A multiresolution prewavelet-based adaptive refinement scheme for RBF approximations of nearly singular problems

A common difficulty in applying radial basis function (RBF) methods to nearly singular problems is either convergence failure or a very slow convergence rate. Motivated by the close connection between RBFs and wavelets, we have demonstrated the efficiency of adaptive distributions based on multiresolution wavelet decomposition for the RBF approximation of nearly singular problems. RBFs are prewavelets per se, RBFs are not orthonormalized. Wavelet decomposition provides a firm mathematical base for projecting a complicated function into a nested sequence of subspaces each of which has a different level of resolution. So, one can analyze the highly localized feature regions of a function at a high level of resolution and simultaneously use a low level of resolution for analyzing the function at flat regions. We also utilize the variable shape parameter scheme to prevent the growth of the well-known ill-conditioning problem in globally supported RBFs. The performance of the proposed method is illustrated in one- and two-dimensional numerical examples with internal or boundary near singularities. Our numerical results show that super convergent behavior can be achieved if the adaptive scheme is utilized to produce the compressed distribution.