Missing point estimation in models described by proper orthogonal decomposition

Astrid, P.; Weiland, S.; Willcox, Karen; Backx, Ton

doi:10.1109/cdc.2004.1430301

Cited by 176 publications

(309 citation statements)

References 5 publications

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“…A more general approach that has been used successfully for nonlinear model reduction is selective sampling of the nonlinear terms combined with interpolation among these samples to recover an approximate nonlinear evaluation. Among this class of methods, the missing point estimation [19] and Gauss Newton with approximated tensors (GNAT) [66] methods both build upon the gappy POD interpolation method [87]; the empirical interpolation method (EIM) [26] and its discrete variant, the discrete empirical interpolation method (DEIM) [67], conduct interpolation on a low-dimensional basis for the nonlinear term. The EIM has been recently extended to the case where A(p) represents a PDE operator [77].…”

Section: Projection-based Model Reductionmentioning

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: Projection-based Model Reductionmentioning

confidence: 99%

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

2015

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“…In order to reduce the online cost of evaluating the nonlinear term(s), several "hyper-reduction" techniques have been proposed, such as DEIM [30] (Discrete Empirical Interpolation Method), DBPIM [12] (Discrete Best Points Interpolation Method), MPE (Missing Point Estimation) [8] and GNAT [5] (Gauss-Newton with Approximated Tensor quantities). In general, most of these methods attempt to approximate the nonlinearity using linear combinations of the POD basis functions…”

Section: "Exploit the Known Structure Of The Solutions"mentioning

confidence: 99%

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

163

179

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This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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“…In a first subset of these strategies, the nonlinear function is reconstructed by interpolation over an other POD basis ("gappy" technique) [5,34,51,36]. The expansion of the nonlinear term reads: Kerfriden …”

Section: Evaluation Of Nonlinear Terms On Reduced Spatial Domains-mentioning

confidence: 99%

“…In [34], is constructed such that the condition number of operator is minimised. In the hyperreduction method [53], the non-zero entries of correspond to the largest entries (in some sense) of the approximated nonlinear vector function.…”

Section: Europe Pmc Funders Author Manuscriptsmentioning

confidence: 99%

A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics

Kerfriden

Goury

Rabczuk

et al. 2013

Computer Methods in Applied Mechanics and Engineering

117

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We propose in this paper a reduced order modelling technique based on domain partitioning for parametric problems of fracture. We show that coupling domain decomposition and projectionbased model order reduction permits to focus the numerical effort where it is most needed: around the zones where damage propagates. No a priori knowledge of the damage pattern is required, the extraction of the corresponding spatial regions being based solely on algebra. The efficiency of the proposed approach is demonstrated numerically with an example relevant to engineering fracture.

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Missing point estimation in models described by proper orthogonal decomposition

Cited by 176 publications

References 5 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

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics

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