Many applications in engineering, science, and statistics require inter- or extrapolation from data and involve systems that contain a huge amount of parameters. Generic examples are computer-based simulations, data mining, or forecasting. Indispensable foundation in every such situation is a mathematical or statistical model. It allows to represent the underlying real-world phenomena - or at least some simplification thereof - in a way suitable for computation and mathematical analysis. The model formulation very often involves multivariate functions f(x1, ...,xd) where the dimension d may be very large. The problem of inter- and extrapolation then is to find a function which fits the given data in a suitable sense. Problems of this kind are so ubiquitous that several mathematical disciplines are devoted to them. Each uses its own language: in approximation theory and numerical analysis, the terms function identification, function recovery, and function reconstruction are common. Statisticians speak of regression, function estimation or function fitting. To “learn a function” is also a widely used phrase in artificial intelligence, machine learning and statistical learning theory.

A special focus of this workshop will be on connections of (classical) approximation theory (AT), asymptotic geometric analysis (AGA) and (high-dimensional) probability theory (HDP), and information-based complexity (IBC). The latter three are relatively young areas and until now AGA & HDP and IBC are essentially independent branches of mathematics dealing with high-dimensional problems, while AT may be seen as a common ancestor and often serves as a source of motivation, mathematical techniques and open problems for the other areas. The goal is to explore further the relations between those fields, in particular, to high-dimensional numerical integration, dispersion of point sets in high-dimensions, and the probabilistic behavior of lacunary systems. Very recent results reveal the deep impact of methods from AGA & HDP in IBC, a connection that is promising, exciting, and shall be further pursued during this workshop.

The conference is a successor of "Approximation and Geometry in High Dimensions," that was held 16-22.08.2022 in Będlewo (https://www.impan.pl/en/activities/banach-center/conferences/22-approximation).

A special one-day memorial session dedicated to Albrecht Pietsch, who recently passed away, will take place during the conference. It is organized by Stefan Heinrich and Thomas Kühn.

Note: The conference is by initation only.