Barbara Zwicknagl

Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning

Sampling inequalities give a precise formulation of the fact that a differentiable function cannot attain large values if its derivatives are bounded and if it is small on a sufficiently dense discrete set. Sampling inequalities can be applied to the difference of a function and its reconstruction in order to obtain (sometimes optimal) convergence orders for very general possibly regularized recovery processes. So far, there are only sampling inequalities for finitely smooth functions, which lead to algebraic convergence orders. In this paper, the case of infinitely smooth functions is investigated, in order to derive error estimates with exponential convergence orders.

Sampling and stability

In Numerical Analysis one often has to conclude that an error function is small everywhere if it is small on a large discrete point set and if there is a bound on a derivative. Sampling inequalities put this onto a solid mathematical basis. A stability inequality is similar, but holds only on a finite–dimensional space of trial functions. It allows bounding a trial function by a norm on a sufficiently fine data sample, without any bound on a high derivative. This survey first describes these two types of inequalities in general and shows how to derive a stability inequality from a sampling inequality plus an inverse inequality on a finite–dimensional trial space. Then the state–of–the–art in sampling inequalities is reviewed, and new extensions involving functions of infinite smoothness and sampling operators using weak data are presented. Finally, typical applications of sampling and stability inequalities for recovery of functions from scattered weak or strong data are surveyed. These include Support Vector Machines and unsymmetric methods for solving partial differential equations.

Multiscale approximation and reproducing kernel Hilbert space methods

We consider reproducing kernels

Scaling Law and Reduced Models for Epitaxially Strained Crystalline Films

K:\Omega\times \Omega \to \mathbb{R}

Full length article: Interpolation and approximation in Taylor spaces

in multiscale series expansion form, i.e., kernels of the form

Study of Island Formation in Epitaxially Strained Films on Unbounded Domains

K\left(\boldsymbol{x},\boldsymbol{y}\right)=\sum_{\ell\in\mathbb{N}}\lambda_\ell\sum_{j\in I_\ell}\phi_{\ell,j}\left(\boldsymbol{x}\right)\phi_{\ell,j}\left(\boldsymbol{y}\right)

Low volume-fraction microstructures in martensites and crystal plasticity

with weights

Domain Formation in Membranes Near the Onset of Instability

\lambda_\ell

Regularized Kernel-Based Reconstruction in Generalized Besov Spaces

and structurally simple basis functions

Deformation concentration for martensitic microstructures in the limit of low volume fraction

\left\{\phi_{\ell,i}\right\}