of degree of freedom (DoF). It also illustrates that EKFDMD requires significant numerical resources for many-degree-of-freedom (many-DoF) problems and that the combination with truncated proper orthogonal decomposition (trPOD) helps us to apply the EKFDMD algorithm to many-DoF problems, though it prevents the algorithm from being fully online. The numerical experiments of a noisy dataset with a small number of DoFs illustrate that EKFDMD can estimate eigenvalues better than or as well as the existing algorithms, whereas EKFDMD can also denoise the original dataset online. In particular, EKFDMD performs better than existing algorithms for the case in which system noise is present. The EKFDMD with trPOD, which unfortunately is not fully online, can be successfully applied to many-DoF problems, including a fluid-problem example, and the results reveal the superior performance of system identification and denoising.

A parallel and streaming Dynamic Mode Decomposition algorithm with finite precision error analysis for large data

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.