Parallel data-driven decomposition algorithm for large-scale datasets: with application to transitional boundary layers

Sayadi, Taraneh; Schmid, Peter J.

doi:10.1007/s00162-016-0385-x

Cited by 63 publications

(21 citation statements)

References 21 publications

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“…In the case of high resolution numerical simulations the ambient space dimension n (determined by the fineness of the grid) can be in millions, and mere memory fetching and storing is the bottleneck that precludes efficient computation, let alone flop count. For instance, [51], [52] report computation with the dimension n of half billion on a massively parallel computer. Tu and Rowley [40] discuss an implementation of DMD that has been designed to reduce computational work and memory requirements; this approach is also adopted in the library modred [53].…”

Section: Memory Efficient Dmdmentioning

confidence: 99%

Data Driven Modal Decompositions: Analysis and Enhancements

Drmač¹,

Mezić²,

Mohr³

2018

SIAM J. Sci. Comput.

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The Dynamic Mode Decomposition (DMD) is a tool of trade in computational data driven analysis of fluid flows. More generally, it is a computational device for Koopman spectral analysis of nonlinear dynamical systems, with a plethora of applications in applied sciences and engineering. Its exceptional performance triggered developments of several modifications that make the DMD an attractive method in data driven framework. This work offers further improvements of the DMD to make it more reliable, and to enhance its functionality. In particular, data driven formula for the residuals allows selection of the Ritz pairs, thus providing more precise spectral information of the underlying Koopman operator, and the well-known technique of refining the Ritz vectors is adapted to data driven scenarios. Further, the DMD is formulated in a more general setting of weighted inner product spaces, and the consequences for numerical computation are discussed in detail. Numerical experiments are used to illustrate the advantages of the proposed method, designated as DDMD RRR (Refined Rayleigh Ritz Data Driven Modal Decomposition). AMS subject classifications: 15A12, 15A23, 65F35, 65L05, 65M20, 65M22, 93A15, 93A30, 93B18, 93B40, 93B60, 93C05, 93C10, 93C15, 93C20, 93C57 Drmač, Mezić, Mohr / Enhanced DMD / AIMdyn Technical Reports 201708.004v1 2 aimdyn:201708.004v1 AIMDYN INC. Drmač, Mezić, Mohr / Enhanced DMD / AIMdyn Technical Reports 201708.004v1 3to assess the quality of each particular Ritz pair. Further, we show how to apply the well known Ritz vector refinement technique to the DMD data driven setting, and we discuss the importance of data scaling. All these modifications are integrated in §3.5 where we propose a new version of the DMD, designated as DDMD RRR (Refined Rauleigh-Ritz Data Driven Modal Decomposition). In §4 we provide numerical examples that show the benefits of our modifications, and we discuss the fine details of software implementation. In §5 we use the Exact DMD [12] to show that our modifications apply to other versions of DMD. In §6, we provide a compressed form of the new DDMD RRR, designed to improve the computational efficiency in case of extremely large dimensions. A matrixroot-free modification of the Forward-Backward DMD [13] is presented in §7. The column scaling used in the new DDMD implementation in §3.5 is just a particular case of a more general weighting scheme that we address in §8. Using the concept of the generalized SVD introduced by Van Loan [14], we define weighted DDMD with the Hilbert space structures in the spaces of the snapshots and the observables (spatial and temporal) given by two positive definite matrices. aimdyn:201708.004v1 AIMDYN INC. aimdyn:201708.004v1 AIMDYN INC. aimdyn:201708.004v1 AIMDYN INC.

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Section: Memory Efficient Dmdmentioning

confidence: 99%

Data Driven Modal Decompositions: Analysis and Enhancements

Drmač¹,

Mezić²,

Mohr³

2018

SIAM J. Sci. Comput.

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“…The goal of all projected DMD methods is to compute Q (which may or may not be orthonormal) and Q H AQ from the snapshot vectors X N 1 alone without knowledge of the linear mapping A. Some possible choices of Q are : matrix of snapshots X N−1 1 (Schmid and Sesterhenn (2008); Schmid (2010); Rowley et al (2009)), left singular vectors of economy SVD of X N−1 1 (Schmid (2010); Sayadi and Schmid (2016)) and orthonormal matrix from QR factorization of X N−1 1 (Hemati et al (2014)).…”

Section: Projected Dmd Methods: Backgroundmentioning

confidence: 99%

A parallel and streaming Dynamic Mode Decomposition algorithm with finite precision error analysis for large data

Anantharamu

Mahesh

2019

Journal of Computational Physics

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A novel technique based on the Full Orthogonalization Arnoldi (FOA) is proposed to perform Dynamic Mode Decomposition (DMD) for a sequence of snapshots. A modification to FOA is presented for situations where the matrix A is unknown, but the set of vectorsThe modified FOA is the kernel for the proposed projected DMD algorithm termed, FOA based DMD. The proposed algorithm to compute DMD modes and eigenvalues i) does not require Singular Value Decomposition (SVD) for snapshot matrices X with κ 2 (X) ≪ 1/ǫ m , where κ 2 (X) is the 2-norm condition number of the snapshot matrix and ǫ m is the relative round-off error or machine epsilon, ii) has an optional rank truncation step motivated by round off error analysis for snapshot matrices X with κ 2 (X) ≈ 1/ǫ m , iii) requires only one snapshot at a time, thus making it a 'streaming' method even with the optional rank truncation step, iv) consumes less memory and requires less floating point operations to obtain the projected matrix than existing projected DMD methods and v) lends itself to easy parallelism as the main computational kernel involves only vector additions, dot products and matrix vector products. The new technique is therefore well-suited for DMD of large datasets on parallel computing platforms. We show both theoretically and using numerical examples that for FOA based DMD without rank truncation, the finite precision error in the computed projection of the linear mapping is O(ǫ m κ 2 (X)). The proposed method is also compared to existing projected DMD methods for computational cost, memory consumption and relative round off error. Error indicators are presented that are useful to decide when to stop acquiring new snapshots. The proposed method is applied to several examples of numerical simulations of fluid flow.

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“…When dealing with massive fluid flows that are too large to read into fast memory, the extension to sequential, distributed, and parallel computing might be inevitable [39]. In particular, it might be necessary to distribute the data across processors which have no access to a shared memory to exchange information.…”

Section: Blocked Randomized Algorithmmentioning

confidence: 99%

Randomized Dynamic Mode Decomposition

Erichson¹,

Mathelin²,

Brunton³

et al. 2019

SIAM J. Appl. Dyn. Syst.

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This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of 'big data'. The idea is to derive a small matrix from the highdimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD (rDMD) algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension.

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Parallel data-driven decomposition algorithm for large-scale datasets: with application to transitional boundary layers

Cited by 63 publications

References 21 publications

Data Driven Modal Decompositions: Analysis and Enhancements

Data Driven Modal Decompositions: Analysis and Enhancements

A parallel and streaming Dynamic Mode Decomposition algorithm with finite precision error analysis for large data

Randomized Dynamic Mode Decomposition

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