A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners

By Karen Veroy, Dimitrios Rovas, Anthony T. Patera. 2002

70 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/1d1bf9c13aee1173bf1d360d51ebb494413894fd

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In Computational Mechanics

In Mechanics of Materials

In Statistical and Nonlinear Physics

Citations in the corpus

EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS (2007, methodology): “More recently, the reduced-basis approach and also associated a posteriori error estimation procedures have been successfully developed for (i) linear elliptic and parabolic PDEs that are affine in the parameter [13, 14, 20, 21, 29, 39] – the bilinear form a(w, v; μ) can be expressed as”
Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity (2005, background): “To obtain a bound for ηN (μ) (limited and close to unity) see [21].”
A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations (2005, background): “various recipes for this construction can be found in [34, 44].”
CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS (2012, background): “In the recent past, reduced basis approximations and associated a posteriori error estimation for linear and at most quadratically nonlinear elliptic and parabolic PDEs honoring this requirement have been successfully developed.(19,20,23,32,30,36,46,48,49) In Ref.”
Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations (2002, background): “First, it is possible to develop more general bound-conditioner-based a posteriori error estimation procedures [113].”
Reduced-basis methods applied to problems in elasticity : analysis and applications (2003, methodology): “7) We then introduce a symmetric, continuous, and coercive bound conditioner [21, 37, 47] C(μ) : Y → Y ′ such that the minimum and maximum eigenvalues ρmin(μ) ≡ min v∈Y 〈A(μ)v, v〉〈C(μ)v, v〉 , (4.”
Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation (2018, background): “For the construction of the coercivity lower bound, aLBðlÞ, various recipes exist (Huynh et al. 2007; Prud’homme et al. 2002; Veroy et al. 2002).”
A general multipurpose interpolation procedure: the magic points (2008, unknown): “…∈ D} , 1 ≤ N ≤ N max , and associated nested Lagrangian [21] reduced-basis spaces n = ≤ ≤ · · The classical reduced-basis approximation [13, 15, 22, 24, 8] is then obtained by a standard Galerkin projection: given Unfortunately, the presence of strong nonlinearity in g does not allow an eﬃcient…”
An efficient reduced‐order modeling approach for non‐linear parametrized partial differential equations (2008, methodology): “This allows us to apply an offline/online computational decomposition [7, 17 , 21] for the generation and simulation of the reduced-order model.”
Port reduction in parametrized component static condensation: approximation and a posteriori error estimation (2013, methodology): “In [29] a multi-reference-parameter approach is also introduced: for a given μ an interpolation between a few inverse reference operators is performed to optimize the conditioner.”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, background): “) The improvement is effected through the introduction of multiple inner products [150]: the focus is on the output effectivity; for the output bound, the inner product is a means to an end — and hence can be optimized.”
Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, background): “Its success is limited to the problems of linear elliptic parabolic PDEs with affine parameters or low-order polynomial nonlinearities [32, 25, 26, 41, 29].”