The dimension weighted fast multipole method for scattered data approximation
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This talk considers scattered data approximation for high dimensional datasets exhibiting anisotropic behavior across coordinate directions. We tailor sparse polynomial interpolation to exploit this anisotropy and derive corresponding kernel approximations, which we embed in a dimension weighted fast multipole method. The dimension weighting lets the fast multipole method scale to significantly higher input dimensionality than conventional black-box fast multipole methods based on tensor product or total degree interpolation. We present a rigorous analysis of the scheme, including error estimates, and demonstrate its performance in numerical studies assessing accuracy and computational cost. Finally, we illustrate the method’s practical benefits by applying it to interpolate an output functional in a shape uncertainty quantification problem.
