# Workshop: Discrepancy, Numerical Integration and Hyperbolic Cross Approximation

**Date:** September 23-27, 2013

**Venue:** Hausdorff Center for Mathematics, University of Bonn

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Organizers: Tino Ullrich and Vladimir Temlyakov

Brief Description:

Many multi-parameter real-world problems are modeled on spaces of multivariate functions in d variables, where d may be very large. Since these problems can almost never be solved analytically, one is interested in suitable model assumptions and good approximate solutions within a reasonable computing time. Already more than fifty years ago classes of functions with bounded mixed derivative had been used in numerical integration and approximation for the first time. In this context numerical integration dealt with two powerful methods, Quasi Monte Carlo Method and Sparse Grids. Approximation methods used polynomials with frequencies from hyperbolic crosses. It was understood later that classes of functions with bounded mixed derivative appear naturally in discrepancy, the small ball problem from probability theory and in problems involving Wiener sheet measure. Furthermore, it has been shown recently that the wave functions of the electronic SchrÃ¶dinger equation possess a bounded mixed derivative. As a consequence, the by now classical model of function spaces with dominating mixed smoothness attracted more and more interest among researchers from different fields. So far, the theory of hyperbolic cross approximation and its applications has developed into a beautiful practically useful theory which still has a number of important open problems to work on. In fact, the basic ideas in approximating and integrating such functions have been mainly developed in the former Soviet Union during the last fifty years. However, many remarkable and highly important theoretical results in this context remained unrecognized in the western scientific community. Since the Institute for Numerical Simulation in Bonn is one of the main centers for the design of Sparse Grid methods in scientific computing, an exchange of ideas is highly appreciated. The proposed workshop aims at bringing together experts from different communities to learn new approaches and techniques from each other. In particular, we aim at identifying possible new research directions and methods that may arise from the combination of different expertises.