A Stable Galerkin Reduced Order Model (Rom) for Compressible Flow

Kalashnikova, Irina; Arunajatesan, Srinivasan

doi:10.5151/meceng-wccm2012-18407

Cited by 5 publications

(10 citation statements)

References 42 publications

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“…[65, §6.1], [66], [67]. As an illustration of the importance of the first step we refer to [68], [69], [70] for examples how different inner products yield completely different results -certainly not desirable situation for real world applications.…”

Section: General Framework: Weighted 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: General Framework: Weighted Dmdmentioning

confidence: 99%

Data Driven Modal Decompositions: Analysis and Enhancements

Drmač¹,

Mezić²,

Mohr³

2018

SIAM J. Sci. Comput.

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“…Unfortunately, balanced truncation becomes computationally intractable for systems of very large dimension (e.g., of size N ≥ 10, 000), and hence is not practical for many systems of physical interest [39]. This is due to the high computational cost of solving the Lyapunov equations (26) and (27) for the reachability and observability Gramians (O(N 3 ) operations). The storage requirements of balanced truncation can be prohibitive as well.…”

Section: Pod Vs Balanced Truncationmentioning

confidence: 99%

“…In [39], Rowley et al show that Galerkin projection preserves the stability of an equilibrium point at the origin if an "energy-based" inner product is employed. In [9,27,26], Barone et al demonstrate that a symmetry transformation leads to a stable formulation for a Galerkin ROM for the linearized compressible Euler equations [9,27] and non-linear compressible Navier-Stokes equations [26] with solid wall and far-field boundary conditions. In [41], Serre et al propose applying the stabilizing projection developed by Barone et al in [9,27] to a skew-symmetric system constructed by augmenting a given linear system with its adjoint system.…”

Section: Introductionmentioning

confidence: 99%

“…The primary objective of the present work is to unify various approaches that fall into the first "class" of ROM stabilization approaches described above (those derived to have an a priori stability guarantee) using the energy method [23] and the concept of "energy-stability". The work is motivated by the observation that many of these approaches, e.g., those presented in [39,9,27,26,41], have certain similarities. In particular:…”

Section: Introductionmentioning

confidence: 99%

“…• The energy inner products derived in [39,9,27,26,41] assume the Galerkin projection step of the model reduction is performed at the level of the continuous PDEs. Is it possible to compute numerically a discrete form of the energy inner product?…”

Section: Introductionmentioning

confidence: 99%

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Construction of energy-stable Galerkin reduced order models.

Kalashnikova¹,

Barone²,

Arunajatesan³

et al. 2013

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This report aims to unify several approaches for building stable projection-based reduced order models (ROMs). Attention is focused on linear time-invariant (LTI) systems. The model reduction procedure consists of two steps: the computation of a reduced basis, and the projection of the governing partial differential equations (PDEs) onto this reduced basis. Two kinds of reduced bases are considered: the proper orthogonal decomposition (POD) basis and the balanced truncation basis. The projection step of the model reduction can be done in two ways: via continuous projection or via discrete projection. First, an approach for building energy-stable Galerkin ROMs for

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Model Order Reduction for Water Quality Dynamics

Wang

Taha

Chakrabarty

et al. 2022

Water Resources Research

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A state‐space representation of water quality (WQ) dynamics describing disinfectant (e.g., chlorine) transport dynamics in drinking water distribution networks has been recently proposed. Such representation is a byproduct of space‐ and time‐discretization of the partial differential equations modeling transport dynamics. This results in a large state‐space dimension even for small networks with tens of nodes. Although such a state‐space model provides a model‐driven approach to predict WQ dynamics, incorporating it into model‐based control algorithms or state estimators for large networks is challenging and at times intractable. To that end, this paper investigates model order reduction (MOR) methods for WQ dynamics with the objective of performing post‐reduction feedback control. The presented investigation focuses on reducing state‐dimension by orders of magnitude, the stability of the MOR methods, and the application of these methods to model predictive control.

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A Stable Galerkin Reduced Order Model (Rom) for Compressible Flow

Cited by 5 publications

References 42 publications

Data Driven Modal Decompositions: Analysis and Enhancements

Data Driven Modal Decompositions: Analysis and Enhancements

Construction of energy-stable Galerkin reduced order models.

Model Order Reduction for Water Quality Dynamics

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