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Article Dans Une Revue SIAM Journal on Numerical Analysis Année : 2016

Convergence analysis of the Generalized Empirical Interpolation Method

Résumé

Let $F$ be a compact set of a Banach space $\mathcal{X}$. This paper analyses the ``Generalized Empirical Interpolation Method'' (GEIM) which, given a function $f\in F$, builds an interpolant $\mathcal{J}_n[f]$ in an $n$-dimensional subspace $X_n \subset \mathcal{X}$ with the knowledge of $n$ outputs $(\sigma_i(f))_{i=1}^n$, where $\sigma_i\in \mathcal{X}'$ and $\mathcal{X}'$ is the dual space of $\mathcal{X}$. The space $X_n$ 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 $(\sigma_i)_{i=1}^n$ from a dictionary $\Sigma\subset \mathcal{X}'$. In this paper, we study the interpolation error $\max_{f\in F} \Vert f-\mathcal{J}_n[f]\Vert_{\mathcal{X}}$ by comparing it with the best possible performance on an $n$-dimensional space, i.e., the Kolmogorov $n$-width of $F$ in $\mathcal{X}$, $d_n(F,\mathcal{X})$. For polynomial or exponential decay rates of $d_n(F,\mathcal{X})$, 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 $\mathcal X$ is a Hilbert space.
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Dates et versions

hal-01032458 , version 1 (22-07-2014)
hal-01032458 , version 2 (26-05-2016)

Identifiants

Citer

Yvon Maday, Olga Mula, Gabriel Turinici. Convergence analysis of the Generalized Empirical Interpolation Method. SIAM Journal on Numerical Analysis, 2016, 54 (3), pp.1713-1731. ⟨10.1137/140978843⟩. ⟨hal-01032458v2⟩
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