Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

Negri, Federico; Manzoni, Andrea; Amsallem, David

doi:10.1016/j.jcp.2015.09.046

Cited by 129 publications

(110 citation statements)

References 57 publications

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“…Since the focus of this section is geometry reduction, we do not discuss the construction of the ROM for the mapped problems (36) and (38).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…We present below the non-interpolatory extension of EIM employed in this paper; the same approach has also been employed in [53]. We refer to [55,36] for two alternatives applicable to vector-valued fields. Given the space W Q = span{ω q } Q q=1 ⊂ W and the points {x i q } Q q=1 ⊂ Ω, we define the least-squares approximation operator I Q : W → W Q such that for all v ∈ W…”

Section: D2 Extension To Vector-valued Fieldsmentioning

confidence: 99%

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A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

Taddei

2020

SIAM J. Sci. Comput.

110

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We propose a general -i.e., independent of the underlying equationregistration method for parameterized Model Order Reduction. Given the spatial domain Ω ⊂ R d and a set of snapshots {u k } n train k=1 over Ω associated with ntrain values of the model parameters µ 1 , . . . , µ n train ∈ P, the algorithm returns a parameter-dependent bijective mapping Φ : Ω × P → R d : the mapping is designed to make the mapped manifold {uµ • Φµ : µ ∈ P} more suited for linear compression methods. We apply the registration procedure, in combination with a linear compression method, to devise low-dimensional representations of solution manifolds with slowlydecaying Kolmogorov N -widths; we also consider the application to problems in parameterized geometries. We present a theoretical result to show the mathematical rigor of the registration procedure. We further present numerical results for several two-dimensional problems, to empirically demonstrate the effectivity of our proposal.

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“…Since the focus of this section is geometry reduction, we do not discuss the construction of the ROM for the mapped problems (36) and (38).…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: D2 Extension To Vector-valued Fieldsmentioning

confidence: 99%

A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

Taddei

2020

SIAM J. Sci. Comput.

110

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show abstract

“…Furthermore, alternative approximations of the system tangent, e.g. [5,29,38], could be investigated. Yet some of these methods require additional high dimensional snapshots of the sparse system tangent which becomes computational infeasible rapidly.…”

Section: Resultsmentioning

confidence: 99%

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

et al. 2017

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Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors-GNAT) is favoured to obtain an optimal projection and a robust reduced model.

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“…In the online stage, the interested output is computed by solving the resultant ROM for many instances of parameters, and the influence of the uncertainty is estimated. Thus, the projection‐based ROM reduces the computational cost associated with the solution of parameter‐dependent full‐order model (FOM) by restricting the solution space to a subspace of much smaller dimension (Negri et al, ).…”

Section: Introductionmentioning

confidence: 99%

A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media

Jiang

2019

Water Resources Research

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In this paper, we develop a reduced generalized multiscale basis method for efficiently solving the parametrized groundwater flow problems in heterogeneous porous media. Recently proposed generalized multiscale finite element (GMsFE) method is one of the accurate and efficient methods to solve the multiscale problems on a coarse grid. However, the GMsFE basis functions usually depend on the random parameters in the parametrized model, which substantially impacts on the computation efficiency when the parametrized equation needs to be solved many times for many instances of parameters. To enhance computation efficiency, we construct reduced generalized multiscale basis functions independent of parameters from the high‐dimensional GMsFE space and generate a reduced‐order multiscale model by projecting the parameterized flow governing equation onto the low‐dimensional reduced GMsFE space. Then, in order to improve the online computation of reduced‐order model, we apply a matrix discrete empirical interpolation method to affinely decompose the nonaffine parametrized systems arising from the discretization of governing equation. Finally, a few numerical experiments are carried out for the parametrized transient flow problems in heterogeneous porous media to illustrate the efficiency and accuracy of the proposed model reduction method. The results show that the proposed reduced generalized multiscale basis method can significantly improve computation efficiency while maintaining comparative accuracy for the groundwater flow problems by selecting a suitable coarse mesh size and an optimal number of reduced GMsFE basis functions at each of coarse nodes.

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Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

Cited by 129 publications

References 57 publications

A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media

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