Convergence analysis of the Generalized Empirical Interpolation Method
| Autor: | Maday, Y., Mula, O., Turinici, G. |
|---|---|
| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | SIAM J. Numer. Anal., 2017, 54-3, 1713-1731 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1137/140978843 |
| Popis: | 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 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. Comment: arXiv admin note: text overlap with arXiv:1204.2290 by other authors |
| Databáze: | arXiv |
| Externí odkaz: |
|