<i>A priori</i>convergence of the Greedy algorithm for the parametrized reduced basis method

Buffa, Annalisa; Maday, Yvon; Patera, Anthony T.; Prud'Homme, Christophe; Turinici, Gabriel

doi:10.1051/m2an/2011056

Cited by 279 publications

(281 citation statements)

References 8 publications

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“…We apply the Weak Greedy algorithm described in Algorithm 1 to form Z N (see also a detailed review -in particular as regard the construction of an error bound that is efficient in the many-query setting -by Rozza et al [25]). The algorithm has been proven to generate an optimal sequence of spaces with respect to the Kolmogorov width of M bk in Binev et al [3], Buffa et al [4], and DeVore et al [7].…”

Section: A Posteriori Error Estimatesmentioning

confidence: 99%

A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics

Maday

Patera

Penn

et al. 2014

Numerical Meth Engineering

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110

187

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SUMMARYWe present a Parametrized-Background Data-Weak (PBDW) formulation of the variational data assimilation (state estimation) problem for systems modeled by partial differential equations. The main contributions are a constrained optimization weak framework informed by the notion of experimentally observable spaces; a priori and a posteriori error estimates for the field and associated linear-functional outputs; Weak Greedy construction of prior (background) spaces associated with an underlying potentially high-dimensional parametric manifold; stability-informed choice of observation functionals and related sensor locations; and finally, output prediction from the optimality saddle in O(M 3 ) operations, where M is the number of experimental observations. We present results for a synthetic Helmholtz acoustics model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory. To conclude, we consider a physical raised-box acoustic resonator chamber: we integrate the PBDW methodology and a Robotic Observation Platform to achieve real-time in situ state estimation of the time-harmonic pressure field; we demonstrate the considerable improvement in prediction provided by the integration of a best-knowledge model and experimental observations; we extract even from these results with real data the numerical trends indicated by the theoretical convergence and stability analyses.

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Section: A Posteriori Error Estimatesmentioning

confidence: 99%

A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics

Maday

Patera

Penn

et al. 2014

Numerical Meth Engineering

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110

187

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“…See also [15] for the first but less sharp estimates. In other words, if the underlying problem allows an efficient and compact reduced basis, the greedy approximation will find an exponentially convergent approximation to it.…”

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confidence: 99%

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

865

1,221

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The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-22470-1International audienceThis book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples

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“…Assuming such a decay, it is not obvious that the particular N − dimensional subspace of W (Y ) constructed with the greedy algorithm enjoys such an approximation property. This has been proved in [35,36]. More precisely the application of [36,Corollary 3] shows the following result.…”

Section: A Priori Error Analysismentioning

confidence: 77%

Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems

Abdulle

Bai

2012

Journal of Computational Physics

108

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A prioriconvergence of the Greedy algorithm for the parametrized reduced basis method

Cited by 279 publications

References 8 publications

A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics

A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems

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