Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis

By C. Gear, James M. Hyman, Panagiotis G Kevrekidid, Ioannis G. Kevrekidis, Olof Runborg, Constantinos Theodoropoulos. 2003

832 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistical and Nonlinear Physics, Numerical Methods & Scientific Computing, Computational Biology & Bioinformatics

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Citations in the corpus

Nonintrusive and Structure Preserving Multiscale Integration of Stiff ODEs, SDEs, and Hamiltonian Systems with Hidden Slow Dynamics via Flow Averaging (2009, methodology): “Two classes of numerical methods have been built on this observation: The equation free method [52, 51] and the Heterogeneous Multiscale Method [30, 32, 29, 5].”
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control (2015, methodology): “Thus, the SINDy algorithm is an equation-free method [52] to identify a dynamical system from data.”
Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control (2021, background): “A wide range of data-driven model discovery methods exist, including equationfree modelling [1], normal form identification [2–4], nonlinear Laplacian spectral analysis [5], Koopman analysis [6,7] and dynamic mode decomposition (DMD) [8–10], symbolic regression [11–15], sparse regression [16,17], Ga”
Data-driven discovery of coordinates and governing equations (2019, background): “Numerous recent approaches leverage machine learning to identify dynamical systems models from data [19–26], with many using neural networks to model time-series data [18, 27–38].”
Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics (2019, methodology): “ad to wrong results in HMM [34]. Another well-known multiscale methodology is the equation-free approach (EFA). EFA is intended to be used when a macroscopic description is unavailable in closed form [35]. In EFA, microscopic simulations are performed only in small spatial domains over a short time ”
Principles of Multiscale Modeling (2012, background): “Also important is the work of Mori and Zwanzig which provides a general formalism for eliminating degrees of freedom in a system [35, 51].”
Modal Analysis of Fluid Flows: An Overview (2017, methodology): “In many contexts, Koopman analysis provides an equation-free method [139] to extract coherent structures and dynamics from data measurements of a complex system; these coherent structures are related to POD modes in fluids [1].”
A Micro-Macro Parareal Algorithm: Application to Singularly Perturbed Ordinary Differential Equations (2012, background): “The coarse propagator required in Algorithms 1 and 2 can be replaced by a coarse projective integration approach [22, 23].”
Data Driven Governing Equations Approximation Using Deep Neural Networks (2018, methodology): “Earlier efforts include methods using symbolic regression ([5, 43]), equation-free modeling [24], heterogeneous multiscale method (HMM) ([15]), artificial neural networks ([19]), nonlinear regression ([50]), empirical dynamic modeling ([46, 53]), nonlinear Laplacian spectral analysis ([18]), automat”
Analysis of the heterogeneous multiscale method for elliptic homogenization problems (2004, methodology): “Other numerical methods that use local microscale solvers to help extract macroscale behavior are found in [26, 27].”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “In addition, there are numerous exciting directions in equation-free modeling (Kevrekidis et al., 2003), including the Perron-Frobenius operator (Froyland and Padberg, 2009), cluster reduced-order models based on probabilistic transition between various system behaviors (Kaiser et al., 2014), and…”
The heterogeneous multiscale method* (2012, methodology): “Other notable examples include the extended multigrid method and the equation-free approach (Brandt 2002, Kevrekidis et al. 2003).”