Research Group of Dr. T. Ullrich
Institute for Numerical Simulation
Research Seminar

Mathematics of Computation

In cooperation with Prof. Ira Neitzel, Dr. Tino Ullrich and Prof. Markus Bachmayr

Upcoming talks

Thursday, 16.11.2107 10:15 (s.t), We6 6.020 Thomas Hangelbroek

Nonlinear and anisotropic approximation with Gaussians

abstract: For approximation with most positive definite kernels -- perhaps most obviously for Gaussian radial basis functions -- the selection of a scale (by tuning of shape parameters) is challenging, yet exigent in the following sense: for constant scale approximation, high approximation rates are possible, but at the cost of conditioning, while for stationary scaling (fixing the scale to fit a grid) the approximation problem can be made very stable (e.g., with interpolation matrices nearing the identity) but approximation power is lost. For this reason, multi-scale strategies, which use kernels at different dilation levels, have been considered. But this aspect of kernel approximation is relatively new, and its theoretical understanding lags far behind traditional techniques multi-scale tools like wavelets and tensor product splines. In this talk we discuss a pair of nonlinear, multi-scale Gaussian approximation problems. The first, considered with Amos Ron, features approximation by Gaussians at multiple, spatially varying scales, and provides correct rates for functions in standard smoothness spaces for nonlinear approxiation. The second, recently considered with Amos Ron and Wolfgang Erb, treats $N$-term Gaussian approximation of cartoon class functions with rates comparable to those of curvelets and shearlets.

Click here for a list of talks given in the past.