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UID:news-1562@dmi.unibas.ch
DTSTAMP:20230926T153730
DTSTART;TZID=Europe/Zurich:20231013T110000
DTEND;TZID=Europe/Zurich:20231013T120000
SUMMARY:Seminar in Numerical Analysis: Markus Weimar (Julius-Maximilians-Universität Würzburg)
LOCATION:
DESCRIPTION:As a rule of thumb in approximation theory, the asymptotic spee
d of convergence of numerical algorithms is governed by the reg
ularity of the objects we like to approximate. Besides classica
l isotropic Sobolev smoothness, in the last decades the notion
of so-called dominating- mixed regularity of functions turned o
ut to be an important concept in numerical analysis. Indeed, it
naturally arises in high-dimensional real-world applications,
e.g., related to the electronic Schrödinger equation. Although
optimal approximation rates for embeddings within the scales
of isotropic or dominating-mixed Lp-Sobolev spaces are well-u
nderstood, not that much is known for embeddings across those
scales (break-of-scale embeddings).\r\nIn this lecture, we fir
st review the Fourier analytic approach towards by now well-est
ablished (Besov and Triebel-Lizorkin) scales of distribution sp
aces that measure either isotropic or dominating-mixed regula
rity. In addition, we introduce new function spaces of hybrid s
moothness which are able to simultaneously capture both types
of regularity at the same time. As a further generalization of
the aforementioned scales, they particularly include standard
Sobolev spaces on domains. On the other hand, our new spaces yi
eld an appropri- ate framework to study break-of-scale embeddin
gs by means of harmonic analysis. We shall present (non-)adapti
ve wavelet-based multiscale algorithms that approximate such em
bed- dings at optimal dimension-independent rates of convergenc
e. Important special cases cover the approximation of functions
having dominating-mixed Sobolev smoothness w.r.t. Lp in the
norm of the (isotropic) energy space H1.\r\nThe talk is based
on a recent paper [1] which represents the first part of a join
t work with Glenn Byrenheid (FSU Jena), Markus Hansen (PU Marbu
rg), and Janina Hübner (RU Bochum).\r\nReferences:\r\n[1] G. B
yrenheid, J. Hübner, and M. Weimar. Rate-optimal sparse approx
imation of compact break-of-scale embeddings. Appl. Comput. Har
mon. Anal. 65:40–66, 2023 (arXiv:2203.10011).\r\nFor further
information about the seminar, please visit this webpage [http
s://dmi.unibas.ch/de/forschung/mathematik/seminar-in-numerical-
analysis/].
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