Anisotropic Weights for RBF-PU Interpolation with Subdomains of Variable Shapes
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.
This research has been accomplished within RITA (Rete ITaliana di Approssimazione) and partially supported by GNCS-INδAM. The first and second authors were partially supported by the 2016–2017 project Metodi numerici e computazionali per le scienze applicate of the Department of Mathematics of the University of Torino. The third author was partially supported by grant NSF-DMS #1522687. The last author is supported by the research project No. BIRD167404.
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