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1
Kawashima, Takahiro, Hayaru Shouno, and Hideitsu Hino. "Bayesian Dynamic Mode Decomposition with Variational Matrix Factorization." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 9 (May 18, 2021): 8083–91. http://dx.doi.org/10.1609/aaai.v35i9.16985.
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Dynamic mode decomposition (DMD) and its extensions are data-driven methods that have substantially contributed to our understanding of dynamical systems. However, because DMD and most of its extensions are deterministic, it is difficult to treat probabilistic representations of parameters and predictions. In this work, we propose a novel formulation of a Bayesian DMD model. Our Bayesian DMD model is consistent with the procedure of standard DMD, which is to first determine the subspace of observations, and then compute the modes on that subspace. Variational matrix factorization makes it possible to realize a fully-Bayesian scheme of DMD. Moreover, we derive a Bayesian DMD model for incomplete data, which demonstrates the advantage of probabilistic modeling. Finally, both of nonlinear simulated and real-world datasets are used to illustrate the potential of the proposed method.
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Schmid, Peter J. "Dynamic Mode Decomposition and Its Variants." Annual Review of Fluid Mechanics 54, no. 1 (January 5, 2022): 225–54. http://dx.doi.org/10.1146/annurev-fluid-030121-015835.
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Dynamic mode decomposition (DMD) is a factorization and dimensionality reduction technique for data sequences. In its most common form, it processes high-dimensional sequential measurements, extracts coherent structures, isolates dynamic behavior, and reduces complex evolution processes to their dominant features and essential components. The decomposition is intimately related to Koopman analysis and, since its introduction, has spawned various extensions, generalizations, and improvements. It has been applied to numerical and experimental data sequences taken from simple to complex fluid systems and has also had an impact beyond fluid dynamics in, for example, video surveillance, epidemiology, neurobiology, and financial engineering. This review focuses on the practical aspects of DMD and its variants, as well as on its usage and characteristics as a quantitative tool for the analysis of complex fluid processes.
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Schmid, Peter J. "Dynamic Mode Decomposition and Its Variants." Annual Review of Fluid Mechanics 54, no. 1 (January 5, 2022): 225–54. http://dx.doi.org/10.1146/annurev-fluid-030121-015835.
Full textAbstract:
Dynamic mode decomposition (DMD) is a factorization and dimensionality reduction technique for data sequences. In its most common form, it processes high-dimensional sequential measurements, extracts coherent structures, isolates dynamic behavior, and reduces complex evolution processes to their dominant features and essential components. The decomposition is intimately related to Koopman analysis and, since its introduction, has spawned various extensions, generalizations, and improvements. It has been applied to numerical and experimental data sequences taken from simple to complex fluid systems and has also had an impact beyond fluid dynamics in, for example, video surveillance, epidemiology, neurobiology, and financial engineering. This review focuses on the practical aspects of DMD and its variants, as well as on its usage and characteristics as a quantitative tool for the analysis of complex fluid processes.
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Heiland, Jan, and Benjamin Unger. "Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition." Mathematics 10, no. 3 (January 28, 2022): 418. http://dx.doi.org/10.3390/math10030418.
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Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.
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Page, Jacob, and Rich R. Kerswell. "Koopman mode expansions between simple invariant solutions." Journal of Fluid Mechanics 879 (September 19, 2019): 1–27. http://dx.doi.org/10.1017/jfm.2019.686.
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A Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with dynamic mode decomposition (DMD). However, there are non-trivial requirements for the Koopman decomposition and DMD to overlap, which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a Stuart–Landau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a cross-over point in state space. Well-designed DMD can extract the two expansions provided that the time window does not contain this cross-over point. We then apply DMD to the Navier–Stokes equations near to a heteroclinic connection in low Reynolds number ($Re=O(100)$) plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and once more indicates the existence of cross-over points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a cross-over point. Our results suggest that in a dynamical system possessing multiple simple invariant solutions, there are generically places in phase space – plausibly hypersurfaces delineating the boundary of a local Koopman expansion – across which the dynamics cannot be represented by a convergent Koopman expansion.
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Albidah, A. B., W. Brevis, V. Fedun, I. Ballai, D. B. Jess, M. Stangalini, J. Higham, and G. Verth. "Proper orthogonal and dynamic mode decomposition of sunspot data." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, no. 2190 (December 21, 2020): 20200181. http://dx.doi.org/10.1098/rsta.2020.0181.
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High-resolution solar observations show the complex structure of the magnetohydrodynamic (MHD) wave motion. We apply the techniques of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) to identify the dominant MHD wave modes in a sunspot using the intensity time series. The POD technique was used to find modes that are spatially orthogonal, whereas the DMD technique identifies temporal orthogonality. Here, we show that the combined POD and DMD approaches can successfully identify both sausage and kink modes in a sunspot umbra with an approximately circular cross-sectional shape. This article is part of the Theo Murphy meeting issue ‘High-resolution wave dynamics in the lower solar atmosphere’.
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Surasinghe, Sudam, and Erik M. Bollt. "Randomized Projection Learning Method for Dynamic Mode Decomposition." Mathematics 9, no. 21 (November 4, 2021): 2803. http://dx.doi.org/10.3390/math9212803.
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A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension.
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Habibi, Milad, Scott T. M. Dawson, and Amirhossein Arzani. "Data-Driven Pulsatile Blood Flow Physics with Dynamic Mode Decomposition." Fluids 5, no. 3 (July 14, 2020): 111. http://dx.doi.org/10.3390/fluids5030111.
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Dynamic mode decomposition (DMD) is a purely data-driven and equation-free technique for reduced-order modeling of dynamical systems and fluid flow. DMD finds a best fit linear reduced-order model that represents any given spatiotemporal data. In DMD, each mode evolves with a fixed frequency and therefore DMD modes represent physically meaningful structures that are ranked based on their dynamics. The application of DMD to patient-specific cardiovascular flow data is challenging. First, the input flow rate is unsteady and pulsatile. Second, the flow topology can change significantly in different phases of the cardiac cycle. Finally, blood flow in patient-specific diseased arteries is complex and often chaotic. The objective of this study was to overcome these challenges using our proposed multistage dynamic mode decomposition with control (mDMDc) method and use this technique to study patient-specific blood flow physics. The inlet flow rate was considered as the controller input to the systems. Blood flow data were divided into different stages based on the inlet flow waveform and DMD with control was applied to each stage. The system was augmented to consider both velocity and wall shear stress (WSS) vector data, and therefore study the interaction between the coherent structures in velocity and near-wall coherent structures in WSS. First, it was shown that DMD modes can exactly represent the analytical Womersley solution for incompressible pulsatile flow in tubes. Next, our method was applied to image-based coronary artery stenosis and cerebral aneurysm models where complex blood flow patterns are anticipated. The flow patterns were studied using the mDMDc modes and the reconstruction errors were reported. Our augmented mDMDc framework could capture coherent structures in velocity and WSS with a fewer number of modes compared to the traditional DMD approach and demonstrated a close connection between the velocity and WSS modes.
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Bronstein, Emil, Aviad Wiegner, Doron Shilo, and Ronen Talmon. "The spatiotemporal coupling in delay-coordinates dynamic mode decomposition." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 12 (December 2022): 123127. http://dx.doi.org/10.1063/5.0123101.
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Dynamic mode decomposition (DMD) is a leading tool for equation-free analysis of high-dimensional dynamical systems from observations. In this work, we focus on a combination of DMD and delay-coordinates embedding, which is termed delay-coordinates DMD and is based on augmenting observations from current and past time steps, accommodating the analysis of a broad family of observations. An important utility of DMD is the compact and reduced-order spectral representation of observations in terms of the DMD eigenvalues and modes, where the temporal information is separated from the spatial information. From a spatiotemporal viewpoint, we show that when DMD is applied to delay-coordinates embedding, temporal information is intertwined with spatial information, inducing a particular spectral structure on the DMD components. We formulate and analyze this structure, which we term the spatiotemporal coupling in delay-coordinates DMD. Based on this spatiotemporal coupling, we propose a new method for DMD components selection. When using delay-coordinates DMD that comprises redundant modes, this selection is an essential step for obtaining a compact and reduced-order representation of the observations. We demonstrate our method on noisy simulated signals and various dynamical systems and show superior component selection compared to a commonly used method that relies on the amplitudes of the modes.
