A dynamic mode decomposition approach for large and arbitrarily sampled systems

Abstract

A Lagrangian subgrid-scale model with dynamic estimation of Lagrangian time scale for large eddy simulation of complex flows Phys. Fluids 24, 085101 (2012); 10.1063/1.4737656The effects of non-normality and nonlinearity of the Navier-Stokes operator on the dynamics of a large laminar separation bubble Detection of coherent structures is of crucial importance for understanding the dynamics of a fluid flow. In this regard, the recently introduced Dynamic Mode Decomposition (DMD) has raised an increasing interest in the community. It allows to efficiently determine the dominant spatial modes, and their associated growth rate and frequency in time, responsible for describing the time-evolution of an observation of the physical system at hand. However, the underlying algorithm requires uniformly sampled and time-resolved data, which may limit its usability in practical situations. Further, the computational cost associated with the DMD analysis of a large dataset is high, both in terms of central processing unit and memory. In this contribution, we present an alternative algorithm to achieve this decomposition, overcoming the above-mentioned limitations. A synthetic case, a two-dimensional restriction of an experimental flow over an open cavity, and a large-scale three-dimensional simulation, provide examples to illustrate the method. C 2015 AIP Publishing LLC.

Methods for data-driven multiscale model discovery for materials

“…DMD was originally designed for data collected at regularly spaced time steps. However, recent advances enable both sparse spatial [61,62] and temporal [63] data collection, as well as irregularly spaced collection times [64].…”

“…A remarkable feature of the DMD algorithm is its modularity for mathematical enhancements. Specifically, the DMD algorithm can be engineered to exploit sparse sampling [61,62], it can be modified to handle inputs and actuation [71], it can be used to more accurately approximate the Koopman operator when using judiciously chosen functions of the state-space [72], it can be easily made computationally scalable [73], and it can easily decompose data into multiscale temporal features in order to produce a multi-resolution DMD (mrDMD) decomposition [74]. Few mathematical architectures are capable of seamlessly integrating such diverse modifications of the dynamical system.…”

On linear and nonlinear aspects of dynamic mode decomposition

“…Their low‐storage method for performing DMD can be updated inexpensively as new data become available. The problem of modal decomposition of large and arbitrarily sampled systems is addressed by Guéniat et al . Their method essentially formulates the problem in an optimization setting and decouples the estimation of the temporal description from the spatial description.…”

“…The dynamic mode decomposition (DMD) is a recently devised method for the search of a small number of basis vectors (dynamic modes) able to describe the total fluid state [1][2][3][4][5][6][7][8][9][10][11][12]. It promises certain advance in the retrieval of flow structures, which provide a low-dimensional approximation of complex unsteady flowfields.…”

“…Their low-storage method for performing DMD can be updated inexpensively as new data become available. The problem of modal decomposition of large and arbitrarily sampled systems is addressed by Guéniat et al [4].…”