Sampling and stability

Authors:
Christian Rieger;Robert Schaback;Barbara Zwicknagl
Affiliations:
Institut für numerische Simulation Hausdorff-Zentrum für Mathematik, Universität Bonn, Bonn, Germany;Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, Germany;Institut für Angewandte Mathematik Hausdorff-Zentrum für Mathematik, Universität Bonn, Bonn, Germany
Venue:
MMCS'08 Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces
Year:
2008

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Abstract

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.