Partition of Unity Interpolation on Multivariate Convex Domains

TL;DR

This paper introduces a new multivariate interpolation algorithm for scattered data in convex domains using kd-tree partitioning, local radial basis functions, and compact weights, with demonstrated efficiency on various datasets.

Contribution

It presents a novel algorithm combining kd-tree partitioning with partition of unity and radial basis functions for interpolation in convex domains.

Findings

Efficient interpolation on various convex domains.

Good performance on Halton data points.

Complexity analysis supports scalability.

Abstract

In this paper we present a new algorithm for multivariate interpolation of scattered data sets lying in convex domains $\Omega \subseteq \RR^N$, for any $N \geq 2$. To organize the points in a multidimensional space, we build a $kd$-tree space-partitioning data structure, which is used to efficiently apply a partition of unity interpolant. This global scheme is combined with local radial basis function approximants and compactly supported weight functions. A detailed description of the algorithm for convex domains and a complexity analysis of the computational procedures are also considered. Several numerical experiments show the performances of the interpolation algorithm on various sets of Halton data points contained in $\Omega$, where $\Omega$ can be any convex domain like a 2D polygon or a 3D polyhedron.