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Wynn, A., D. S. Pearson, B. Ganapathisubramani, and P. J. Goulart. "Optimal mode decomposition for unsteady flows." Journal of Fluid Mechanics 733 (September 24, 2013): 473–503. http://dx.doi.org/10.1017/jfm.2013.426.
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AbstractA new method, herein referred to as optimal mode decomposition (OMD), of finding a linear model to describe the evolution of a fluid flow is presented. The method estimates the linear dynamics of a high-dimensional system which is first projected onto a subspace of a user-defined fixed rank. An iterative procedure is used to find the optimal combination of linear model and subspace that minimizes the system residual error. The OMD method is shown to be a generalization of dynamic mode decomposition (DMD), in which the subspace is not optimized but rather fixed to be the proper orthogonal decomposition (POD) modes. Furthermore, OMD is shown to provide an approximation to the Koopman modes and eigenvalues of the underlying system. A comparison between OMD and DMD is made using both a synthetic waveform and an experimental data set. The OMD technique is shown to have lower residual errors than DMD and is shown on a synthetic waveform to provide more accurate estimates of the system eigenvalues. This new method can be used with experimental and numerical data to calculate the ‘optimal’ low-order model with a user-defined rank that best captures the system dynamics of unsteady and turbulent flows.
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Alford-Lago, D. J., C. W. Curtis, A. T. Ihler, and O. Issan. "Deep learning enhanced dynamic mode decomposition." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 3 (March 2022): 033116. http://dx.doi.org/10.1063/5.0073893.
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Koopman operator theory shows how nonlinear dynamical systems can be represented as an infinite-dimensional, linear operator acting on a Hilbert space of observables of the system. However, determining the relevant modes and eigenvalues of this infinite-dimensional operator can be difficult. The extended dynamic mode decomposition (EDMD) is one such method for generating approximations to Koopman spectra and modes, but the EDMD method faces its own set of challenges due to the need of user defined observables. To address this issue, we explore the use of autoencoder networks to simultaneously find optimal families of observables, which also generate both accurate embeddings of the flow into a space of observables and submersions of the observables back into flow coordinates. This network results in a global transformation of the flow and affords future state prediction via the EDMD and the decoder network. We call this method the deep learning dynamic mode decomposition (DLDMD). The method is tested on canonical nonlinear data sets and is shown to produce results that outperform a standard DMD approach and enable data-driven prediction where the standard DMD fails.
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Darling, Keir, and Lawrence M. Widrow. "Eigenfunctions of Galactic phase space spirals from dynamic mode decomposition." Monthly Notices of the Royal Astronomical Society 490, no. 1 (September 13, 2019): 114–23. http://dx.doi.org/10.1093/mnras/stz2539.
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ABSTRACT We investigate the spatiotemporal structure of simulations of the homogeneous slab and isothermal plane models for the vertical motion in the Galactic disc. We use dynamic mode decomposition (DMD) to compute eigenfunctions of the simulated distribution functions for both models, referred to as DMD modes. In the case of the homogeneous slab, we compare the DMD modes to the analytic normal modes of the system to evaluate the feasibility of DMD in collisionless self-gravitating systems. This is followed by the isothermal plane model, where we focus on the effect of self-gravity on phase mixing. We compute DMD modes of the system for varying relative dominance of mutual interaction and external potential, so as to study the corresponding variance in mode structure and lifetime. We find that there is a regime of relative dominance, at approximately 4 : 1 external potential to mutual interaction where the DMD modes are spirals in the (z, vz) plane, and are nearly un-damped. This leads to the proposition that a system undergoing phase mixing in the presence of weak-to-moderate self-gravity can have persisting spiral structure in the form of such modes. We then conclude with the conjecture that such a mechanism may be at work in the phase space spirals observed in Gaia Data Release 2, and that studying more complex simulations with DMD may aid in understanding both the timing and form of the perturbation that lead to the observed spirals.
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Noack, Bernd R., Witold Stankiewicz, Marek Morzyński, and Peter J. Schmid. "Recursive dynamic mode decomposition of transient and post-transient wake flows." Journal of Fluid Mechanics 809 (November 21, 2016): 843–72. http://dx.doi.org/10.1017/jfm.2016.678.
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A novel data-driven modal decomposition of fluid flow is proposed, comprising key features of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The first mode is the normalized real or imaginary part of the DMD mode that minimizes the time-averaged residual. The $N$th mode is defined recursively in an analogous manner based on the residual of an expansion using the first $N-1$ modes. The resulting recursive DMD (RDMD) modes are orthogonal by construction, retain pure frequency content and aim at low residual. Recursive DMD is applied to transient cylinder wake data and is benchmarked against POD and optimized DMD (Chen et al., J. Nonlinear Sci., vol. 22, 2012, pp. 887–915) for the same snapshot sequence. Unlike POD modes, RDMD structures are shown to have purer frequency content while retaining a residual of comparable order to POD. In contrast to DMD, with exponentially growing or decaying oscillatory amplitudes, RDMD clearly identifies initial, maximum and final fluctuation levels. Intriguingly, RDMD outperforms both POD and DMD in the limit-cycle resolution from the same snapshots. Robustness of these observations is demonstrated for other parameters of the cylinder wake and for a more complex wake behind three rotating cylinders. Recursive DMD is proposed as an attractive alternative to POD and DMD for empirical Galerkin models, in particular for nonlinear transient dynamics.
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Landa, Vlad, and Yuval Reuveni. "Assessment of Dynamic Mode Decomposition (DMD) Model for Ionospheric TEC Map Predictions." Remote Sensing 15, no. 2 (January 6, 2023): 365. http://dx.doi.org/10.3390/rs15020365.
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In this study, we assess the Dynamic Mode Decomposition (DMD) model applied with global ionospheric vertical Total Electron Content (vTEC) maps to construct 24-hour global ionospheric vTEC map forecasts using the available International GNSS Service (IGS) 2-hours cadence vTEC maps. In addition, we examine the impact of a EUV 121.6 nm time series data source with the DMD control (DMDc) framework, which shows an improvement in the vTEC Root Mean Square Error (RMSE) values compared with the IGS final solution vTEC maps. Both the DMD and DMDc predictions present close RMSE scores compared with the available CODE 1-day predicted ionospheric maps, both for quiet and disturbed solar activity. Finally, we evaluate the predicted global ionospheric vTEC maps with the East-North-Up (ENU) coordinate system errors metric, as an ionospheric correction source for L1 single-frequency GPS/GNSS Single Point Positioning (SPP) solutions. Based on these findings, we argue that the commonly adopted vTEC map comparison RMSE metric fails to correctly reflect an informative impact with L1 single-frequency positioning solutions using dual-frequency ionospheric corrections.
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Gao, Mei, Xiao-Qun Cao, Bai-Nian Liu, Zi-Hang Han, Shi-Cheng Hou, and Guo-Gui Yang. "Comparison of the Atmospheric 200 hPa Jet’s Analyses between Proper Orthogonal Decomposition and Advanced Dynamic Mode Decomposition Method." Advances in Meteorology 2020 (September 17, 2020): 1–15. http://dx.doi.org/10.1155/2020/8383657.
