The dimension weighted fast multipole method for scattered data approximation

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.