A sharp upper bound for sampling numbers in L_2

Matthieu Dolbeault (Sorbonne U, Paris)

May 10. 2022, 13:50 — 14:40

For a class F of complex-valued functions on a set D, we define its sampling numbers g_n

as the minimal worst-case error on F, measured in L_2, that can be achieved with an optimal algorithm

based on n function evaluations. We prove that there is a universal constant > 0 such that

the sampling numbers of the unit ball F of every separable reproducing kernel Hilbert space are bounded by

the tails of the the Kolmogorov widths (or approximation numbers) of F in L_2.

We also obtain similar upper bounds for more general classes F, including compact subsets of

the space of continuous functions on a bounded domain D.

Moreover, we provide examples where these bounds are attained up to a constant.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Computational Uncertainty Quantification: Mathematical Foundations, Methodology & Data (Thematic Programme)
Organizer(s):
Clemens Heitzinger (TU Vienna)
Fabio Nobile (EPFL Lausanne)
Robert Scheichl (U Heidelberg)
Christoph Schwab (ETH Zürich)
Sara van de Geer (ETH Zürich)
Karen Willcox (U of Texas, Austin)