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In this paper, a frequently employed technique named the sparsity-promoting dynamic mode decomposition (SPDMD) is proposed to analyze the velocity fields of atmospheric motion. The dynamic mode decomposition method (DMD) is an effective technique to extract dynamic information from flow fields that is generated from direct experiment measurements or numerical simulation and has been broadly employed to study the dynamics of the flow, to achieve a reduced-order model (ROM) of the complex high dimensional flow field, and even to predict the evolution of the flow in a short time in the future. However, for standard DMD, it is hard to determine which modes are the most physically relevant, unlike the proper orthogonal decomposition (POD) method which ranks the decomposed modes according to their energy content. The advanced modal decomposition method SPDMD is a variant of the standard DMD, which is capable of determining the modes that can be used to achieve a high-quality approximation of the given field. It is novel to introduce the SPDMD to analyze the atmospheric flow field. In this study, SPDMD is applied to extract essential dynamic information from the 200 hPa jet flow, and the decomposed results are compared with the POD method. To further demonstrate the extraction effect of POD/SPDMD methods on the 200 hPa jet flow characteristics, the POD/SPDMD reduced-order models are constructed, respectively. The results show that both modal decomposition methods successfully extract the underlying coherent structures from the 200 hPa jet flow. And the DMD method provides additional information on the modal properties, such as temporal frequency and growth rate of each mode which can be used to identify the stability of the modes. It is also found that a fewer order of modes determined by the SPDMD method can capture nearly the same dynamic information of the jet flow as the POD method. Furthermore, from the quantitative comparisons between the POD and SPDMD reduced-order models, the latter provides a higher precision than the former, especially when the number of modes is small.
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Nguyen, Van Duc, Viet Dung Duong, Minh Hoang Trinh, Hoang Quan Nguyen, and Dang Thai Son Nguyen. "Low order modeling of dynamic stall using vortex particle method and dynamic mode decomposition." International Journal of Micro Air Vehicles 15 (January 2023): 175682932211479. http://dx.doi.org/10.1177/17568293221147923.
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Low order modelings are performed in this paper, including iterative Brinkman penalized vortex method (IBVM) and data-driven dynamic mode decomposition (DMD) for dynamic stall study of symmetric airfoil. The data are extracted from IBVM as input for flow field reconstruction using combinations of DMD dominant modes, representing extracted flow features. The primary mode together with its harmonics, and the mean mode are termed to be dominant for the airfoil wake duplication at fixed angles of attack ([Formula: see text]) ranging from [Formula: see text] to [Formula: see text]. For the dynamic stall duplication, at small and large pitching amplitudes, the nearfield and farfield vorticty contours from the DMD generally agree well with those from the IBVM. In addition, the lift coefficient from the DMD collapses well with that from the IBVM and the experiment.
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Li, Binghua, Jesús Garicano-Mena, Yao Zheng, and Eusebio Valero. "Dynamic Mode Decomposition Analysis of Spatially Agglomerated Flow Databases." Energies 13, no. 9 (April 28, 2020): 2134. http://dx.doi.org/10.3390/en13092134.
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Dynamic Mode Decomposition (DMD) techniques have risen as prominent feature identification methods in the field of fluid dynamics. Any of the multiple variables of the DMD method allows to identify meaningful features from either experimental or numerical flow data on a data-driven manner. Performing a DMD analysis requires handling matrices V ∈ R n p × N , where n p and N are indicative of the spatial and temporal resolutions. The DMD analysis of a complex flow field requires long temporal sequences of well resolved data, and thus the memory footprint may become prohibitively large. In this contribution, the effect that principled spatial agglomeration (i.e., reduction in n p via clustering) has on the results derived from the DMD analysis is investigated. We compare twelve different clustering algorithms on three testcases, encompassing different flow regimes: a synthetic flow field, a R e D = 60 flow around a cylinder cross section, and a R e τ ≈ 200 turbulent channel flow. The performance of the clustering techniques is thoroughly assessed concerning both the accuracy of the results retrieved and the computational performance. From this assessment, we identify DBSCAN/HDBSCAN as the methods to be used if only relatively high agglomeration levels are affordable. On the contrary, Mini-batch K-means arises as the method of choice whenever high agglomeration n p ˜ / n p ≪ 1 is possible.
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Yin, Jingwei, Bing Liu, Guangping Zhu, and Zhinan Xie. "Moving Target Detection Using Dynamic Mode Decomposition." Sensors 18, no. 10 (October 15, 2018): 3461. http://dx.doi.org/10.3390/s18103461.
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It is challenging to detect a moving target in the reverberant environment for a long time. In recent years, a kind of method based on low-rank and sparse theory was developed to study this problem. The multiframe data containing the target echo and reverberation are arranged in a matrix, and then, the detection is achieved by low-rank and sparse decomposition of the data matrix. In this paper, we introduce a new method for the matrix decomposition using dynamic mode decomposition (DMD). DMD is usually used to calculate eigenmodes of an approximate linear model. We divided the eigenmodes into two categories to realize low-rank and sparse decomposition such that we detected the target from the sparse component. Compared with the previous methods based on low-rank and sparse theory, our method improves the computation speed by approximately 4–90-times at the expense of a slight loss of detection gain. The efficient method has a big advantage for real-time processing. This method can spare time for other stages of processing to improve the detection performance. We have validated the method with three sets of underwater acoustic data.
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Bagheri, Shervin. "Koopman-mode decomposition of the cylinder wake." Journal of Fluid Mechanics 726 (June 11, 2013): 596–623. http://dx.doi.org/10.1017/jfm.2013.249.
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AbstractThe Koopman operator provides a powerful way of analysing nonlinear flow dynamics using linear techniques. The operator defines how observables evolve in time along a nonlinear flow trajectory. In this paper, we perform a Koopman analysis of the first Hopf bifurcation of the flow past a circular cylinder. First, we decompose the flow into a sequence of Koopman modes, where each mode evolves in time with one single frequency/growth rate and amplitude/phase, corresponding to the complex eigenvalues and eigenfunctions of the Koopman operator, respectively. The analytical construction of these modes shows how the amplitudes and phases of nonlinear global modes oscillating with the vortex shedding frequency or its harmonics evolve as the flow develops and later sustains self-excited oscillations. Second, we compute the dynamic modes using the dynamic mode decomposition (DMD) algorithm, which fits a linear combination of exponential terms to a sequence of snapshots spaced equally in time. It is shown that under certain conditions the DMD algorithm approximates Koopman modes, and hence provides a viable method to decompose the flow into saturated and transient oscillatory modes. Finally, the relevance of the analysis to frequency selection, global modes and shift modes is discussed.
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Towne, Aaron, Oliver T. Schmidt, and Tim Colonius. "Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis." Journal of Fluid Mechanics 847 (May 29, 2018): 821–67. http://dx.doi.org/10.1017/jfm.2018.283.
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We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.
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Yang, Yang, Xiao Liu, and Zhihao Zhang. "Analysis of V-Gutter Reacting Flow Dynamics Using Proper Orthogonal and Dynamic Mode Decompositions." Energies 13, no. 18 (September 17, 2020): 4886. http://dx.doi.org/10.3390/en13184886.
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The current work is focused on investigating the potential of data-driven post-processing techniques, including proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) for flame dynamics. Large-eddy simulation (LES) of a V-gutter premixed flame was performed with two Reynolds numbers. The flame transfer function (FTF) was calculated. The POD and DMD were used for the analysis of the flame structures, wake shedding frequency, etc. The results acquired by different methods were also compared. The FTF results indicate that the flames have proportional, inertial, and delay components. The POD method could capture the shedding wake motion and shear layer motion. The excited DMD modes corresponded to the shear layer flames’ swing and convect motions in certain directions. Both POD and DMD could help to identify the wake shedding frequency. However, this large-scale flame oscillation is not presented in the FTF results. The negative growth rates of the decomposed mode confirm that the shear layer stabilized flame was more stable than the flame possessing a wake instability. The corresponding combustor design could be guided by the above results.
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Seo, Jong-Hyeon, Ichiro Tsuda, Young Ju Lee, Akio Ikeda, Masao Matsuhashi, Riki Matsumoto, Takayuki Kikuchi, and Hunseok Kang. "Pattern Recognition in Epileptic EEG Signals via Dynamic Mode Decomposition." Mathematics 8, no. 4 (April 1, 2020): 481. http://dx.doi.org/10.3390/math8040481.
