Kolmogorov widths under holomorphic mappings
Résumé
If $L$ is a bounded linear operator mapping the Banach space $X$ into the Banach space $Y$ and $K$ is a compact set
in $X$, then the Kolmogorov widths of the image $L(K)$ do not exceed those of $K$ multiplied by
the norm of $L$. We extend this result from linear maps
to holomorphic mappings $u$ from $X$ to $Y$ in the following sense:
when the $n$ widths of
$K$ are $O(n^{-r})$ for some $r>1$, then those of
$u(K)$ are $O(n^{-s})$ for any $s < r-1$,
We then use these results to prove various theorems about Kolmogorov widths of
manifolds consisting of solutions to certain parametrized PDEs.
Results of this type are important in the numerical analysis of reduced
bases and other reduced modeling methods, since the best possible performance
of such methods is governed by the rate of decay of the
Kolmogorov widths of the solution manifold.
Domaines
Analyse numérique [cs.NA]
Origine : Fichiers produits par l'(les) auteur(s)