Anisotropic weights for RBF-PU interpolation with subdomains of variable shapes

Abstract: The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has already been proved to be an effective tool for solving interpolation or collocation problems when large data sets are considered. It decomposes the original domain into several subdomains or patches so that only linear systems of relatively small size need to be solved. In research on such partition of unity methods, such subdomains usually consist of spherical patches of a fixed radius. However, for particular data sets, such as track data, ellipsoidal patches seem to be more suitable. Therefore, in this paper, we propose a scheme based on a priori error estimates for selecting the sizes of such variable ellipsoidal subdomains. We jointly solve for both these domain decomposition parameters and the anisotropic RBF shape parameters on each subdomain to achieve superior accuracy in comparison to the standard partition of unity method.