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In this paper, we propose a new method based on the dynamic mode decomposition (DMD) to find a distinctive contrast between the ictal and interictal patterns in epileptic electroencephalography (EEG) data. The features extracted from the method of DMD clearly capture the phase transition of a specific frequency among the channels corresponding to the ictal state and the channel corresponding to the interictal state, such as direct current shift (DC-shift or ictal slow shifts) and high-frequency oscillation (HFO). By performing classification tests with Electrocorticography (ECoG) recordings of one patient measured at different timings, it is shown that the captured phenomenon is the unique pattern that occurs in the ictal onset zone of the patient. We eventually explain how advantageously the DMD captures some specific characteristics to distinguish the ictal state and the interictal state. The method presented in this study allows simultaneous interpretation of changes in the channel correlation and particular information for activity related to an epileptic seizure so that it can be applied to identification and prediction of the ictal state and analysis of the mechanism on its dynamics.
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Ryu, Tony, Wael H. Ali, and Pierre F. Lermusiaux. "Dynamic mode decomposition for underwater acoustics on autonomous platforms." Journal of the Acoustical Society of America 152, no. 4 (October 2022): A156. http://dx.doi.org/10.1121/10.0015873.
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A sufficiently accurate representation of the ocean physics is needed for reliable underwater acoustics forecasting. Ocean physics are often simulated by high-dimensional data-assimilative numerical ocean simulations. Due to operational constraints such as computing, memory, communication-bandwidth, etc., it is infeasible to run these numerical simulations onboard the platforms. In this work, we employ the onboard data-assimilative incremental, Low-Rank Dynamic Mode Decomposition (iLRDMD) for forecast transmission, onboard computation, and data assimilation. The iLRDMD is an adaptive reduced-order model (ROM) that uses proper orthogonal decomposition (POD) and DMD for compression and transmission of high-fidelity ocean forecast to communication-disadvantaged platforms. It provides reduced-order onboard forecasting and also enables efficient updates of the DMD model itself from inputs from remote centers. Finally, when sparse observations are made by the platform, the iLRDM uses Bayesian data assimilation to correct both the forecasts and the DMD model. We demonstrate the use of deterministic and data-assimilative iLRDMD forecasted ocean fields for underwater acoustics computations in real ocean examples. We further explore the use of our adaptive ROM for joint reduction of the ocean physics and acoustics fields.
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Hall, Brenton T., Ching-Shan Chou, and Jen-Ping Chen. "Using the Nonuniform Dynamic Mode Decomposition to Reduce the Storage Required for PDE Simulations." International Journal of Aerospace Engineering 2019 (January 3, 2019): 1–15. http://dx.doi.org/10.1155/2019/8291616.
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Partial Differential Equation simulations can produce large amounts of data. These datasets are very slow to transfer, for example, from an off-site supercomputer to a local research facility. There have been many model reduction techniques that have been proposed and utilized over the past three decades. Two of the most popular techniques are the Proper Orthogonal Decomposition and Dynamic Mode Decomposition. Nonuniform Dynamic Mode Decomposition (NU-DMD) is one of the newest techniques as it was introduced in 2015 by Guéniat et al. In this paper, the NU-DMD’s mathematics are explained in detail, and three versions of the NU-DMD’s algorithm are outlined. Furthermore, different numerical experiments were performed on the NU-DMD to ascertain its behavior with respect to errors, memory usage, and computational efficiency. It was shown that the NU-DMD could reduce an advection-diffusion simulation to 6.0075% of its original memory storage size. The NU-DMD was also applied to a computational fluid dynamics simulation of a NASA single-stage compressor rotor, which resulted in a reduced model of the simulation (using only three of the five simulation variables) that used only about 4.67% of the full simulation’s storage with an overall average percent error of 8.90%. It was concluded that the NU-DMD, if used appropriately, could be used to possibly reduce a model that uses 400 GB of memory to a model that uses as little as 18.67 GB with less than 9% error. Further conclusions were made about how to best implement the NU-DMD.
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Blubaugh, Frank. "Surrogate modeling in structural vibration problems with dynamic mode decomposition." Journal of the Acoustical Society of America 152, no. 4 (October 2022): A134. http://dx.doi.org/10.1121/10.0015794.
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Solving large-scale vibration problems presents a difficult challenge for complicated geometries that is only addressable through the use of large-scale compute resources. These problems not only require extensive compute time and cost, but are also expensive in engineering hours for modeling and analysis. While optimization techniques can help limit engineering time in the loop, the computational requirements of these models have made applying traditional optimization techniques to this class of problem untenable. Dynamic Mode Decomposition (DMD) is an approach that can help bridge this gap by building high-fidelity surrogate models allowing for the inline development of a surrogate model to dramatically reduce the computational time and complexity of a problem. This approach enables fast frequency sweeping while capturing the dominant dynamics of the model. DMD has historically been applied to other high-data volume models such as computational fluid dynamics, medical imaging and controls, as well as less structured problems including sociology and market behavior. This paper will cover the fundamentals of Dynamic Mode Decomposition, the extension of the theory to apply the technique to vibration problems, the demonstration of a potential workflow, and show how this technique performs on simple test cases illustrating performance speed ups for classic problems from existing literature.
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Nedzhibov, Gyurhan H. "Dynamic Mode Decomposition: A New Approach for Computing the DMD Modes and Eigenvalues." Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application 14, no. 1-2 (2022): 5–16. http://dx.doi.org/10.56082/annalsarscimath.2022.1-2.5.
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Nedzhibov, Gyurhan. "On Alternative Algorithms for Computing Dynamic Mode Decomposition." Computation 10, no. 12 (December 1, 2022): 210. http://dx.doi.org/10.3390/computation10120210.
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Dynamic mode decomposition (DMD) is a data-driven, modal decomposition technique that describes spatiotemporal features of high-dimensional dynamic data. The method is equation-free in the sense that it does not require knowledge of the underlying governing equations. The main purpose of this article is to introduce new alternatives to the currently accepted algorithm for calculating the dynamic mode decomposition. We present two new algorithms which are more economical from a computational point of view, which is an advantage when working with large data. With a few illustrative examples, we demonstrate the applicability of the introduced algorithms.
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Ye, Zhi-Xian, Yi-Yang Jiang, Ze-Nan Tian, Shao-Chang Mo, Yuan-Qi Fang, Jian-Feng Zou, and Yao Zheng. "Analysis of synthetic jet PIV experiment based on dynamic mode decomposition." International Journal of Modern Physics B 34, no. 14n16 (June 1, 2020): 2040103. http://dx.doi.org/10.1142/s0217979220401037.
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Synthetic Jet Actuator (SJA) works cyclically with the directionally transportation of fluid near the exit, and the paper presents the performance of a loudspeaker driven SJA in static flow field. Time-Resolved Particle Image Velocimetry (TR-PIV) system is used to measure the flow field characteristics near the SJA slot exit with input voltage changing, and the flow field snapshots obtained by TR-PIV are modality analyzed by Dynamic Mode Decomposition (DMD) method. The PIV experiments show that by varying input voltage at fixed oscillating frequency, the loudspeaker diaphragm vibration displacement is the parameter that affects the jet velocity and the performance of the SJA. The traveling vortex vanishes at high voltage due to the interaction between vortex structures and the synthetic main jet. In DMD method, the first three-order modes can characterize the main information of the original flow field with reverting the flow field snapshot sequence. It indicates that the DMD method is applicable in the SJA flow field research and the reduced order model can effectively simplify the analysis of flow field.
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Mendez, M. A., M. Balabane, and J. M. Buchlin. "Multi-scale proper orthogonal decomposition of complex fluid flows." Journal of Fluid Mechanics 870 (May 15, 2019): 988–1036. http://dx.doi.org/10.1017/jfm.2019.212.
