The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

Maday, Yvon; Mula, Olga; Patera, Anthony T.; Yano, Masayuki

doi:10.1016/j.cma.2015.01.018

Cited by 64 publications

(64 citation statements)

References 21 publications

(44 reference statements)

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“…that pair with each V n and always ensure an inf-sup constant larger than β * . This strategy bears similarity to the generalized empirical interpolation method [15] where, at each step, one adds a new function to V n and a new linear functional to W m . In that case, we always have m = n, but no theoretical guarantee that β(V n , W n ) remains bounded away from zero.…”

Section: Pointwise Evaluationmentioning

confidence: 99%

Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models

Binev¹,

Cohen²,

Mula³

et al. 2018

SIAM/ASA J. Uncertainty Quantification

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We consider the problem of optimal recovery of an unknown function u in a Hilbert space V from measurements of the form j (u), j = 1, . . . , m, where the j are known linear functionals on V . We are motivated by the setting where u is a solution to a PDE with some unknown parameters, therefore lying on a certain manifold contained in V . Following the approach adopted in [12,5], the prior on the unknown function can be described in terms of its approximability by finite-dimensional reduced model spaces (V n ) n≥1 where dim(V n ) = n. Examples of such spaces include classical approximation spaces, e.g. finite elements or trigonometric polynomials, as well as reduced basis spaces which are designed to match the solution manifold more closely. The error bounds for optimal recovery under such priors are of the form µ(V n , W m )ε n , where ε n is the accuracy of the reduced model V n and µ(V n , W m ) is the inverse of an inf-sup constant that describe the angle between V n and the space W m spanned by the Riesz representers of ( 1 , . . . , m ). This paper addresses the problem of properly selecting the measurement functionals, in order to control at best the stability constant µ(V n , W m ), for a given reduced model space V n . Assuming that the j can be picked from a given dictionary D we introduce and analyze greedy algorithms that perform a sub-optimal selection in reasonable computational time. We study the particular case of dictionaries that consist either of point value evaluations or local averages, as idealized models for sensors in physical systems. Our theoretical analysis and greedy algorithms may therefore be used in order to optimize the position of such sensors.

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Section: Pointwise Evaluationmentioning

confidence: 99%

Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models

Binev¹,

Cohen²,

Mula³

et al. 2018

SIAM/ASA J. Uncertainty Quantification

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show abstract

“…RA N D O MUN I F O R M M is a stochastic sequential procedure: we simply draw points sequentially from the uniform density over . GEIM[10,28]:PR O C E S S U M Á GEIM M . We select,in a greedy manner from L , a sequence of observation functionals aimed to minimize the interpolation error associated with the approximation space Z N max U .…”

mentioning

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

103

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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“…We observe that the upper bound in (35) is decreasing for λ → 0, and for λ = 0 the upper bound coincides with the operator norm Q 0 U * op = 1 cos(ϕ T ,U ) . Unfortunately we do not know yet how to obtain a meaningful bound on the operator norm of Q λ .…”

Section: 3mentioning

confidence: 55%

Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections

Berger

Gröchenig

Matz

2018

J Fourier Anal Appl

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We study reconstruction operators on a Hilbert space that are exact on a given reconstruction subspace. Among those the reconstruction operator obtained by the least squares fit has the smallest operator norm, and therefore is most stable with respect to noisy measurements. We then construct the operator with the smallest possible quasi-optimality constant, which is the most stable with respect to a systematic error appearing before the sampling process (model uncertainty). We describe how to vary continuously between the two reconstruction methods, so that we can trade stability for quasi-optimality. As an application we study the reconstruction of a compactly supported function from nonuniform samples of its Fourier transform. 4

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The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

Cited by 64 publications

References 21 publications

Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models

Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models

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

Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections

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