Session I.1 - Multiresolution and Adaptivity in Numerical PDEs
Poster
Rate-Optimal Sparse Approximation of Compact Break-of-Scale Embeddings
Markus Weimar
JMU Wuerzburg, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
The poster addresses the approximation problem of functions in new scales of function spaces with hybrid smoothness. In these scales we combine classical (isotropic) regularity measured in $L_p$ with so-called dominating mixed smoothness which arises in high-dimemsional real-world applications, e.g., related to the electronic Schrödinger equation. Sharp dimension-independent rates of convergence for linear and nonlinear best approximations using $n$ hyperbolic wavelets are presented. Important special cases include the approximation of function having dominating mixed smoothness w.r.t. $L_p$ in the norm of the isotropic energy space $H^1$.
The presented results are based on a recent paper [1] which represents the first part of a long term research project.
[1] G. Byrenheid, J. Hübner, and M. Weimar. Rate-optimal sparse approximation of compact break-of-scale embeddings. Appl. Comput. Harmon. Anal. 65:40--66, 2023 (arXiv:2203.10011).
Joint work with Glenn Byrenheid (FSU Jena, Germany), Janina Hübner (RUB, Germany) and Markus Hansen (PU Marburg, Germany).