Localized Discrete Empirical Interpolation Method

Peherstorfer, Benjamin; Butnaru, Daniel; Willcox, Karen; Bungartz, Hans‐Joachim

doi:10.1137/130924408

Cited by 225 publications

(167 citation statements)

References 32 publications

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“…In the following discussion, the parameter p belongs to a single domain Ω ⊂ R d . Recent work has proposed approaches to split Ω into multiple subdomains and construct reduced models in each subdomain [10,76,82,121,190,221]. Any of the model generation strategies described in the following can be applied in a partitioning setting by replacing Ω with the corresponding subdomain.…”

Section: Parameterized Reduced Model Generationmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Parameterized Reduced Model Generationmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…Unlike the offline adaptation, online adaptive methods modify the reduced system during the computations in the integration stage. Most of the existing global online adaptive methods [13][14][15] rely only on precomputed quantities from the offline stage. We consider here a different approach whereby online adaptivity is performed by incorporating new data that becomes available in the online stage.…”

Section: Of 21mentioning

confidence: 99%

Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media

2016

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Abstract:We propose an online adaptive local-global POD-DEIM model reduction method for flows in heterogeneous porous media. The main idea of the proposed method is to use local online indicators to decide on the global update, which is performed via reduced cost local multiscale basis functions. This unique local-global online combination allows (1) developing local indicators that are used for both local and global updates (2) computing global online modes via local multiscale basis functions. The multiscale basis functions consist of offline and some online local basis functions. The approach used for constructing a global reduced system is based on Proper Orthogonal Decomposition (POD) Galerkin projection. The nonlinearities are approximated by the Discrete Empirical Interpolation Method (DEIM). The online adaption is performed by incorporating new data, which become available at the online stage. Once the criterion for updates is satisfied, we adapt the reduced system online by changing the POD subspace and the DEIM approximation of the nonlinear functions. The main contribution of the paper is that the criterion for adaption and the construction of the global online modes are based on local error indicators and local multiscale basis function which can be cheaply computed. Since the adaption is performed infrequently, the new methodology does not add significant computational overhead associated with when and how to adapt the reduced basis. Our approach is particularly useful for situations where it is desired to solve the reduced system for inputs or controls that result in a solution outside the span of the snapshots generated in the offline stage. Our method also offers an alternative of constructing a robust reduced system even if a potential initial poor choice of snapshots is used. Applications to single-phase and two-phase flow problems demonstrate the efficiency of our method.

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“…These considerations have motivated the recent development of novel local model reduction approaches in which smaller local subspaces are defined and the reduced-order models marches from one subspace to another one within each single simulation [1][2][3]. Local subspaces can be defined in time [1,2], parameter space [4][5][6], solution features [7] or state-space [3,6,[8][9][10].…”

Section: Introductionmentioning

confidence: 99%

PEBL-ROM: Projection-error based local reduced-order models

Amsallem

Haasdonk

2016

Adv. Model. and Simul. in Eng. Sci.

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Projection-based model order reduction (MOR) using local subspaces is becoming an increasingly important topic in the context of the fast simulation of complex nonlinear models. Most approaches rely on multiple local spaces constructed using parameter, time or state-space partitioning. State-space partitioning is usually based on Euclidean distances. This work highlights the fact that the Euclidean distance is suboptimal and that local MOR procedures can be improved by the use of a metric directly related to the projections underlying the reduction. More specifically, scale-invariances of the underlying model can be captured by the use of a true projection error as a dissimilarity criterion instead of the Euclidean distance. The capability of the proposed approach to construct local and compact reduced subspaces is illustrated by approximation experiments of several data sets and by the model reduction of two nonlinear systems.

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Localized Discrete Empirical Interpolation Method

Cited by 225 publications

References 32 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media

PEBL-ROM: Projection-error based local reduced-order models

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