Radial basis function partition of unity methods for scattered data interpolation: A review

Radial basis function (RBF) methods are among the most flexible and accurate meshfree tools for scattered data approximation and numerical solution of partial differential equations (PDEs). One of the main drawbacks of global RBF interpolation is due to the possible presence of dense and ill-conditioned linear systems that arise with large data sets or nearly flat kernels. Localized variants have therefore been developed to improve scalability while retaining high accuracy. In this context, RBF partition of unity (RBF-PU) methods (RBF-PUMs) have emerged as a remarkably effective strategy, since the computational domain is covered by overlapping subdomains or patches, on each of which a local RBF approximant is constructed and blended by compactly supported weight functions forming a partition of unity. This review is primarily devoted to RBF-PU interpolation for scattered data. We first recall the basic theory of global RBF interpolation and classical PUMs. We then present the RBF-PU interpolation framework, including the construction of regular covers, weight functions, local approximants, and the assembly of the global interpolant. Theoretical properties such as convergence rates, stability, conditioning, and the role of the shape parameter and patch radii are discussed. A substantial part of the review is devoted to algorithmic issues: generation of node sets and covers, fast neighbor search, stable evaluation techniques and preconditioning, as well as parameter selection and adaptivity. PDE-oriented RBF-PUMs, including collocation, least squares and direct formulations, are treated as important extensions and applications of the same localization principle. Finally, we summarize recent trends such as adaptive and anisotropic RBF-PU algorithms, data-driven and Bayesian parameter tuning, numerical cubature, graph- and manifold-based extensions, nonlinear and rational PUM formulations, and outline several open problems and future research directions.