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A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
This approach is an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes, and if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation.
Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis
- C. W. Gear, J. Hyman, Panagiotis G Kevrekidid, I. Kevrekidis, O. Runborg, C. Theodoropoulos
- Computer Science
- 1 December 2003
A framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/stochastic) level of description to perform modeling tasks at a coarse (macroscopic, systems) level, and can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form is presented.
Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum
We show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters, one containing the "stiff," or fast,…
Low‐dimensional models for complex geometry flows: Application to grooved channels and circular cylinders
Two‐dimensional unsteady flows in complex geometries that are characterized by simple (low‐dimensional) dynamical behavior are considered. Detailed spectral element simulations are performed, and the…
A kernel-based method for data-driven koopman spectral analysis
A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high-dimensional state spaces is presented, using a set of scalar observables that are defined implicitly by the feature map associated with a user-defined kernel function.
Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck Operators
A diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian is presented.
Inherent noise can facilitate coherence in collective swarm motion
A coarse-grained approach to the study of directional switching in a self-propelled particle model assuming an underlying one-dimensional Fokker–Planck equation for the mean velocity of the particles is adopted, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group.
Diffusion maps, spectral clustering and reaction coordinates of dynamical systems
Bistability and Oscillations in the Huang-Ferrell Model of MAPK Signaling
A methodology for the statistical analysis of mechanistic signaling models based on the combination of random parameter search and continuation algorithms is developed and it is argued that this type of analysis should accompany nonlinear multiparameter studies of stationary as well as transient features in network dynamics.
Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator.
An iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network and can effectively and efficiently adapt the trainable Dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori.