Radial functions depend only on the distance from the center of the domain, for example, Gaussians. Interpolations using radial functions of the same shape lead to a non-singular coefficient matrix. Unfortunately, when using adaptive mesh, the shape of radial functions is not the same. This in turn leads to an ill-conditioned system. The authors give sufficient conditions to guarantee a diagonally dominant interpolation matrix when using radial functions with different shapes.

A radial function for which the first derivative is negative and the second derivative is non-negative is called convex. The authors focus on Wendland functions, a set of compactly supported radial basis functions (RBFs) (see Wendland [1]).

They show that the interpolation matrix is non-singular under a local geometric property of the neighborhood of each center. The interpolation matrices guarantee incomplete lower upper (LU) factorization. The authors give MATLAB code to construct the interpolation matrix with differently scaled RBFs. They give several examples in 2D and 3D surface reconstruction (see also Morse et al. [2]). The algorithm was demonstrated to be efficient for large-scale problems but not very accurate. Error estimates were not given.