Interpolation of Lipschitz functions

• Published in: Journal of computational and applied mathematics Vol. 196; no. 1; pp. 20 - 44
• Main Author: Beliakov, Gleb
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.11.2006; Elsevier
• Subjects: Approximations and expansions; Central algorithm; Exact sciences and technology; Lipschitz approximation; Mathematical analysis; Mathematics; Multivariate approximation; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Optimal interpolation; Scattered data interpolation; Sciences and techniques of general use; Optimal interpolation; Multivariate approximation; 41A30; 41A63; 65D05; 41A50; Lipschitz approximation; 41A05; Scattered data interpolation; Central algorithm; 65D05; 41A05; 41A30; 41A50; 41A63; Finite volume method; Numerical approximation; Radial basis function; Multivariate interpolation; Numerical analysis; Scattered data interpolation; Lipschitz approximation; Optimal interpolation; Central algorithm; Multivariate approximation; Efficiency; Lipschitz function
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2005.08.011

This paper describes a new computational approach to multivariate scattered data interpolation. It is assumed that the data is generated by a Lipschitz continuous function f. The proposed approach uses the central interpolation scheme, which produces an optimal interpolant in the worst case scenario. It provides best uniform error bounds on f, and thus translates into reliable learning of f. This paper develops a computationally efficient algorithm for evaluating the interpolant in the multivariate case. We compare the proposed method with the radial basis functions and natural neighbor interpolation, provide the details of the algorithm and illustrate it on numerical experiments. The efficiency of this method surpasses alternative interpolation methods for scattered data.