Model Order Reduction by Proper Orthogonal Decomposition

Gräßle, Carmen; Hinze, Michael; Volkwein, Stefan

doi:10.48550/arxiv.1906.05188

Cited by 2 publications

(3 citation statements)

References 69 publications

(118 reference statements)

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“…A recent mathematical discussion on POD-based ROMs is given in Ref. [41] for nonlinear parametric and time-dependent PDEs. In brief, GP makes use of the orthonormality of the POD modes to reduce the full order model (FOM) (i.e., governing equations) from partial differential equations (PDEs) to a simpler set of ordinary differential equations (ODEs) involving the POD temporal coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…To recover the missing information and account for the small-scale dissipation effects of the truncated POD modes, they introduced a Tikhonov regularization to build a calibrated ROM. In [41], linearizing and projecting the nonlinearity onto the POD space was proposed, allowing for solving the resulting evolution equations explicitly without spatial interpolation.…”

Section: Introductionmentioning

confidence: 99%

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Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive

Ahmed

San

Bistrian

et al. 2020

Numerical Methods in Fluids

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We investigate the sensitivity of reduced order models (ROMs) to training data resolution as well as sampling rate. In particular, we consider proper orthogonal decomposition (POD), coupled with Galerkin projection (POD-GP), as an intrusive reduced order modeling technique. For non-intrusive ROMs, we consider two frameworks. The first is using dynamic mode decomposition (DMD), and the second is based on artificial neural networks (ANNs). For ANN, we utilized a residual deep neural network, and for DMD we have studied two versions of DMD approaches; one with hard thresholding and the other with sorted bases selection. Also, we highlight the differences between mean-subtracting the data (i.e., centering) and using the data without mean-subtraction. We tested these ROMs using a system of 2D shallow water equations for four different numerical experiments, adopting combinations of sampling rates and resolutions. For these cases, we found that the DMD basis obtained with hard threshodling is sensitive to sampling rate, performing worse at very high rates. The sorted DMD algorithm helps to mitigate this problem and yields more stabilized and converging solution. Furthermore, we demonstrate that both DMD approaches with mean subtraction provide significantly more accurate results than DMD with mean-subtracting the data. On the other hand, POD is relatively insensitive to sampling rate and yields better representation of the flow field. Meanwhile, resolution has little effect on both POD and DMD performances. Therefore, we preferred to train the neural network in POD space, since it is more stable and better represents our dataset. Numerical results reveal that an artificial neural network on POD subspace (POD-ANN) performs remarkably better than POD-GP and DMD in capturing system dynamics, even with a small number of modes.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive

Ahmed

San

Bistrian

et al. 2020

Numerical Methods in Fluids

View full text Add to dashboard Cite

show abstract

“…On the other hand, snapshot-based model reduction techniques tailor a reduced space that best fits the problem by extracting the underlying coherent structures that controls the major dynamical evolution we are interested in. Proper orthogonal decomposition (POD) is a very popular and well-established approach extracting the modes which most contributes to the total variance [45,46]. In fluid dynamics applications, where we are mostly interested in the velocity field, those modes contain the largest amount of kinetic energy [47,48].…”

Section: Introductionmentioning

confidence: 99%

Memory embedded non-intrusive reduced order modeling of non-ergodic flows

et al. 2019

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Generating a digital twin of any complex system requires modeling and computational approaches that are efficient, accurate, and modular. Traditional reduced order modeling techniques are targeted at only the first two but the novel non-intrusive approach presented in this study is an attempt at taking all three into account effectively compared to their traditional counterparts. Based on dimensionality reduction using proper orthogonal decomposition (POD), we introduce a long short-term memory (LSTM) neural network architecture together with a principal interval decomposition (PID) framework as an enabler to account for localized modal deformation, which is a key element in accurate reduced order modeling of convective flows. Our applications for convection dominated systems governed by Burgers, Navier-Stokes, and Boussinesq equations demonstrate that the proposed approach yields significantly more accurate predictions than the POD-Galerkin method, and could be a key enabler towards near real-time predictions of unsteady flows. and perhaps much beyond. With the recent wave of digitization, reduced order modeling can be viewed as one of the key enablers to bring the promise of the digital twinning concept closer to reality [38]. Therefore, there is a continuous demand for the development of accurate reduced order models for complex physical phenomena. In projection-based ROMs, the most widely used technique, the discrete high-dimensional operators are projected onto a lower-dimensional space, so that the problem can be solved more efficiently in this reduced space [39][40][41][42][43][44].One of the very early-developed and well-known approaches to build this reduced space is Fourier analysis. However, it assumes universal basis functions (or modes) which have no specific relation to the physical system. On the other hand, snapshot-based model reduction techniques tailor a reduced space that best fits the problem by extracting the underlying coherent structures that controls the major dynamical evolution we are interested in. Proper orthogonal decomposition (POD) is a very popular and well-established approach extracting the modes which most contributes to the total variance [45,46]. In fluid dynamics applications, where we are mostly interested in the velocity field, those modes contain the largest amount of kinetic energy [47,48]. That is why POD is usually classified as an energy-based decomposition method. Another popular approach for model order reduction is the dynamic mode decomposition (DMD) [49][50][51][52][53][54] which generates a number of modes, each characterized by an oscillating frequency and growth/decay rate. In the present study, we are interested in the application of POD for dimensionality reduction.POD generates a set of spatial orthonormal basis functions, each containing a significant amount of total energy. To obtain a reduced representation of a system, the first few modes are selected, and the remaining are truncated assuming their contribution to the system's behavior is minimum (i.e.,...

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Model Order Reduction by Proper Orthogonal Decomposition

Cited by 2 publications

References 69 publications

Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive

Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive

Memory embedded non-intrusive reduced order modeling of non-ergodic flows

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