A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations

By Karen Veroy, Christophe Prud'Homme, Dimitrios Rovas, Anthony T. Patera. 2003

323 citations recorded by the source corpus.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/0145464d8d12125eabdfbf990d3296ccb3f07233

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In Computational Mechanics

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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, unknown): “Within these contexts, projection-based model-reduction methods have been successfully applied to problems in structural dynamics [3], aerodynamics [9, 10, 11, 12, 13] and aeroelasticity [14, 15, 16, 17, 18, 19], among others.”
A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD (2012, background): “In previous works [3, 4] exponential convergence with respect to the number N of basis functions is proved for a one-dimensional parameter case, and numerical experiments [7–9] illustrate the same behavior (or even faster) even in situations where the dimension of the parameter space P is larger.”
Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space (2008, methodology): “In this section, we present a general projection framework for model reduction and discuss computation of the test basis using a minimum-residual approach.”
An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations (2004, unknown): “…est ensuite couplée avec une méthode de discrétisation en base réduite pour l’approximation de la solution du problème La méthode en base réduite est alors (cid:2) (cid:2) max est un jeu de paramètres bien choisis et g M (x µ) = 1 m m classiques en base réduite proposés par exemple dans [6,8].”
Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations (2007, methodology): “we also ask that these bounds be sharp — ∆N (x) ≈ |sN (x)−s(x)| — so as not to mislead the design process (yielding over-designed, suboptimal solutions) or the adaptive approximation procedure [34].”
A low‐cost, goal‐oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems (2011, methodology): “Efficient online solution methods exist when the state equations are linear [33, 34] or quadratic [35, 36, 37] in the state and are affine in functions of the input parameters.”
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (2015, unknown): “It was further developed in reduced basis models for elliptic PDEs in [218], the steady incompressible Navier–Stokes equations in [217], and parabolic PDEs in [111, 112].”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “One option is to employ linearization of the parametric dependence (Daniel et al. 2004; Grepl and Patera 2005; Veroy et al. 2003).”
Reduced basis methods for partial differential equations with stochastic influences (2013, methodology): “, the solution for the parameter in the training set that is so far worst approximated is added [73, 98].”
Approximation of high-dimensional parametric PDEs * (2015, methodology): “5) This greedy algorithm was introduced, for the case when V is a Hilbert space in [89] and subsequently extensively studied in [9, 67, 68].”