Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Jürgen Prestin
Universität Lübeck
Quadrature rules for scattered data on spherical triangles
In this talk we present the construction of quadrature rules on arbitrary triangulations of the sphere
which are exact for polynomials of some fixed degree. In the first part we study quadrature on some
preassigned nodes so that we are able to compute integrals over triangles for arbitrary polynomials.
In a second part we apply Cholesky decomposition methods to obtain the weights for scattered data.
For our numerical tests we used Mathematica where we carried out all calculations in high accuracy or
even with exact numbers. So we were able to overcome a lot of instability problems particularly for very
small and thin triangles. Finally, we compare our local quadrature rules on triangulations and some small
polynomial degree of exactness with global formulas on the whole sphere and high degree of polynomial
exactness. Particularly, for clustered data the local methods seem to be better. This is joint work with
Judith Beckmann (University of Lübeck) and Hrushikesh N. Mhaskar (California State University).
Holger Rauhut
Universität Bonn
Recovery of functions in high dimensions via compressive sensing
Compressive sensing predicts that sparse vectors can be recovered efficiently from highly undersampled
measurements. It is known in particular that multivariate sparse trigonometric polynomials can be recovered
from a small number of random samples. Classical methods for recovering functions in high spatial
dimensions usually suffer the curse of dimension, that is, the number of samples scales exponentially in
the dimension (the number of variables of the function). We introduce a new model of functions in high
dimensions that uses “sparsity with respect to dimensions”. More precisely, we assume that the function
is very smooth in most of the variables, and is allowed to be rather rough in only a small but unknown set
of variables. This translates into a certain sparsity model on the Fourier coefficients. Using techniques
from compressive sensing, we are able to recover functions in this model class efficiently from a small
number of samples. In particular, this number scales only logarithmically in the spatial dimension - in
contrast to the exponential scaling in classical methods.
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