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.

翻译：本文关注具有各维度各向异性行为的高维数据集的散乱数据逼近问题。通过将稀疏多项式插值适配于这种特定情形，我们推导出高效的退化核近似，并将其应用于维度加权快速多极子方法。相比基于插值的标准黑箱快速多极子方法，该维度加权快速多极子方法能够处理更多维度。本文提供了方法的详尽分析，包括严格的误差估计。大量数值结果验证了该方法的精度与计算成本。作为重要应用，我们将该方法应用于形状不确定性量化问题。