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Data-driven decompositions are becoming essential tools in fluid dynamics, allowing for tracking the evolution of coherent patterns in large datasets, and for constructing low-order models of complex phenomena. In this work, we analyse the main limits of two popular decompositions, namely the proper orthogonal decomposition (POD) and the dynamic mode decomposition (DMD), and we propose a novel decomposition which allows for enhanced feature detection capabilities. This novel decomposition is referred to as multi-scale proper orthogonal decomposition (mPOD) and combines multi-resolution analysis (MRA) with a standard POD. Using MRA, the mPOD splits the correlation matrix into the contribution of different scales, retaining non-overlapping portions of the correlation spectra; using the standard POD, the mPOD extracts the optimal basis from each scale. After introducing a matrix factorization framework for data-driven decompositions, the MRA is formulated via one- and two-dimensional filter banks for the dataset and the correlation matrix respectively. The validation of the mPOD, and a comparison with the discrete Fourier transform (DFT), DMD and POD are provided in three test cases. These include a synthetic test case, a numerical simulation of a nonlinear advection–diffusion problem and an experimental dataset obtained by the time-resolved particle image velocimetry (TR-PIV) of an impinging gas jet. For each of these examples, the decompositions are compared in terms of convergence, feature detection capabilities and time–frequency localization.
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Fathi, Mojtaba, Ali Bakhshinejad, Ahmadreza Baghaie, and Roshan D’Souza. "Dynamic Denoising and Gappy Data Reconstruction Based on Dynamic Mode Decomposition and Discrete Cosine Transform." Applied Sciences 8, no. 9 (September 1, 2018): 1515. http://dx.doi.org/10.3390/app8091515.
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Dynamic Mode Decomposition (DMD) is a data-driven method to analyze the dynamics, first applied to fluid dynamics. It extracts modes and their corresponding eigenvalues, where the modes are spatial fields that identify coherent structures in the flow and the eigenvalues describe the temporal growth/decay rates and oscillation frequencies for each mode. The recently introduced compressed sensing DMD (csDMD) reduces computation times and also has the ability to deal with sub-sampled datasets. In this paper, we present a similar technique based on discrete cosine transform to reconstruct the fully-sampled dataset (as opposed to DMD modes as in csDMD) from sub-sampled noisy and gappy data using l 1 minimization. The proposed method was benchmarked against csDMD in terms of denoising and gap-filling using three datasets. The first was the 2-D time-resolved plot of a double gyre oscillator which has about nine oscillatory modes. The second dataset was derived from a Duffing oscillator. This dataset has several modes associated with complex eigenvalues which makes them oscillatory. The third dataset was taken from the 2-D simulation of a wake behind a cylinder at Re = 100 and was used for investigating the effect of changing various parameters on reconstruction error. The Duffing and 2-D wake datasets were tested in presence of noise and rectangular gaps. While the performance for the double-gyre dataset is comparable to csDMD, the proposed method performs substantially better (lower reconstruction error) for the dataset derived from the Duffing equation and also, the 2-D wake dataset according to the defined reconstruction error metrics.
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Cherubini, Stefania, Giovanni De Cillis, Onofrio Semeraro, Stefano Leonardi, and Pietro De Palma. "Data Driven Modal Decomposition of the Wake behind an NREL-5MW Wind Turbine." International Journal of Turbomachinery, Propulsion and Power 6, no. 4 (November 25, 2021): 44. http://dx.doi.org/10.3390/ijtpp6040044.
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The wake produced by a utility-scale wind turbine invested by a laminar, uniform inflow is analyzed by means of two different modal decompositions, the proper orthogonal decomposition (POD) and the dynamic mode decomposition (DMD), in its sparsity-promoting variant. The turbine considered is the NREL-5MW at tip-speed ratio λ=7 and a diameter-based Reynolds number of the order 108. The flow is simulated through large eddy simulation, where the forces exerted by the blades are modeled using the actuator line method, whereas tower and nacelle are modeled employing the immersed boundary method. The main flow structures identified by both modal decompositions are compared and some differences emerge that can be of great importance for the formulation of a reduced-order model. In particular, a high-frequency mode directly related to the tip vortices is found using both methods, but it is ranked differently. The other dominant modes are composed by large-scale low-frequency structures, but with different frequency content and spatial structure. The most energetic 200 POD modes account for ≈20% only of the flow kinetic energy. While using the same number of DMD modes, it is possible to reconstruct the flow field to within 80% accuracy. Despite the similarities between the set of modes, the comparison between these modal-decomposition techniques points out that an energy-based criterion such as that used in the POD may not be suitable for formulating a reduced-order model of wind turbine wakes, while the sparsity-promoting DMD appears able to perform well in reconstructing the flow field with only a few modes.
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Li, Li, Jingjing Luo, Yang Li, Lei Zhang, and Yuzhu Guo. "Phase Analysis of Event-Related Potentials Based on Dynamic Mode Decomposition." Mathematics 10, no. 23 (November 22, 2022): 4406. http://dx.doi.org/10.3390/math10234406.
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Real-time detection of event-related potentials (ERPs) and exploration of ERPs generation mechanisms are vital to practical application of brain–computer interfaces (BCI). Traditional methods for ERPs analysis often fall into time domain, time–frequency domain, or spatial domain. Methods which can reveal spatiotemporal interactions by simultaneously analyzing multi-channel EEG signals may provide new insights into ERP research and is highly desired. Additionally, although phase information has been investigated to describe the phase consistency of a certain frequency component across different ERP trials, it is of research significance to analyze the phase reorganization across different frequency components that constitute a single-trial ERP signal. To address these problems, dynamic mode decomposition (DMD) was applied to decompose multi-channel EEG into a series of spatial–temporal coherent DMD modes, and a new metric, called phase variance distribution (PVD) is proposed as an index of the phase reorganization of DMD modes during the ERP in a single trial. Based on the PVD, a new error-related potential (ErrP) detection method based on symmetric positive defined in Riemann manifold is proposed to demonstrate the significant PVD differences between correct and error trials. By including the phase reorganization index, the 10-fold cross-validation results of an ErrP detection task showed that the proposed method is 4.98%, 27.99% and 7.98% higher than the counterpart waveform-based ErrP detection method in the terms of weighted accuracy rate, precision and recall of the ErrP class, respectively. The resulting PVD curve shows that with the occurrence of ERP peaks, the phases of different frequency rhythms are getting to aligned and yields a significant smaller PVD. Since the DMD modes of different frequencies characterize spatiotemporal coherence of multi-channel EEG at different functional regions, the new phase reorganization index, PVD, may indicate the instantaneous phase alignment of different functional networks and sheds light on a new interpretation of ERP generation mechanism.
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Abdurakipov, Sergey, Vladimir Dulin, and Dmitriy Markovich. "Experimental Investigation of Coherent Structure Dynamics in a Submerged Forced Jet." Siberian Journal of Physics 8, no. 1 (March 1, 2013): 56–64. http://dx.doi.org/10.54362/1818-7919-2013-8-1-56-64.
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The present work investigates the dynamics of coherent structures, including their scales and intensity, in an initial region of a submerged round forced jet by a Particle Image Velocimetry (PIV) technique for measurements of instantaneous velocity fields and statistical analysis tool Dynamic Mode Decomposition (DMD). The PIV measurements were carried out with 1,1 kHz acquisition rate. Application of DMD to the measured set of the velocity fields provided information about dominant frequencies, contained in DMD spectrum, of velocity fluctuations in different flow regions and about scales of the corresponding spatial coherent structures, contained in DMD modes. Additional calculations of time-spectra from turbulent fluctuations showed good agreement between frequencies of the main harmonics and characteristic frequencies of the dominant dynamic modes. Superposition of relevant DMD modes approximately described nonlinear interaction of coherent structures: vortex formation, their quasi-periodic pairing with modulation amplitude of generated harmonics
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Babalola, Oluwaseyi Paul, and Vipin Balyan. "WiFi Fingerprinting Indoor Localization Based on Dynamic Mode Decomposition Feature Selection with Hidden Markov Model." Sensors 21, no. 20 (October 13, 2021): 6778. http://dx.doi.org/10.3390/s21206778.
