Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations

By Yvon Maday, Anthony T. Patera, Gabriel Turinici. 2002

148 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/a63bd0c22064149eb845f42c6d7817378726b69f

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In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

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 methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity (2005, background): “We remind to the main result in [ 6 , 7]: For N Ncrit 1 + e ln( max + 1) and all 2 D we nd:”
A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations (2002, background): “where λ is the maximum eigenvalue of a0 relative to a1 .( Noteλ is perforce bounded thanks to our assumption of continuity and coercivity; the possibility of a continuous spectrum does not, in practice, pose any problems.) We can then prove [ 12 ] that, for N> N crit ≡ 2e ln( λµ max +1 ),”
On the stability of the reduced basis method for Stokes equations in parametrized domains (2007, methodology): “For the general a priori convergence properties of reduced basis method see, for example, [11,12].”
Greedy Algorithms for Reduced Bases in Banach Spaces (2012, methodology): “This greedy algorithm was introduced, for the case X is a Hilbert space, in the reduced basis method [5, 6] for solving a family of PDEs.”
A general multipurpose interpolation procedure: the magic points (2008, unknown): “The classical reduced-basis approximation [13, 15, 22, 24, 8] is then obtained by a standard Galerkin projection: given μ ∈ D, uN (μ) ∈Wu N satisfies μa0(uN (μ), v) + ∫”
An efficient reduced‐order modeling approach for non‐linear parametrized partial differential equations (2008, methodology): “†“Affine parameter dependence” means that the parametrized differential operator can be expressed as a sum of products of parameter-dependent functions and parameter-independent operators [7, 15, 16, 17].”
A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners (2002, background): “The sensitivity of these results to the point distribution is not too great; in [ 11 , 12] we consider a \log distribution on the average" with little detriment to the nal result.”
Reduced basis methods for Stokes equations in domains with non-affine parameter dependence (2009, methodology): “... dependent – approximationstagebasedonGalerkinprojection,used to find the solution for a new set of parameters by a combination of previously computed and stored solutions, allows computational economies of several orders of magnitude (see [3] for application of this strategy within the reduced-b”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, background): “The analysis in this section is a variation — in some cases a specialization, in other cases a generalization — of the development in [90, 91].”
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”
Approximation of high-dimensional parametric PDEs * (2015, background): “20) satisfies a so-called ‘saturation property’, which is analogous to that which is sometimes established in order to prove convergence of adaptive finite element methods; see Dörfler (1996), Morin et al. (2000) and Binev et al.”