Partition of unity interpolation on multivariate convex domains

In this paper, we present an algorithm for multivariate interpolation of scattered data sets lying in convex domains [Formula: see text], for any [Formula: see text]. To organize the points in a multidimensional space, we build a [Formula: see text]-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 (RBF) 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 [Formula: see text], where [Formula: see text] can be any convex domain, like a 2D polygon or a 3D polyhedron. Finally, an application to topographical data contained in a pentagonal domain is presented.

Positive Approximation and Interpolation Using Compactly Supported Radial Basis Functions

We discuss the problem of constrained approximation and interpolation of scattered data by using compactly supported radial basis functions, subjected to the constraint of preserving positivity. The approaches are presented to compute positive approximation and interpolation by solving the two corresponding optimization problems. Numerical experiments are provided to illustrate that the proposed method is flexible.

A New Fast Iterative Method for Interpolation of Multivariate Scattered Data

This paper presents Iterative Scalable Smoothing (ISS), a new itera- tive multi-scale method for multivariate interpolation of scattered data. Each iteration step in the process reduces the residues of the current interpolation result by appli- cation of a smoothing operator to a piecewise constant function that interpolates the residues of the current interpolant, and by adding the resulting function to the current approximation, which is initially set to zero. The convergence of the method is proved and conditions for the di®erentiability of the convergence result are given. For a uni- form mesh an e±cient algorithm is constructed, for which the numerical complexity is estimated. Several 2D numerical examples illustrate the theoretical results. By a 3D test with several test-functions and random nodes it is shown that the accuracy of the proposed method is comparable with the quadratic modiffcation of Shepard's method, which is known to be more accurate than triangle-based methods. Then, in 1D tests, by stochastic simulation with random nodes and random functions the ISS method is compared with the Cubic Splines method, Shepard's method and the Kriging method. We also compare the stability of these methods with respect to noisy data. For the special case of regular nodes, properties of the method are verified by comparing its 1D response function with the response function of the cubic spline and the perfect interpolator (the sinc-function). Special attention is paid to the e®ect of the tuning parameter