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Over the years, WiFi received signal strength indicator (RSSI) measurements have been widely implemented for determining the location of a user’s position in an indoor environment, where the GPS signal might not be received. This method utilizes a huge RSSI dataset collected from numerous access points (APs). The WiFi RSSI measurements are nonlinear with distance and are largely influenced by interference in the indoor environment. Therefore, machine learning (ML) techniques such as a hidden Markov model (HMM) are generally utilized to efficiently identify a trend of RSSI values, which corresponds to locations around a region of interest. Similar to other ML tools, the performance and computing cost of the HMM are dependent on the feature dimension since a large quantity of RSSI measurements are required for the learning process. Hence, this article introduces a feature extraction method based on dynamic mode decomposition (DMD) for the HMM to effectively model WiFi fingerprint indoor localization. The DMD is adopted since it decomposes RSSIs to meaningful spatial and temporal forms over a given time. Here, the mode forms are analytically reconstructed to produce low-dimensional feature vectors, which are used with the HMM. The localization performance of the proposed HMM-DMD is compared with other well-known ML algorithms for WiFi fingerprinting localization using simulations. The results show that the HMM-DMD algorithm yields a significant localization performance improvement, accuracy, and reasonable processing time in comparison with the state-of-the-art algorithms.
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Alcayaga, Leonardo, Gunner Chr Larsen, Mark Kelly, and Jakob Mann. "Identification of large-scale atmospheric structures under different stability conditions using Dynamic Mode Decomposition." Journal of Physics: Conference Series 2265, no. 2 (May 1, 2022): 022006. http://dx.doi.org/10.1088/1742-6596/2265/2/022006.
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Abstract We investigate the characteristic of large-scale coherent motions over a large horizontal domain using the Dynamic Mode Decomposition (DMD) spectral analysis algorithm applied on measurements from two long-range pulsed lidars. We show the results and advantages of this methodology on six cases representative of three thermal stratification conditions at two heights relevant for wind energy: near-neutral, unstable and stable stratification at 50m and 200m above ground level. For these cases the DMD algorithm show three types of structures: streaks near the surface for near-neutral for neutral stratification, large-scale convective rolls for the unstable cases and sheet-like rotational patches for stable conditions. The DMD algorithm also shows the stationary effects of the terrain on the flow at 50m above ground level, within the atmospheric surface layer. The possibility of isolating terrain effects from coherent motions makes DMD attractive for studying complex atmospheric flow phenomena as well as to have more realistic input for wind farm flow simulations.
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Assoum, Hassan H., Jana Hamdi, Marwan Alkheir, Kamel Abed Meraim, Anas Sakout, Bachar Obeid, and Mouhammad El Hassan. "Tomographic Particle Image Velocimetry and Dynamic Mode Decomposition (DMD) in a Rectangular Impinging Jet: Vortex Dynamics and Acoustic Generation." Fluids 6, no. 12 (November 27, 2021): 429. http://dx.doi.org/10.3390/fluids6120429.
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Impinging jets are encountered in ventilation systems and many other industrial applications. Their flows are three-dimensional, time-dependent, and turbulent. These jets can generate a high level of noise and often present a source of discomfort in closed areas. In order to reduce and control such mechanisms, one should investigate the flow dynamics that generate the acoustic field. The purpose of this study is to investigate the flow dynamics and, more specifically, the coherent structures involved in the acoustic generation of these jets. Model reduction techniques are commonly used to study the underlying mechanisms by decomposing the flow into coherent structures. The dynamic mode decomposition (DMD) is an equation-free method that relies only on the system’s data taken either through experiments or through numerical simulations. In this paper, the DMD technique is applied, and the spatial modes and their frequencies are presented. The temporal content of the DMD’s modes is then correlated with the acoustic signal. The flow is generated by a rectangular jet impinging on a slotted plate (for a Reynolds number Re = 4458) and its kinematic field is obtained via the tomographic particle image velocimetry technique (TPIV). The findings of this research highlight the coherent structures signature in the DMD’s spectral content and show the cross correlations between the DMD’s modes and the acoustic field.
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Yu, Paulo, and Vibhav Durgesh. "Modal Decomposition Techniques: Application in Coherent Structures for a Saccular Aneurysm Model." Fluids 7, no. 5 (May 9, 2022): 165. http://dx.doi.org/10.3390/fluids7050165.
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Aneurysms are localized expansions of blood vessels which can be fatal upon rupture. Studies have shown that aneurysm flows exhibit complex flow phenomena which consist of single or multiple vortical structures that move within the flow cycle. Understanding the complex flow behaviors of aneurysms remain challenging. Thus, the goal of this study is to quantify the flow behavior and extract physical insights into aneurysm flows using advance data decomposition methods, Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD). The velocity field data were obtained by performing 2D Particle Image Velocimetry (2D PIV) on the mid-plane of an idealized, rigid, saccular aneurysm model. The input flow conditions were set to Rep=50 and 150 for a fixed α=2 using a precisely controlled piston pump system. POD was used to quantify the spatial features of the flows, while DMD was used to obtain insight on the dynamics. The results obtained from POD and DMD showed the capability of both methods to quantify the flow field, with the modes obtained providing different insights into the flow evolution in the aneurysm. The curve-fitting step of the POD time-varying coefficients, and the appropriate selection of DMD modes based on their energy contribution, allowed the mathematical flow models from POD and DMD to reconstruct flow fields at any given time step. This can be used for validation of numerical or computational data.
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Ngo, Thien-Thu, VanDung Nguyen, Xuan-Qui Pham, Md-Alamgir Hossain, and Eui-Nam Huh. "Motion Saliency Detection for Surveillance Systems Using Streaming Dynamic Mode Decomposition." Symmetry 12, no. 9 (August 21, 2020): 1397. http://dx.doi.org/10.3390/sym12091397.
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Intelligent surveillance systems enable secured visibility features in the smart city era. One of the major models for pre-processing in intelligent surveillance systems is known as saliency detection, which provides facilities for multiple tasks such as object detection, object segmentation, video coding, image re-targeting, image-quality assessment, and image compression. Traditional models focus on improving detection accuracy at the cost of high complexity. However, these models are computationally expensive for real-world systems. To cope with this issue, we propose a fast-motion saliency method for surveillance systems under various background conditions. Our method is derived from streaming dynamic mode decomposition (s-DMD), which is a powerful tool in data science. First, DMD computes a set of modes in a streaming manner to derive spatial–temporal features, and a raw saliency map is generated from the sparse reconstruction process. Second, the final saliency map is refined using a difference-of-Gaussians filter in the frequency domain. The effectiveness of the proposed method is validated on a standard benchmark dataset. The experimental results show that the proposed method achieves competitive accuracy with lower complexity than state-of-the-art methods, which satisfies requirements in real-time applications.
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Wüstner, Daniel. "Dynamic Mode Decomposition of Fluorescence Loss in Photobleaching Microscopy Data for Model-Free Analysis of Protein Transport and Aggregation in Living Cells." Sensors 22, no. 13 (June 23, 2022): 4731. http://dx.doi.org/10.3390/s22134731.
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The phase separation and aggregation of proteins are hallmarks of many neurodegenerative diseases. These processes can be studied in living cells using fluorescent protein constructs and quantitative live-cell imaging techniques, such as fluorescence recovery after photobleaching (FRAP) or the related fluorescence loss in photobleaching (FLIP). While the acquisition of FLIP images is straightforward on most commercial confocal microscope systems, the analysis and computational modeling of such data is challenging. Here, a novel model-free method is presented, which resolves complex spatiotemporal fluorescence-loss kinetics based on dynamic-mode decomposition (DMD) of FLIP live-cell image sequences. It is shown that the DMD of synthetic and experimental FLIP image series (DMD-FLIP) allows for the unequivocal discrimination of subcellular compartments, such as nuclei, cytoplasm, and protein condensates based on their differing transport and therefore fluorescence loss kinetics. By decomposing fluorescence-loss kinetics into distinct dynamic modes, DMD-FLIP will enable researchers to study protein dynamics at each time scale individually. Furthermore, it is shown that DMD-FLIP is very efficient in denoising confocal time series data. Thus, DMD-FLIP is an easy-to-use method for the model-free detection of barriers to protein diffusion, of phase-separated protein assemblies, and of insoluble protein aggregates. It should, therefore, find wide application in the analysis of protein transport and aggregation, in particular in relation to neurodegenerative diseases and the formation of protein condensates in living cells.
