In this paper we propose a new efficient interpolation tool, extremely suitable for large scattered data sets. The partition of unity method is used and performed by blending Radial Basis Functions (RBFs) as local approximants and using locally supported weight functions. In particular we present a new space-partitioning data structure based on a partition of the underlying generic domain in blocks. This approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points, leading to an optimized searching routine. Complexity analysis and numerical experiments in two-and three-dimensional interpolation support our findings. Some applications to geometric modelling are also considered. Moreover, the associated software package written in Matlab is here discussed and made available to the scientific community.

Abstract. In this paper we propose a fast algorithm for trivariate interpolation, which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube-partition searching procedure. More precisely, we construct a cube structure, which partitions the domain and strictly depends on the size of its subdomains, so that the new searching procedure and, accordingly, the resulting algorithm enable us to efficiently deal with a large number of nodes. Complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm.

Abstract. 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.

Kernel-based approximation methods-often in the form of radial basis functions-have been used for many years now and usually involve setting up a kernel matrix which may be ill-conditioned when the shape parameter of the kernel takes on extreme values, i.e., makes the kernel "flat". In this paper we present an algorithm we refer to as the Hilbert-Schmidt SVD and use it to emphasize two important points which-while not entirely new-present a paradigm shift under way in the practical application of kernel-based approximation methods: (i) it is not necessary to form the kernel matrix (in fact, it might even be a bad idea to do so), and (ii) it is not necessary to know the kernel in closed form. While the Hilbert-Schmidt SVD and its two implications apply to general positive definite kernels, we introduce in this paper a class of so-called iterated Brownian bridge kernels which allow us to keep the discussion as simple and accessible as possible.

a b s t r a c tIn this paper a new efficient algorithm for spherical interpolation of large scattered data sets is presented. The solution method is local and involves a modified spherical Shepard's interpolant, which uses zonal basis functions as local approximants. The associated algorithm is implemented and optimized by applying a nearest neighbour searching procedure on the sphere. Specifically, this technique is mainly based on the partition of the sphere in a suitable number of spherical zones, the construction of spherical caps as local neighbourhoods for each node, and finally the employment of a spherical zone searching procedure. Computational cost and storage requirements of the spherical algorithm are analyzed. Moreover, several numerical results show the good accuracy of the method and the high efficiency of the proposed algorithm.