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Convergence analysis of the Generalized Empirical Interpolation Method

Maday, Yvon; Mula, Olga; Turinici, Gabriel (2016), Convergence analysis of the Generalized Empirical Interpolation Method, SIAM Journal on Numerical Analysis, 54, 3, p. 1713-1731. 10.1137/140978843

Type
Article accepté pour publication ou publié
External document link
http://hal.upmc.fr/hal-01032458
Date
2016
Journal name
SIAM Journal on Numerical Analysis
Volume
54
Number
3
Publisher
SIAM
Published in
Paris
Pages
1713-1731
Publication identifier
10.1137/140978843
Metadata
Show full item record
Author(s)
Maday, Yvon

Mula, Olga cc

Turinici, Gabriel cc
Abstract (EN)
Let F be a compact set of a Banach space X. This paper analyses the ``Generalized Empirical Interpolation Method'' (GEIM) which, given a function f∈F, builds an interpolant Jn[f] in an n-dimensional subspace Xn⊂X with the knowledge of n outputs (σi(f))ni=1, where σi∈X′ and X′ is the dual space of X. The space Xn is built with a greedy algorithm that is \textit{adapted} to F in the sense that it is generated by elements of F itself. The algorithm also selects the linear functionals (σi)ni=1 from a dictionary Σ⊂X′. In this paper, we study the interpolation error maxf∈F∥f−Jn[f]∥X by comparing it with the best possible performance on an n-dimensional space, i.e., the Kolmogorov n-width of F in X, dn(F,X). For polynomial or exponential decay rates of dn(F,X), we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where X is a Hilbert space.
Subjects / Keywords
convergence rates; generalized empirical interpolation; interpolation; reduced basis; reduced order model; empirical interpolation

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