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Li, Cruz Y., Zengshun Chen, Tim K. T. Tse, Asiri U. Weerasuriya, Xuelin Zhang, Yunfei Fu, and Xisheng Lin. "A parametric and feasibility study for data sampling of the dynamic mode decomposition: range, resolution, and universal convergence states." Nonlinear Dynamics 107, no. 4 (January 12, 2022): 3683–707. http://dx.doi.org/10.1007/s11071-021-07167-8.
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AbstractScientific research and engineering practice often require the modeling and decomposition of nonlinear systems. The dynamic mode decomposition (DMD) is a novel Koopman-based technique that effectively dissects high-dimensional nonlinear systems into periodically distinct constituents on reduced-order subspaces. As a novel mathematical hatchling, the DMD bears vast potentials yet an equal degree of unknown. This effort investigates the nuances of DMD sampling with an engineering-oriented emphasis. It aimed at elucidating how sampling range and resolution affect the convergence of DMD modes. We employed the most classical nonlinear system in fluid mechanics as the test subject—the turbulent free-shear flow over a prism—for optimal pertinency. We numerically simulated the flow by the dynamic-stress Large-Eddies Simulation with Near-Wall Resolution. With the large-quantity, high-fidelity data, we parametrized and identified four global convergence states: Initialization, Transition, Stabilization, and Divergence with increasing sampling range. Results showed that Stabilization is the optimal state for modal convergence, in which DMD output becomes independent of the sampling range. The Initialization state also yields sufficient accuracy for most system reconstruction tasks. Moreover, defying popular beliefs, over-sampling causes algorithmic instability: as the temporal dimension, n, approaches and transcends the spatial dimension, m (i.e., m < n), the output diverges and becomes meaningless. Additionally, the convergence of the sampling resolution depends on the mode-specific dynamics, such that the resolution of 15 frames per cycle for target activities is suggested for most engineering implementations. Finally, a bi-parametric study revealed that the convergence of the sampling range and resolution are mutually independent.
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Muscari, C., P. Schito, A. Viré, A. Zasso, D. van der Hoek, and JW van Wingerden. "Physics informed DMD for periodic Dynamic Induction Control of Wind Farms." Journal of Physics: Conference Series 2265, no. 2 (May 1, 2022): 022057. http://dx.doi.org/10.1088/1742-6596/2265/2/022057.
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Abstract Dynamic Induction Control (DIC) is a novel, exciting branch of Wind Farm Control. It makes use of time-varying control inputs to increase wake mixing, and consequently improve the velocity recovery rate of the flow and the power production of downstream turbines. The Pulse and the Helix are two promising DIC strategies that rely on sinusoidal excitations of the collective pitch and individual pitch of the blades, respectively. While their beneficial effects are evident in simulations and wind tunnel tests, we do not yet fully understand the physics behind them. We perform a systematic analysis of the dynamics of pulsed and helicoidal wakes by applying a data-driven approach to the analysis of data coming from Large Eddy Simulations (LES). Specifically, Dynamic Mode Decomposition (DMD) is used to extract coherent patterns from high-dimensional flow data. The periodicity of the excitation is exploited by adding a novel physics informed step to the algorithm. We then analyze the power spectral density of the resulting DMD modes as a function of the Strouhal number for different pitch excitation frequencies and amplitudes. Finally, we show the evolution in time and space of the dominant modes and comment on the recognizable patterns. By focusing on the modes that contribute the most to the flow dynamics, we gather insight on what causes the increased wake recovery rate in DIC techniques. This knowledge can then be used for the optimization of the signal parameters in complex layouts and conditions.
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Pascarella, Gaetano, Ioannis Kokkinakis, and Marco Fossati. "Analysis of Transition for a Flow in a Channel via Reduced Basis Methods." Fluids 4, no. 4 (December 5, 2019): 202. http://dx.doi.org/10.3390/fluids4040202.
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The study of the flow mechanisms leading to transition in a planar channel flow is investigated by means of a reduced basis method known as Dynamic Mode Decomposition (DMD). The problem of identification of the most relevant DMD modes is addressed in terms of the ability to (i) provide a fairly accurate reconstruction of the flow field, and (ii) match the most relevant flow structures at the beginning of the transition region. A comparative study between a natural method of selection based on the energetic content of the modes and a new one based on the temporal dynamics of the modes is here presented.
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Li, Yi-bin, Chang-hong He, and Jian-zhong Li. "Study on Flow Characteristics in Volute of Centrifugal Pump Based on Dynamic Mode Decomposition." Mathematical Problems in Engineering 2019 (April 16, 2019): 1–15. http://dx.doi.org/10.1155/2019/2567659.
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To investigate the unsteady flow characteristics and their influence mechanism in the volute of centrifugal pump, the Reynolds time-averaged N-S equation, RNG k-ε turbulence model, and structured grid technique are used to numerically analyze the transient flow-field characteristics inside the centrifugal pump volute. Based on the quantified parameters of flow field in the volute of centrifugal pump, the velocity mode contours and oscillation characteristics of the mid-span section of the volute of centrifugal pump are obtained by dynamic mode decomposition (DMD) for the nominal and low flow-rate condition. The research shows that the first-order average flow mode extracted by DMD is the dominant flow structure in the flow field of the volute. The second-order and third-order modes are the most important oscillation modes causing unsteady flow in the volute, and the characteristic frequency of the two modes is consistent with the blade passing frequency and the 2x blade passing frequency obtained by the fast Fourier transform (FFT). By reconstructing the internal flow field of the volute with the blade passing frequency for the nominal flow-rate condition, the periodic variation of the unsteady flow structure in the volute under this frequency is visually reproduced, which provides some ideas for the study of the unsteady structure in the internal flow field of centrifugal pumps.
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Magionesi, Francesca, Giulio Dubbioso, Roberto Muscari, and Andrea Di Mascio. "Modal analysis of the wake past a marine propeller." Journal of Fluid Mechanics 855 (September 19, 2018): 469–502. http://dx.doi.org/10.1017/jfm.2018.631.
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Modal decomposition techniques are used to analyse the wake field past a marine propeller achieved by previous numerical simulations (Muscari et al. Comput. Fluids, vol. 73, 2013, pp. 65–79). In particular, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are used to identify the most energetic modes and those that play a dominant role in the inception of the destabilization mechanisms. Two different operating conditions, representative of light and high loading conditions, are considered. The analysis shows a strong dependence of temporal and spatial scales of the process on the propeller loading and correlates the spatial shape of the modes and the temporal scales with the evolution and destabilization mechanisms of the wake past the propeller. At light loading condition, due to the stable evolution of the wake, both POD and DMD describe the flow field by the non-interacting evolution of the tip and hub vortex. The flow is mainly associated with the ordered convection of the tip vortex and the corresponding dominant modes, identified by both decompositions, are characterized by spatial wavelengths and frequencies related to the blade passing frequency and its multiples, whereas the dynamic of the hub vortex has a negligible contribution. At high loading condition, POD and DMD identify a marked separation of the flow field close to the propeller and in the far field, as a consequence of wake breakdown. The tonal modes are prevalent only near to the propeller, where the flow is stable; on the contrary, in the transition region a number of spatial and temporal scales appear. In particular, the phenomenon of destabilization of the wake, originated by the coupling of consecutive tip vortices, and the mechanisms of hub–tip vortex interaction and wake meandering are identified by both POD and DMD.
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Noack, Bernd R. "From snapshots to modal expansions – bridging low residuals and pure frequencies." Journal of Fluid Mechanics 802 (August 1, 2016): 1–4. http://dx.doi.org/10.1017/jfm.2016.416.
