"Rate-optimal sparse approximation of compact break-of-scale embeddings"Hübner, Janina\noindent Approximation theory tells us that the expected rate of convergence of numerical methods is closely related to the regularity of the object we want to approximate. Besides classical \emph{isotropic} Sobolev smoothness, the notion of \emph{dominating mixed regularity} of functions turned out to be an important concept in numerical analysis. Although the theory of embeddings within those scales seems to be well-understood, not that much is known about \emph{break-of-scale embeddings}.\\ In the talk we define new function spaces of Besov- and Triebel-Lizorkin-type with hybrid smoothness, that include both scales of function spaces discussed above. We present some embeddings and construct explicit (non-)linear algorithms based on hyperbolic wavelets that yield sharp dimension-independent rates of convergence.\\ The poster is based on recent joint work [1] with Glenn Byrenheid and Markus Weimar.\\ \noindent References:\\ $[1]$ G. Byrenheid, J. Hübner, and M. Weimar. \emph{Rate-optimal sparse approximation of compact break-of-scale embeddings}, preprint arXiv:2203.10011, (2022). |
« back