Interpolating functions with gradient discontinuities via Variably Scaled Kernels

In kernel–based methods, how to handle the scaling or the choice of the shape parameter is a well– documented but still an open problem. The shape or scale parameter can be tuned by the user according to the applications, and it plays a crucial role both for the accuracy of the method and its stability. In [7], the Variably Scaled Kernels (VSKs) were introduced. The idea is to vary the scale in a continuous way by letting the scale parameter be an additional coordinate. In this way a scale function c(x) is introduced and this allows varying scales without leaving the firm grounds of kernel–based interpolation. In [7] several numerical experiments are devoted to show that the method performs better than the standard fixed–scale basis and the proper use of VSKs, i.e. a proper choice of the scale function c, can lead to a more stable and better shape–preserving approximation. When the scale function c is chosen to depend on critical shape properties of the data, the interpolant reproduces the underlying phenomenon in a much more faithful way. This is the starting point of this paper. The goal here is to recover from scattered data, in one and two dimensions, functions with discontinuities. The paper is devoted to gradient discontinuities that in general are more difficult to handle. The idea is to use translates of basis functions that change their smoothness locally according to the position of the discontinuities and we show that this can be achieved by suitable choices of the scale function.