An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations

By M. Barrault, Yvon Maday, Ngoc Cuong Nguyen, Anthony T. Patera. 2004

1,702 citations recorded by the source corpus.

Subfields: Computational Mechanics, Mechanics of Materials, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9f43ca2b0b1d7576fa40ceb7ded3d94def908f2a

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In Computational Mechanics

In Mechanics of Materials

In Statistical and Nonlinear Physics

Citations in the corpus

Parameter multi‐domain ‘hp’ empirical interpolation (2012, methodology): “The empirical interpolation method (EIM) was first introduced in [1, 2] as a tool within the reduced basis (RB) framework [3] for parametrized PDEs.”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, background): “[122] M.”
Data-driven reduced order modeling for time-dependent problems (2019, methodology): “The empirical interpolation method (EIM) [6] and its discrete variants [12, 28], also referred to as hyper-reductions [28], are developed to deal with this problem by recovering an affine expansion of the differential operator in a non-affine case.”
Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations (2011, methodology): “In particular, approaches have been developed to select points that minimize the error in the reconstructed snapshots [27, 30], the difference between the reconstructed snapshots and their optimal projections [25, 28], and the condition number of the normal-equations matrix used for reconstruction…”
A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations (2005, methodology): “This affine parameter dependence is crucial for the computational efficiency of the proposed method; however, see [6, 41] for extensions to the non-affine and nonlinear case.”
Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, methodology): “This approach is a discrete variant of the empirical interpolation method (EIM) introduced by Barrault, Maday, Nguyen, and Patera [7], which was originally posed in an empirically derived finite-dimensional function space.”
Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective (2014, methodology): “There are some techniques, which provide efficient estimates of f(V z(t)) as the (discrete) empirical interpolation methods (DEIM or EIM) [21, 56], the missing point estimation method [8], or a best points interpolation method [146].”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, background): “Another goal could be to interpolate sampled solutions for rapidly solving the parametrized linear system with respect to a parameter configuration that is not known a priori, similar to the reduced basis method [9].”
Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, methodology): “Let us finally mention another model reduction technique, called the “Reduced Basis” method [ 11 , 79, 80], which has been initially introduced for parametric analysis.”
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations (2007, methodology): “This extension is based on the Empirical Interpolation Method (EIM) [ 18 ]: a collateral RB space for the offending nonaffine coefficient functions; an interpolation system that avoids costly (N -dependent) projections; and several (from less rigorous/simple to completely rigorous/very cumbersome) a”
A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs (2020, methodology): “The same diﬃculty might also aﬀect the use of hyper-reduction techniques, such as the (discrete) empirical interpolation [29, 30], mandatory in order to assemble the operators appearing in the ROM (8) without relying on expensive N h -dimensional arrays.”
A multilevel projection‐based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics (2017, methodology): “It was also very recently equipped in [20, 21] with the empirical interpolation method [22, 23] and related variants for hyper reduction – albeit at the microscale level only – which nevertheless enabled the speedup of multiscale computations based on the concept of a locally attached microstructure”
Interpolation among reduced‐order matrices to obtain parameterized models for design, optimization and probabilistic analysis (2009, methodology): “A more general approach is to employ the coefficient-function approximation of [1, 2], which replaces nonlinear parametric dependencies with a reduced-basis expansion and then uses interpolation to efficiently compute the coefficients of that expansion for new parameter values.”
CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS (2012, background): “We recall that (i) the set of all characteristic functions {V m }m=1 is a basis for W g M , and (ii) the Lebesgue constant ΛM satisfies ΛM ≤ 2 − 1.(4,18) In applications, the actual asymptotic behavior of ΛM is much lower, as we shall observe subsequently.”
EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS (2007, methodology): “method” ‐ coecient function approximation — introduced in [ 4 ].”