Fast reduced order modeling technique for large scale LTV systems

Astrid, P.

doi:10.23919/acc.2004.1383697

Cited by 42 publications

(21 citation statements)

References 4 publications

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“…We note that the number of POD, DEIM and DMD are always the same in Figure 5 Since the POD-DMD is faster than other method it is natural to look at the performance with different number of basis functions. Figure 5.4 shows the error for a fixed number of basis functions = {5, 10, 15} and k ∈ [1,40]. As we can see the POD-DMD performs better than POD-DEIM when = 5, 10.…”

Section: Test 3: Semi-linear Parabolic Equationmentioning

confidence: 93%

Nonlinear Model Order Reduction via Dynamic Mode Decomposition

Alla¹,

Kutz²

2017

SIAM J. Sci. Comput.

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Abstract. We propose a new technique for obtaining reduced order models for nonlinear dynamical systems. Specifically, we advocate the use of the recently developed Dynamic Mode Decomposition (DMD), an equation-free method, to approximate the nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix that correlates spatial features while simultaneously associating the activity with periodic temporal behavior. With this decomposition, one can obtain a fully reduced dimensional surrogate model and avoid the evaluation of the nonlinear term in the online stage. This allows for an impressive speed up of the computational cost, and, at the same time, accurate approximations of the problem. We present a suite of numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.Key words. nonlinear dynamical systems, proper orthogonal decomposition, dynamic mode decomposition, data-driven modeling, reduced-order modeling, dimensionality reduction AMS subject classifications. 65L02, 65M02, 37M05, 62H251. Introduction. Reduced-order models (ROMs) are of growing importance in scientific computing as they provide a principled approach to approximating highdimensional PDEs with low-dimensional models. Indeed, the dimensionality reduction provided by ROMs help reduce the computational complexity and time needed to solve large-scale, engineering systems [24,3], enabling simulation based scientific studies not possible even a decade ago. One of the primary challenges in producing the low-rank dynamical system is efficiently projecting the nonlinearity of the governing PDEs (inner products) [2,6] on to the proper orthogonal decomposition (POD) [18,10,30] basis. This fact was recognized early on in the ROM community, and methods such as gappy POD [8,31,32] where proposed to more efficiently enable the task. More recently, the empirical interpolation method (EIM) [2], and the simplified discrete empirical interpolation method (DEIM) [6] for the proper orthogonal decomposition (POD) [18,10,30], have provided a computationally efficient method for discretely (sparsely) sampling and evaluating the nonlinearity. These broadly used and highly-successful methods ensure that the computational complexity of ROMs scale favorably with the rank of the approximation, even for complex nonlinearities. As an alternative to the EIM/DEIM architecture, we propose using the recently developed Dynamic Mode Decomposition (DMD) for producing low-rank approximations of the PDE nonlinearities. DMD provides a decomposition of data into spatio-temporal modes that correlates the data across spatial features (like POD), but also associates the correlated data to unique temporal Fourier modes, allowing for a computationally efficient regression of the nonlinear terms to a least-square fit linear dynamics approximation. We demonstrate that the POD-DMD method produces a viable ROM architecture, scaling favorable in computational efficiency relative to

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Section: Test 3: Semi-linear Parabolic Equationmentioning

confidence: 93%

Nonlinear Model Order Reduction via Dynamic Mode Decomposition

Alla¹,

Kutz²

2017

SIAM J. Sci. Comput.

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“…These grid point selection procedures were later improved by incorporating a greedy algorithm from [95]. The applications of the MPE method are primarily in the context of a linear time varying system arising from FV discretization of a nonlinear computational fluid dynamic model for a glass melting furnace [6,5,8,7]. It has also been used in modeling heat transfer in electrical circuits [89] and in subsurface flow simulation [17].…”

Section: Techniques For Nonlinearitiesmentioning

confidence: 99%

“…This procedure can be viewed as performing the Galerkin projection onto the truncated POD basis via a specially constructed inner product as defined in [9] that evaluates only at selected grid points instead of computing the usual L 2 inner product. Two heuristic methods for selecting these spatial grid points are introduced in the thesis [6] (also in subsequent publications, e.g [5,8,7]) by aiming to minimize aliasing effects in using only partial spatial points. This was shown to be equivalent to a criterion for preserving the orthogonality of the restricted POD basis vectors, which is further translated into a criterion for controlling condition number growth.…”

Section: Techniques For Nonlinearitiesmentioning

confidence: 99%

Nonlinear Model Reduction via Discrete Empirical Interpolation

Chaturantabut¹,

Sorensen²

2010

SIAM J. Sci. Comput.

1,658

1,778

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“…Examples of hyper-reduction methods are: Gappy-POD [91], Missing Point Estimation (MPE) [92], Discrete Empirical Interpolation Method (DEIM) [93], Dynamic Mode Decomposition (DMD) [94] or numerical linearization [95]; for a comparison see [96]. These methods construct (linear) reduced representations of the nonlinearity in a data-driven manner.…”

Section: On Hyper-reductionmentioning

confidence: 99%

emgr—The Empirical Gramian Framework

Himpe

2018

Algorithms

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System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality. These so-called system Gramians were developed in linear system theory for applications such as model order reduction of control systems. Empirical Gramians are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation. The empirical Gramian framework -emgrimplements the empirical Gramians in a uniform and configurable manner, with applications such as Gramian-based (nonlinear) model reduction, decentralized control, sensitivity analysis, parameter identification and combined state and parameter reduction. Keywords

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Fast reduced order modeling technique for large scale LTV systems

Cited by 42 publications

References 4 publications

Nonlinear Model Order Reduction via Dynamic Mode Decomposition

Nonlinear Model Order Reduction via Dynamic Mode Decomposition

Nonlinear Model Reduction via Discrete Empirical Interpolation

emgr—The Empirical Gramian Framework

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