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Data-driven low-order modelling has been enjoying rapid advances in fluid mechanics. Arguably, Sirovich (Q. Appl. Maths, vol. XLV, 1987, pp. 561–571) started these developments with snapshot proper orthogonal decomposition, a particularly simple method. The resulting reduced-order models provide valuable insights into flow physics, allow inexpensive explorations of dynamics and operating conditions, and enable model-based control design. A winning argument for proper orthogonal decomposition (POD) is the optimality property, i.e. the guarantee of the least residual for a given number of modes. The price is unpleasant frequency mixing in the modes which complicates their physical interpretation. In contrast, temporal Fourier modes and dynamic mode decomposition (DMD) provide pure frequency dynamics but lose the orthonormality and optimality property of POD. Sieber et al. (J. Fluid Mech., vol. 792, 2016, pp. 798–828) bridge the least residual and pure frequency behaviour with an ingenious interpolation, called spectral proper orthogonal decomposition (SPOD). This article puts the achievement of the TU Berlin authors in perspective, illustrating the potential of SPOD and the challenges ahead.
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Wang, Liang, Liying Li, and Song Fu. "A comparative study of DES type methods for mild flow separation prediction on a NACA0015 airfoil." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 11 (November 6, 2017): 2528–43. http://dx.doi.org/10.1108/hff-07-2016-0263.
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Purpose The purpose of this paper is to numerically investigate the mildly separated flow phenomena on a near-stall NACA0015 airfoil, by using Detached-Eddy Simulation (DES) type methods. It includes a comparison of different choices of underlying Reynolds-averaged Navier–Stokes model as well as subgrid-scale stress model in Large-Eddy simulation mode. Design/methodology/approach The unsteady flow phenomena are simulated by using delayed DES (DDES) and improved DDES (IDDES) methods, with an in-house computational fluid dynamics solver. Characteristic frequencies in different flow regions are extracted using fast Fourier transform. Dynamic mode decomposition (DMD) method is applied to uncover the critical dynamic modes. Findings Among all the DES type methods investigated in this paper, only the Spalart–Allmaras-based IDDES captures the separation point as measured in the experiments. The classical vortex-shedding and the shear-layer flapping modes for airfoil flows with shallow separation are also found from the IDDES results by using DMD. Originality/value The value of this paper lies in the assessment of five different DES-type models through the detailed investigation of the Reynolds stresses as well as the separation and reattachment.
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Li, Cruz Y., Zengshun Chen, Tim K. T. Tse, Asiri Umenga Weerasuriya, Xuelin Zhang, Yunfei Fu, and Xisheng Lin. "A parametric and feasibility study for data sampling of the dynamic mode decomposition: Spectral insights and further explorations." Physics of Fluids 34, no. 3 (March 2022): 035102. http://dx.doi.org/10.1063/5.0082640.
Full textAbstract:
The present work extends the parametric investigation on the sampling nuances of dynamic mode decomposition (DMD) under Koopman analysis. Through turbulent wakes, the study corroborated the generality of universal convergence states for all DMD implementations. It discovered implications of sampling range and resolution—determinants of spectral discretization by discrete bins and the highest resolved frequency range, respectively. The work reaffirmed the necessity of the convergence state for sampling independence, too. Results also suggested that the observables derived from the same flow may contain dynamically distinct information, thus altering the DMD output. Surface pressure and vortex fields are optimal for characterizing the structure and the flow field, respectively. Pressure, velocity magnitude, and turbulence kinetic energy also suffice for general applications, but Reynolds stresses and velocity components shall be avoided. Mean-subtraction is recommended for the best approximations of Koopman eigen tuples. Furthermore, the parametric investigation on truncation discovered some low-energy states that dictate a system's temporal integrity. The best practice for order reduction is to avoid truncation and employ dominant mode selection on a full-state subspace, though large-degree truncation supports fair data reconstruction with low computational cost. Finally, this work demonstrated synthetic noise resulting from pre-decomposition interpolation. In unavoidable interpolations to increase the spatial dimension n, high-order schemes are recommended for better retention of original dynamics. Finally, the observations herein, derived from inhomogeneous anisotropic turbulence, offer constructive references for DMD on fluid systems, if not also for others beyond fluid mechanics.
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Shi, Zheyu, Kaiwen Zhou, Chen Qin, and Xin Wen. "Experimental Study of Dynamical Airfoil and Aerodynamic Prediction." Actuators 11, no. 2 (February 2, 2022): 46. http://dx.doi.org/10.3390/act11020046.
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Dynamic stall is a critical limiting factor for airfoil aerodynamics and a challenging problem for active flow control. In this experimental study, dynamic stall was measured by high-frequency surface pressure tapes and pressure-sensitive paint (PSP). The influence of the oscillation frequency was examined. Dynamic mode decomposition (DMD) with time-delay embedding was proposed to predict the pressure field on the oscillating airfoil based on scattered pressure measurements. DMD with time-delay embedding was able to reconstruct and predict the dynamic stall based on scattered measurements with much higher accuracy than standard DMD. The reconstruction accuracy of this method increased with the number of delay steps, but this also prolonged the computation time. In summary, using the Koopman operator obtained by DMD with time-delay embedding, the future dynamic pressure on an oscillating airfoil can be accurately predicted. This method provides powerful support for active flow control of dynamic stall.
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Bao, Anqi, Eduardo Gildin, Abhinav Narasingam, and Joseph S. Kwon. "Data-Driven Model Reduction for Coupled Flow and Geomechanics Based on DMD Methods." Fluids 4, no. 3 (July 19, 2019): 138. http://dx.doi.org/10.3390/fluids4030138.
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Learning reservoir flow dynamics is of primary importance in creating robust predictive models for reservoir management including hydraulic fracturing processes. Physics-based models are to a certain extent exact, but they entail heavy computational infrastructure for simulating a wide variety of parameters and production scenarios. Reduced-order models offer computational advantages without compromising solution accuracy, especially if they can assimilate large volumes of production data without having to reconstruct the original model (data-driven models). Dynamic mode decomposition (DMD) entails the extraction of relevant spatial structure (modes) based on data (snapshots) that can be used to predict the behavior of reservoir fluid flow in porous media. In this paper, we will further enhance the application of the DMD, by introducing sparse DMD and local DMD. The former is particularly useful when there is a limited number of sparse measurements as in the case of reservoir simulation, and the latter can improve the accuracy of developed DMD models when the process dynamics show a moving boundary behavior like hydraulic fracturing. For demonstration purposes, we first show the methodology applied to (flow only) single- and two-phase reservoir models using the SPE10 benchmark. Both online and offline processes will be used for evaluation. We observe that we only require a few DMD modes, which are determined by the sparse DMD structure, to capture the behavior of the reservoir models. Then, we applied the local DMDc for creating a proxy for application in a hydraulic fracturing process. We also assessed the trade-offs between problem size and computational time for each reservoir model. The novelty of our method is the application of sparse DMD and local DMDc, which is a data-driven technique for fast and accurate simulations.
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Corrochano, Adrián, Donnatella Xavier, Philipp Schlatter, Ricardo Vinuesa, and Soledad Le Clainche. "Flow Structures on a Planar Food and Drug Administration (FDA) Nozzle at Low and Intermediate Reynolds Number." Fluids 6, no. 1 (December 24, 2020): 4. http://dx.doi.org/10.3390/fluids6010004.
Full textAbstract:
In this paper, we present a general description of the flow structures inside a two-dimensional Food and Drug Administration (FDA) nozzle. To this aim, we have performed numerical simulations using the numerical code Nek5000. The topology patters of the solution obtained, identify four different flow regimes when the flow is steady, where the symmetry of the flow breaks down. An additional case has been studied at higher Reynolds number, when the flow is unsteady, finding a vortex street distributed along the expansion pipe of the geometry. Linear stability analysis identifies the evolution of two steady and two unsteady modes. The results obtained have been connected with the changes in the topology of the flow. Finally, higher-order dynamic mode decomposition has been applied to identify the main flow structures in the unsteady flow inside the FDA nozzle. The highest-amplitude dynamic mode decomposition (DMD) modes identified by the method model the vortex street in the expansion of the geometry.