Optimal pointwise sampling for $L^2$ approximation - Université Paris Cité Accéder directement au contenu
Article Dans Une Revue Journal of Complexity Année : 2022

Optimal pointwise sampling for $L^2$ approximation

Albert Cohen
  • Fonction : Auteur
Matthieu Dolbeault

Résumé

Given a function $u\in L^2=L^2(D,\mu)$, where $D\subset \mathbb R^d$ and $\mu$ is a measure on $D$, and a linear subspace $V_n\subset L^2$ of dimension $n$, we show that near-best approximation of $u$ in $V_n$ can be computed from a near-optimal budget of $Cn$ pointwise evaluations of $u$, with $C>1$ a universal constant. The sampling points are drawn according to some random distribution, the approximation is computed by a weighted least-squares method, and the error is assessed in expected $L^2$ norm. This result improves on the results in [6,8] which require a sampling budget that is sub-optimal by a logarithmic factor, thanks to a sparsification strategy introduced in [17,18]. As a consequence, we obtain for any compact class $\mathcal K\subset L^2$ that the sampling number $\rho_{Cn}^{\rm rand}(\mathcal K)_{L^2}$ in the randomized setting is dominated by the Kolmogorov $n$-width $d_n(\mathcal K)_{L^2}$. While our result shows the existence of a randomized sampling with such near-optimal properties, we discuss remaining issues concerning its generation by a computationally efficient algorithm.

Mots clés

Fichier principal
Vignette du fichier
CD2021.pdf (232.93 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03221181 , version 1 (07-05-2021)

Licence

Paternité

Identifiants

  • HAL Id : hal-03221181 , version 1

Citer

Albert Cohen, Matthieu Dolbeault. Optimal pointwise sampling for $L^2$ approximation. Journal of Complexity, 2022, 68, pp.101602. ⟨hal-03221181⟩
72 Consultations
259 Téléchargements

Partager

Gmail Facebook X LinkedIn More