Optimal selection of local approximants in RBF-PU interpolation
This work provides a practical solution for tuning hyperparameters in RBF-PU interpolation, benefiting researchers and practitioners in scattered data interpolation.
The paper addresses the optimal selection of local subdomain size and shape parameter in RBF-PU interpolation to improve accuracy. The proposed method, based on minimizing an a priori error estimate, achieves high accuracy even for non-homogeneous data, as demonstrated by extensive numerical experiments.
The Partition of Unity (PU) method, performed with local Radial Basis Function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered.