Interpolation with variably scaled kernels

Abstract

Within kernel-based interpolation and its many applications, the handling of the scaling or the shape parameter is a well-documented but unsolved problem. We consider native spaces whose kernels allow us to change the kernel scale of a d-variate interpolation problem locally, depending on the requirements of the application. The trick is to define a scale function c on the domain Omega aS, R-d to transform an interpolation problem from data locations x(j) in R-d to data locations (x(j), c(x(j))) and to use a fixed-scale kernel on Rd+1 for interpolation there. The (d+1)-variate solution is then evaluated at (x, c(x)) for x a R-d to give a d-variate interpolant with a varying scale. A large number of examples show how this can be done in practice to get results that are better than the fixed-scale technique, with respect to both condition number and error. The background theory coincides with fixed-scale interpolation on the submanifold of Rd+1 given by the points (x, c(x)) of the graph of the scale function c.