Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation

作     者：Albrecht, Kristof Entzian, Juliane Iske, Armin 

作者机构：Univ Hamburg Dept Math Hamburg Germany 

出 版 物：《ADVANCES IN COMPUTATIONAL MATHEMATICS》 (Adv. Comput. Math.)

年 卷 期：2025年第51卷第2期

页      面：1-27页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Universitt Hamburg (1037) [GRK 2583] Deutsche Forschungsgemeinschaft (DFG) 

主　　题：Kernel-based approximation Product kernels Greedy algorithms 

摘      要：This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (Trans. Am. Math. Soc. 68, 337-404 1950), the product of positive semi-definite kernel functions is again positive semi-definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for grid-like samples. We use this case to show that the product of positive definite kernel functions is again positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.