Sparsity and Computation

Monday, 10.10-11.00 How can we obtain tractability? Erich Novak University of Jena We study numerical problems, like integration and approximation, defined for classes Fd of functions of d variables, say f : [0, 1] d → R for f ∈ Fd. The problem is polynomially tractable if the computing time is bounded by Cdqε−p for all d ∈ N, where ε > 0 is the error bound. We start with an example that shows that the order of convergence can be excellent and still the problem is not polynomially tractable. Hence we have to reconsider most of the classical error bounds in numerical analysis. We deal mainly with the question of how we can obtain tractability. • Can tractability be obtained by strong smoothness assumptions? • Can tractability be obtained by sparsity, finite order weights or special structure? • Can tractability be obtained by randomization? (The presentation is based on joint work with Henryk Woźniakowski, in particular our book “Tractability of Multivariate Problems”, European Math. Society. Volume I appeared in 2008, Volume II will appear in 2010. ) Monday, 11.30-12.00 Geometric separation by single-pass alternating thresholding Gitta Kutyniok University of Osnabrück Modern data is customarily of multimodal nature, and analysis tasks typically require separation into the single components such as, for instance, in neurobiological imaging a separation into spines (pointlike structures) and dendrites (curvilinear structures). Although a highly ill-posed problem, inspiring empirical results show that the morphological difference of these components sometimes allows a very precise separation.