Data Driven Modal Decompositions: Analysis and Enhancements

Drmač, Zlatko; Mezić, Igor; Mohr, Ryan

doi:10.1137/17m1144155

Cited by 52 publications

(65 citation statements)

References 69 publications

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“…All of the algorithms used gave very similar results. This suggests that the sea ice concentration data dynamical behavior is "well behaved" in the sense that the resulting condition number is sufficiently small that any of the various approximations of the Koopman decomposition are valid here 17 and thus supports the conclusion that the KMD results obtained here are physically meaningful and not numerical artifacts.…”

Section: Resultssupporting

confidence: 79%

Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman mode decomposition

Hogg¹,

Fonoberova²,

Mezić

2020

Sci Rep

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Sea ice cover in the Arctic and Antarctic is an important indicator of changes in the climate, with important environmental, economic and security consequences. The complexity of the spatio-temporal dynamics of sea ice makes it difficult to assess the temporal nature of the changes—e.g. linear or exponential—and their precise geographical loci. In this study, Koopman Mode Decomposition (KMD) is applied to satellite data of sea ice concentration for the Northern and Southern hemispheres to gain insight into the temporal and spatial dynamics of the sea ice behavior and to predict future sea ice behavior. We observe spatial modes corresponding to the mean and annual variation of Arctic and Antarctic sea ice concentration and observe decreases in the mean sea ice concentration from early to later periods, as well as corresponding shifts in the locations that undergo significant annual variation in sea ice concentration. We discover exponentially decaying spatial modes in both hemispheres and discuss their precise spatial extent, and also perform predictions of future sea ice concentration. The Koopman operator-based, data-driven decomposition technique gives insight into spatial and temporal dynamics of sea ice concentration not apparent in traditional approaches.

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Section: Resultssupporting

confidence: 79%

Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman mode decomposition

Hogg¹,

Fonoberova²,

Mezić

2020

Sci Rep

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“…The observable value used as the KMD input was the MTF "fullness," defined as the ratio of the occupancy of each MTF to its capacity (see Fig 3 for example time series). The KMD algorithm used was the DMD_RRR method [23].…”

Section: Koopman Mode Analysis Of Mtf Simulation Resultsmentioning

confidence: 99%

“…In this paper, we use a variant of DMD recently introduced in Drmać et al ([23], Algorithm 2) which includes an optional refinement procedure for the computed modes and eigenvalues as well as an explicit error term specifying the accuracy of the computed spectrum. We discuss the basic DMD algorithm here and refer the reader to [23] for details on the refinement procedure.…”

Section: Koopman Mode Decomposition (Kmd)mentioning

confidence: 99%

Koopman Mode Analysis of agent-based models of logistics processes

Hogg¹,

Fonoberova²,

Mezić

et al. 2019

PLoS ONE

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Modern logistics processes and systems can feature extremely complicated dynamics. Agent Based Modeling is emerging as a powerful modeling tool for design, analysis and control of such logistics systems. However, the complexity of the model itself can be overwhelming and mathematical meta-modeling tools are needed that aggregate information and enable fast and accurate decision making and control system design. Here we present Koopman Mode Analysis (KMA) as such a tool. KMA uncovers exponentially growing, decaying or oscillating collective patterns in dynamical data. We apply the methodology to two problems, both of which exhibit a bifurcation in dynamical behavior, but feature very different dynamics: Medical Treatment Facility (MTF) logistics and ship fueling (SF) logistics. The MTF problem features a transition between efficient operation at low casualty rates and inefficient operation beyond a critical casualty rate, while the SF problem features a transition between short mission life at low initial fuel levels and sustained mission beyond a critical initial fuel level. Both bifurcations are detected by analyzing the spectrum of the associated Koopman operator. Mathematical analysis is provided justifying the use of the Dynamic Mode Decomposition algorithm in punctuated linear decay dynamics that is featured in the SF problem.

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“…If Algorithm 2.1 uses k < m, then S k is a Rayleigh quotient of C m , and the columns of Z k are not the same Ritz vectors as in W m from (2.6). For more details on this connection, we refer to [12].…”

Section: Reconstruction Using Schmid's Dmd -An Alternative To Workingmentioning

confidence: 99%

Data Driven Koopman Spectral Analysis in Vandermonde--Cauchy Form via the DFT: Numerical Method and Theoretical Insights

Drmač¹,

Mezić²,

Mohr³

2019

SIAM J. Sci. Comput.

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The goals and contributions of this paper are twofold. It provides a new computational tool for data driven Koopman spectral analysis by taking up the formidable challenge to develop a numerically robust algorithm by following the natural formulation via the Krylov decomposition with the Frobenius companion matrix, and by using its eigenvectors explicitly -these are defined as the inverse of the notoriously ill-conditioned Vandermonde matrix. The key step to curb ill-conditioning is the discrete Fourier transform of the snapshots; in the new representation, the Vandermonde matrix is transformed into a generalized Cauchy matrix, which then allows accurate computation by specially tailored algorithms of numerical linear algebra. The second goal is to shed light on the connection between the formulas for optimal reconstruction weights when reconstructing snapshots using subsets of the computed Koopman modes. It is shown how using a certain weaker form of generalized inverses leads to explicit reconstruction formulas that match the abstract results from Koopman spectral theory, in particular the Generalized Laplace Analysis.AMS subject classifications: 15A12, 15A23, 65F35, 65L05, 65M20, 65M22, 93A15, 93A30, 93B18, 93B40, 93B60, 93C05, 93C10, 93C15, 93C20, 93C57In this paper, we revisit the natural formulation of finite dimensional Koopman spectral analysis in terms of Krylov bases and the Frobenius companion matrix. In this formulation, reviewed in §2.1 below, a spectral approximation of A is obtained by the Rayleigh-Ritz extraction using the primitive Krylov basis X m = (f 1 , . . . , f m ). This means that the Rayleigh quotient of A is the Frobenius companion matrix, whose eigenvector matrix is the inverse of the Vandermonde matrix V m , parametrized by its eigenvalues λ i . The DMD (Koopman) modes, that is, the Ritz vectors z i , are then computed as. . , m. Unfortunately, Vandermonde matrices can have extremely high conditions numbers, so a straightforward implementation of this algebraically elegant scheme can lead to inaccurate results in finite precision computation. Most other DMD-variants bypass this issue by computing an orthonormal basis from the snaphshots using a truncated singular value decomposition [34,33,38]. We note that the original formulation of DMD was based on the SVD ([34], reviewed in Algorithm 2.1 in this paper), whereas the first connection between DMD and Koopman operator theory was formulated in terms of the companion matrix [32]. In Rowley et al. [32], though, there was no consideration of the deep numerical issues related to working with Vandermonde matrices which has likely contributed to the prevalence of the SVD-based variants of DMD. There is, however, a certain intrinsic elegance in the decomposition of the snapshots in terms of the spectral structure of the companion matrix in addition to a stronger connection to Generalized Laplace Analysis (GLA) theory [27,26] of Koopman operator theory than the SVD-based DMD variants have.Following this natural formulation, we present a DMD alg...

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Data Driven Modal Decompositions: Analysis and Enhancements

Cited by 52 publications

References 69 publications

Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman mode decomposition

Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman mode decomposition

Koopman Mode Analysis of agent-based models of logistics processes

Data Driven Koopman Spectral Analysis in Vandermonde--Cauchy Form via the DFT: Numerical Method and Theoretical Insights

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