Data-dependent approximation through RBF

Abstract: In this article we present a modification of classical Radial Basis Function (RBF) interpolation techniques aimed at reducing oscillations near discontinuities in one and two dimensions. Our approach introduces an adaptive mechanism by varying the shape parameter of the RBFs and making it data-dependent, forcing it to tend to infinity in the vicinity of discontinuities. This modification results in kernel functions that locally resemble %Kronecker delta functions, effectively minimizing spurious oscillations. To detect discontinuities, we employ smoothness indicators: for grid-based data, these are computed as undivided second-order differences squared. For scattered data, we use least squares approximations of the Laplacian multiplied by the square of the mean local separation of the stencil points, and then squared. These indicators guide the adaptive adjustment of the shape parameter. We prove the invertibility of the resulting interpolation matrix and propose a solution strategy that maintains the condition number comparable to that of a system where points near discontinuities are excluded. Numerical experiments in one and two dimensions demonstrate that the proposed method significantly reduces oscillations near discontinuities across various kernel types, whether locally or globally supported. At the same time, the interpolation accuracy and matrix conditioning in smooth regions remain essentially unchanged, as measured by the infinity norm of the error and the condition number.