Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems

Antil, Harbir; Heinkenschloss, Matthias; Sorensen, Danny C.

doi:10.1007/978-3-319-02090-7_4

Cited by 41 publications

(76 citation statements)

References 22 publications

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“…For modeling problems with strong nonlinearities, the cost of evaluating the reduced order models still depends on the size of the full order model and therefore is still expensive. The discrete empirical interpolation method (DEIM) described in detail in [59] represents an option to further approximate the nonlinearities in the projection-based reduced order strategies. The application of a DEIM-ROM strategy combined with the DMD method proposed in this paper represents a subject that we will defer to a future study.…”

Section: Numerical Results For Swirling Flow Problemmentioning

confidence: 99%

The method of dynamic mode decomposition in shallow water and a swirling flow problem

Bistrian

Navon

2016

Numerical Methods in Fluids

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SUMMARYIn this paper, a new vector-filtering criterion for dynamic modes selection is proposed that is able to extract dynamically relevant flow features from dynamic mode decomposition of time-resolved experimental or numerical data. We employ a novel modes selection criterion in parallel with the classic selection based on modes amplitudes, in order to analyze which of these procedures better highlight the coherent structures of the flow dynamics. Numerical tests are performed on two distinct problems. The efficiency of the proposed criterion is proved in retaining the most influential modes and reducing the size of the dynamic mode decomposition model. By applying the proposed filtering mode technique, the flow reconstruction error is shown to be significantly reduced.

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Section: Numerical Results For Swirling Flow Problemmentioning

confidence: 99%

The method of dynamic mode decomposition in shallow water and a swirling flow problem

Bistrian

Navon

2016

Numerical Methods in Fluids

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“…The key to the accuracy of this method is to select the indices of the discrete PDE that are most important to produce an accurate representation of the nonlinear component of the solution projected on the reduced space. The efficiency requirement is clearly satisfied when the nonlinear function is a PDE discretized using the finite element method [1].…”

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confidence: 99%

“…With a finite element discretization, a component F i (u) depends on the components u j for which the intersection of the support of the basis functions have measure that is nonzero. See [1] for additional discussion of this point. The elements that must be tracked in the DEIM computations are referred to as the sample mesh.…”

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confidence: 99%

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Numerical solution of the parameterized steady-state Navier–Stokes equations using empirical interpolation methods

Elman

Forstall

2017

Computer Methods in Applied Mechanics and Engineering

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Abstract. Reduced-order modeling is an efficient approach for solving parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem at significantly lower cost, producing an accurate estimate of the solution. For nonlinear problems, however, standard methods do not achieve the desired cost savings. Empirical interpolation methods represent a modification of this methodology used for cases of nonlinear operators or nonaffine parameter dependence. These methods identify points in the discretization necessary for representing the nonlinear component of the reduced model accurately, and they incur online computational costs that are independent of the spatial dimension N . We will show that empirical interpolation methods can be used to significantly reduce the costs of solving parameterized versions of the Navier-Stokes equations, and that iterative solution methods can be used in place of direct methods to further reduce the costs of solving the algebraic systems arising from reduced-order models.Key words. reduced basis, empirical interpolation, iterative methods, preconditioning 1. Introduction. Methods of reduced-order modeling are designed to obtain the numerical solution of parameterized partial differential equations (PDEs) efficiently. In settings where solutions of parameterized PDEs are required for many parameters, such as uncertainty quantification, design optimization, and sensitivity analysis, the cost of obtaining high-fidelity solutions at each parameter may be prohibitive. In this scenario, reduced-order models can often be used to keep computation costs low by projecting the model onto a space of smaller dimension with minimal loss of accuracy.We begin with a brief statement of the reduced basis method for constructing a reduced-order model. Consider an algebraic system of equations G(u) = 0 where u := u(ξ) is an unknown vector of dimension N , and ξ is vector of m input parameters. We are interested in the case where this system arises from the discretization of a PDE and N is large, as would be the case for a high-fidelity discretization. We will refer to this system as the full model. We would like to compute solutions for many parameters ξ. Reduced basis methods compute a (relatively) small number of full model solutions, u(ξ 1 ), . . . u(ξ k ), known as snapshots, and then for other parameters, ξ = ξ j , construct approximations of u(ξ) in the space spanned by {u(ξ j )}

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“…The solution to the bottleneck of the nonlinear term is found in the Discrete Empirical Interpolation (DEIM) [1,3]. In fact, the POD-DEIM reduction is reduced to the approximation of the nonlinear part of the internal force, which requires only a few components (dofs) of the nonlinear internal force vector [5,6].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Model Reduction for Truss Frame Based on POD-DEIM

Cui¹,

Gong²,

Yu³

et al. 2017

Proceedings of the 2017 International Conference on Applied Mathematics, Modelling and Statistics Application (AMMSA 2017)

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Abstract-Nonlinear model simplification is an important part of many applications, such as flow control, optimization and statistical analysis. In this article, we summarize the POD-DEIM method to solve the nonlinear system of the truss frame .And use an example of a clamped beam to illustrate the DEIM specific simplification process with images which can intuitively observe the image of the specific selection process. The application of the method to the truss frame to verify the correctness of the study. The results show that POD-DEIM is able to approximate fully nonlinear behavior of geometrically nonlinear finite element models with good accuracy.

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Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems

Cited by 41 publications

References 22 publications

The method of dynamic mode decomposition in shallow water and a swirling flow problem

The method of dynamic mode decomposition in shallow water and a swirling flow problem

Numerical solution of the parameterized steady-state Navier–Stokes equations using empirical interpolation methods

Nonlinear Model Reduction for Truss Frame Based on POD-DEIM

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