Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems

By L. Machiels, Yvon Maday, I. B. Oliveira, Anthony T. Patera, Dimitrios Rovas. 2000

166 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/007360b799cf2ee7da2e20058a31e6d94195d62a

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In Computational Mechanics

In Computational Theory and Mathematics

In Statistical and Nonlinear Physics

Citations in the corpus

Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity (2005, background): “For further information and a detailed understanding of bound conditioner families (and special cases) we suggest to see [5, 11, 12].”
Reduced-basis output bound methods for parabolic problems (2006, methodology): “This is indeed the case when a certain decomposition exists for all the parameter-dependent linear and bilinear forms [11, 19].”
Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation (2010, background): “The affine representation (2) permits a “Construction-Evaluation” decomposition Balmes (1996); Machiels et al. (2000) ; Prud’homme et al. (2002) of computational effort that greatly reduces the marginal cost — relevant in the real-time and many-query contexts — of both the RB output evaluation, (7)”
Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations (2007, background): “Our second ingredient is a posteriori error estimators [22, 30]: rigorous and sharp bounds for the error in the reduced-basis predictions of the outputs of interest; from these error estimators, we can develop rigorous upper and lower bounds — which we denote “output bounds” — for the true output.”
Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations (2002, background): “This is indeed the case when a certain decomposition exists for all the parameter-dependent linear and bilinear forms [61].”
Certified Real-Time Solution of Parametrized Partial Differential Equations (2005, methodology): “Typically [8, 9], and in some very special cases provably [10], uN (μ), ψN (μ), and sN (μ) converge to u(μ),ψ(μ), and s(μ) uniformly and extremely rapidly – thanks to the smoothness in μ – and thus we may achieve the desired accuracy for N , N du N .”
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].”
A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners (2002, result): “Our work [10,12,16] differs from these earlier efforts in several important ways: first, we develop (in some cases, provably) global approximation spaces; second, we introduce rigorous a posteriori error estimators ; and third, we exploit off-line/on-line computational decompositions (see [5] for an”
Reduced-basis methods applied to problems in elasticity : analysis and applications (2003, background): “1 Bound Conditioner We again introduce a symmetric, continuous, and coercive bound conditioner [21, 37, 47] C(μ) : Y → Y ′ such that the eigenvalues ρmin(μ) = min v∈Y 〈A(μ)v, v〉〈C(μ)v, v〉 , (5.”
A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations (2005, methodology): “Computational procedure In this section we develop offline-online computational procedures in order to fully exploit the dimension reduction of the problem [3, 17, 22, 34].”
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations (2007, methodology): “Here, implicitly, the interpretation of the BRR theory is unfortunately less compelling due to the (guaranteed) multiplicity of often nearby solutions; hence the a posteriori error estimators for eigenvalue problems [ 84 ,113] are currently less than satisfactory.”
CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS (2012, methodology): “Computational procedure We summarize here the offline-online procedure.(3,22,27,36) We first express y N,M (μ) as”
A general multipurpose interpolation procedure: the magic points (2008, unknown): “…= u , µ uN ∈ 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…”