. Workshop on Function Approximation December 1 , 2016 – December 2 , 2016

s – Lectures Thursday, December 1, 9:00-10:30 Lecture 1: Spline approximation Stefan Takacs RICAM, Computational Methods for PDEs Thomas Takacs JKU Linz, Institute of Applied Geometry In the first part of the lecture we discuss standard approximation properties for B-spline and NURBS spaces. B-splines are piecewise polynomial functions of a given prescribed smoothness. NURBS stands for non-uniform rational B-splines and denotes rational functions, where both the numerator and the denominator are B-splines. We consider approximation error bounds for functions in Sobolev spaces. We start with classical approximation properties for univariate functions and then extend the results to tensor-product spaces as well as to mapped NUBRS spaces, so called isogeometric function spaces. The presented results determine the convergence properties of isogeometric methods, a family of numerical methods which use isogeometric functions for the solution of partial differential equations. In the second part of the lecture we present improved approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness given in standard Sobolev norms and semi-norms. The presented estimates do not depend on the polynomial degree of the splines but only on the grid size. We will see that the approximation lives in a subspace of the classical B-spline space. We show that for this subspace, there is an inverse inequality which is also independent of the polynomial degree. As the approximation error estimate and the inverse inequality show complementary behavior, the results can be used to construct fast iterative methods for solving problems arising from isogeometric discretizations of partial differential equations. Thursday, December 1, 11:00-12:30 Lecture 2: Multivariate approximation Christian Irrgeher RICAM, Multivariate Algorithms and Quasi-Monte Carlo Methods Peter Kritzer RICAM, Multivariate Algorithms and Quasi-Monte Carlo Methods We start the lecture by introducing the general setting, which is considered in the field of informationbased complexity. That is, we consider operators S : F → G between two (function) spaces F,G and we study algorithms to approximate such operators S. These algorithms can be categorized with respect to the sort of information which they use. To analyze the “quality” of an algorithm we can consider different settings, e.g. the worst-case setting or the average-case setting. In this lecture we mainly focus on the worst-case setting. To this end, we give the definitions of the relevant errors as well as of the information complexity. Furthermore, we introduce various notions of tractability which help to figure out the dependence on the dimension. Then we present some general properties of linear problems which show that it is sufficient to restrict oneself to a simple class of algorithms to achieve optimal convergence results. Moreover, we give a general result on optimal algorithms for the Hilbert space case. In the second part of the lecture we turn to multivariate L2-approximation in the Korobov space. The Korobov space is a Sobolev-type space of periodic functions. We show that, in general,