  The present article is concerned scattered data approximation for higher
dimensional data sets which exhibit an anisotropic behavior in the different
dimensions. Tailoring sparse polynomial interpolation to this specific
situation, we derive very efficient degenerate kernel approximations which we
then use in a dimension weighted fast multipole method. This dimension weighted
fast multipole method enables to deal with many more dimensions than the
standard black-box fast multipole method based on interpolation. A thorough
analysis of the method is provided including rigorous error estimates. The
accuracy and the cost of the approach are validated by extensive numerical
results. As a relevant application, we apply the approach to a shape
uncertainty quantification problem.
