Nodal Representations for Kernel-Based Multilevel Interpolation

Abstract

We study the kernel-based multilevel method for approximating or learning a multivariate function from scattered data, motivated in part by recent applications in sparse grid methods. For nested families of point sets, we derive a nodal representation of the multilevel interpolant that depends explicitly on the values of the target function. This representation allows us to characterize the range of the associated interpolation operator and to construct a cardinal basis for this space. We prove that the resulting basis functions exhibit exponential decay, analogous to the localization properties known for certain kernel-based Lagrange functions. We further analyze the computational cost of the resulting formulation. For non-nested families of point sets, we derive a generalized nodal representation.