A Trivariate Interpolation Algorithm Using a Cube-Partition Searching Procedure

Abstract

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.

Partition of unity interpolation on multivariate convex domains

“…Here, Ω is any convex domain like a 2D polygon (e.g., a triangle or a hexagon) or a 3D polyhedron (e.g., a pyramid or a cylinder). Note that this algorithm for convex hulls extends our previous works on the topic [3,4,5,6].…”

OpenCL Based Parallel Algorithm for RBF-PUM Interpolation

“…A change of this type may influence more or less significantly the approximation results in term of both accuracy and stability. In fact, a cover with small (large) partitions results in worse (better) approximations, even if it turns out to be computationally cheaper (more expensive); for further details we refer to [7,17].…”

“…The basic idea of PUM consists of decomposing the domain into several sub-domains forming a covering of the original domain, then constructing a local RBF interpolant on each sub-domain. Originally, the RBF-PUM (also known as RBF-PU method) has been introduced in the context of PDEs [2,25], but now it is also an effective and very used tool in the field of approximation theory and its applications [7,8,9,22,36,39].…”

Two and Three Dimensional Partition of Unity Interpolation by Product-Type Functions

“…They are characterized by the partition of the domain Ω in square or cube cells, enabling us to use efficient searching procedures. The basic versions of these interpolation algorithms have been proposed and widely tested in [15,16]. Here, such algorithms have been modified and efficiently updated to locally apply the product-type interpolants.…”