Multiresolution Dynamic Mode Decomposition

By J. Nathan Kutz, Xing Fu, Steven L. Brunton. SIAM Journal on Applied Dynamical Systems, 2015

368 citations recorded by the source corpus.

We demonstrate that the integration of the recently developed dynamic mode decomposition (DMD) with a multiresolution analysis allows for a decomposition method capable of robustly separating complex systems into a hierarchy of multiresolution time-scale components. A one-level separation allows for background (low-rank) and foreground (sparse) separation of dynamical data, or robust principal component analysis. The multiresolution DMD (mrDMD) is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank terms whose temporal coefficients in time are known. DMD modes with temporal frequencies near the origin (zero-modes) are interpreted as background (low-rank) portions of the given dynamics, and the terms with temporal frequencies bounded away from the origin are their sparse counterparts. The mrDMD method is demonstrated on several examples involving multiscale dynamical data, showing excellent decomposition results, ...

Subfields: Computational Mechanics, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/229dcf2ef3b637c0768fb33c3a9749ba5d2b8d6e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Molecular Biology

In Statistical and Nonlinear Physics

Citations in the corpus

Multi-scale proper orthogonal decomposition of complex fluid flows (2018, methodology): “This is a major difference between the mPOD and the SPOD by Sieber et al. (2016), in which the filtering procedure can eventually result in a nonorthogonal temporal basis, or the mrDMD by Kutz et al. (2016b), in which the loss of orthogonality can be produced by growing/decaying modes.”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, methodology): “In fact, modal decomposition based on the Koopman operator has been utilized in various engineering tasks, including robotic control [10], image processing [11], and nonlinear system identification [12].”
Generalizing Koopman Theory to Allow for Inputs and Control (2016, positive): “2, has been previously extended to handle data from multi-scale processes [27].”
Data-driven discovery of Koopman eigenfunctions for control (2017, methodology): “The DMD algorithm has also been extended to include sparsity and compressed sensing [78]– [82], actuation and control [83], multi-resolution analysis [84], de-noising [85], [86], and streaming variants [87].”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, background): “systems [9], including clustering [90], network theory [91], [92], DMD-based model reduction [93], [94], and local POD subspaces [95], to name a few. Recently, a multiresolution DMD has been proposed [96], where a low-dimensional subspace may locally characterize the attractor, despite a high-dimens”
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator (2016, methodology): “The initial success of this algorithm in the context of fluid mechanics motivated an ongoing line of research on data-driven analysis of high-dimensional and complex systems using Koopman operator theory and, consequently, a large number of DMD-type algorithms have been proposed in recent years for ”
Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks (2018, methodology): “hese approaches due to advances in computing power, algorithmic innovations and the ample availability of data. A major beneciary of this convergence are data-driven dimensionality reduction methods [7, 2, 3, 5, 6, 4, 8], model identication procedures [11, 10, 12, 13, 14, 15, 16, 9] and forecasting ”