The dimension weighted fast multipole method for scattered data approximation

This article is concerned with scattered data approximation for higher dimensional data sets which exhibit an anisotropic behavior with respect to their different dimensions. Tailoring sparse polynomial interpolation to this specific situation, we derive degenerate kernel approximations which are integrated into a dimension weighted fast multipole method. This dimension weighted fast multipole method enables to deal with considerably more dimensions than the standard black box fast multipole method based on tensor product or total degree interpolation. A thorough analysis of the method is provided including rigorous error estimates. The accuracy and the cost of the approach are validated by numerical studies. As a relevant application, we apply the approach to the interpolation of an output functional of a shape uncertainty quantification problem.