EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

By Martin A. Grepl, Yvon Maday, Ngoc Cuong Nguyen, Anthony T. Patera. 2007

589 citations recorded by the source corpus.

Subfields: Computational Mechanics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/bd672f0339ff296188ed6b1b7a9e9add6fcb8788

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Citations in the corpus

The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows (2012, methodology): “The ‘empirical interpolation’ method developed for linear elliptic problems with non-affine parameter dependence [22], as well as for nonlinear elliptic and parabolic problems [23], reduces the computational cost associated with nonlinearities by interpolating the governing nonlinear function at a f”
Structure‐preserving, stability, and accuracy properties of the energy‐conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models (2015, methodology): “The empirical interpolation method (EIM) introduced later in [20] follows the same general idea.”
A Multiscale Neural Network Based on Hierarchical Matrices (2018, methodology): “The development of algorithms and heuristics to reduce the cost is an 28 area of active research [4, 18, 37, 15, 14].”
Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective (2014, methodology): “For other time-domain and snapshot-based nonlinear MOR methods like POD and the reduced basis method [100], the nonlinearity f(V z(t)) must be directly evaluated.”
Non-linear Petrov-Galerkin methods for reduced order modelling of the Navier-Stokes equations using a mixed finite element pair (2013, background): “A comparison of a number of POD formulations including the greedy reduced order approximation (ROA), the reduced-basis approach (RBA) of [24,31] and the standard Galerkin projection approach has been provided in [30].”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, methodology): “They can be effectively applied also to nonlinear problems [37, 57, 58], even if this in turn introduces both numerical and theoretical complications, and many open research issues are still to be faced.”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “Another basis generation approach is the reducedbasis method [182, 86, 183, 181], where orthogonalized carefully selected snapshots”
Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation (2009, unknown): “duced basis approximations must be considered [3,6,11,32] that in turn introduce both numerical and theoretical complications.”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “…nonlinear terms in the reduced-order model with selective spatial sampling, or the coefficient-function approximation (Barrault et al. 2004; Grepl et al. 2007), which replaces nonlinear parametric dependencies with a reduced-basis expansion and then uses interpolation to efficiently…”
The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation (2015, background): “…interest for the present work is the Empirical Interpolation Method (EIM, [2], [7], [8]) that has been developed in the broad framework where the functions f to approximate belong to a compact set F of continuous functions ( = C 0 ( Ω )). is The structure of F is supposed to make any f ∈ F be…”
Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, methodology): “empirical interpolation method) in order to recover the “affinity” assumption [1, 10,16,34].”
Computational Reduction for Parametrized PDEs: Strategies and Applications (2012, unknown): “The reader interested in time-dependent problems can instead refer to [19, 21] or to the more recent review provided in [45].”
Data-driven operator inference for nonintrusive projection-based model reduction (2016, methodology): “The reduced basis method [3, 8] derives a reduced basis with a greedy approach [9] based on computationally cheap a posteriori error estimators of intermediate reduced models.”
Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control (2013, methodology): “Proper orthogonal decomposition, model-order reduction, apriori and a-posteriori error analysis, linear-quadratic optimal control, primal-dual active set strategy.”
Multifidelity importance sampling (2016, background): “Surrogate models with rigorous and efficient a posteriori error estimators are currently available only for a limited class of problems [22, 23].”
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.”