Journal ArticleDOI
An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations
TLDR
Barrault et al. as discussed by the authors presented an efficient reduced-basis discretization procedure for partial differential equations with nonaffine parameter dependence, replacing non-affine coefficient functions with a collateral reducedbasis expansion, which then permits an affine offline-online computational decomposition.About:
This article is published in Comptes Rendus Mathematique.The article was published on 2004-11-01. It has received 1265 citations till now. The article focuses on the topics: Discretization & Numerical analysis.read more
Citations
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Journal ArticleDOI
Nonlinear Model Reduction via Discrete Empirical Interpolation
TL;DR: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs).
Journal ArticleDOI
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
TL;DR: Model reduction aims to reduce the computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior as mentioned in this paper. But model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books.
Journal ArticleDOI
Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations
TL;DR: (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations are considered.
Journal ArticleDOI
Generalized multiscale finite element methods (GMsFEM)
TL;DR: The main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space.
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Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations
TL;DR: In this article, the authors acknowledge the partial support of the National Science Foundation Graduate Fellowship and the National Defense Science and Engineering Graduate Fellowship for a research grant from King Abdullah University of Science and Technology (KAUST) and Stanford University.
References
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Journal ArticleDOI
Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods
Christophe Prud'Homme,Dimitrios V. Rovas,Karen Veroy,Luc Machiels,Yvon Maday,Anthony T. Patera,Gabriel Turinici +6 more
TL;DR: The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
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Reduced Basis Technique for Nonlinear Analysis of Structures
Ahmed K. Noor,Jeanne M. Peters +1 more
TL;DR: In this paper, a reduced basis technique and a computational algorithm are presented for predicting the nonlinear static response of structures, where a total Lagrangian formulation is used and the structure is discretized by using displacement finite element models.
Proceedings ArticleDOI
A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations
TL;DR: In this paper, a technique for the prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence is presented, where the essential components are (i) rapidly convergent global reduced-basis approximations -Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/
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Automatic choice of global shape functions in structural analysis
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A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations
TL;DR: It is shown that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space, thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges.