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Numerical Methods for ODEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Dynamic mode decomposition (DMD) approximates the Koopman operator with a best-fit linear model; it can be used, for example, to identify low-order dynamics of a system. Specifically, (208.3) is approximated by xk+1≈Axk where A is a matrix. The value of A is found as follows: given data {xk}k=1m from (208.2), the matrix X=x1x2…xm−1 and its time-shifted version XS=x2x3…xm are created. Then, using the model XS≈AX, the value of A is given by A=argminBXS−BXF=XSX†
The performance of different unsteady Reynolds-averaged Navier-Stokes models mined by Fuzzy C-means algorithm with Fourier coefficient-based distance on the constructed vortex structures around a single building
Published in Journal of Asian Architecture and Building Engineering, 2023
Fangli Du, Huiyuan Shen, Zhenjun Ma, Kehua Li, Kunrang Liu
Data analysis is an efficient method that can mine the characteristics of flow structure, such as the Dynamic Mode Decomposition (DMD) method (Li et al. 2021; Li, Tse, and Hu 2020). The DMD method is one of the tools with great potential. The transient changing information of the large-scale vortices structure characterized by transient ensemble mean velocity and the Q criterion value could be stored in the transient database easily. The clustering analysis is one of the commonly used data partitioning technologies particularly for assigning data objects into clusters/groups so that the objects in one group are similar to each other while different from the other groups. Fuzzy C-means (FCM) algorithm shows the higher segmentation efficiency than the other partitioning methodologies with fuzzy local information. However, FCM algorithm could not be directly used to analyze the turbulence fluctuations in the time sequence because the vortices in the turbulence are developing and transporting with the time/length (Christopher et al. 2020), only for the turbulence with statistical magnitude and magnitude at one time-section (Fan et al. 2019; Xu et al. 2020). The Fourier coefficient-based distance was successfully used to analyze the sequence databases (Agrawal, Faloutsos, and Swami 1993).
Dynamic mode decomposition type algorithms for modeling and predicting queue lengths at signalized intersections with short lookback
Published in Journal of Intelligent Transportation Systems, 2023
Kazi Redwan Shabab, Shakib Mustavee, Shaurya Agarwal, Mohamed H. Zaki, Sajal K. Das
Uncovering complex traffic dynamics from high-fidelity data requires “interpretable” data-driven techniques. Dynamic mode decomposition (DMD) is a purely data-driven technique that estimates a locally linear representation of complex nonlinear dynamics. The critical point about the method is that it does not require any prior information about the system or its internal physics to capture its dynamics. Hence, it is comparable to gray box models of system identification. DMD and related algorithms provide approximate system identification, unlike purely data-driven statistical models, and machine learning algorithms. The identification of complex dynamical systems as approximate linear dynamics has several benefits. Among them is the simplicity in understanding the system, short-term prediction, pattern identification, and the applicability of linear control algorithms. There have been a few studies attempting to utilize DMD-based algorithms for applications in ITS. DMD-based algorithm was used in image-based traffic flow visualization (Schmid, 2011), short-term traffic flow prediction (Yu et al., 2021), highway traffic dynamics analysis (X. Wang & Sun, 2023), and estimation of traffic mobility pattern (Li & Yang, 2022).
Dynamic mode decomposition of a direct numerical simulation of a turbulent premixed planar jet flame: convergence of the modes
Published in Combustion Theory and Modelling, 2018
Temistocle Grenga, Jonathan F. MacArt, Michael E. Mueller
Dynamic Mode Decomposition (DMD) is a powerful method for analysing the dynamics of nonlinear systems using data generated either computationally or experimentally. The method, first developed by Schmid [1], is based on Koopman modes and identifies the low-order dynamics describing the flow field that are actually governed by the infinite-dimensional Navier–Stokes equations. The DMD modes are spatial fields representing coherent structures in the flow associated with a distinct oscillatory frequency. Later, Schmid and collaborators published several papers [2–4] about the use of DMD to study nonlinear fluid dynamics through the analysis of approximated linear systems.