LocalPoly interpolation: Generalizing tricubic for Cn continuity in M‐dimensional spaces

Abstract

AbstractTricubic interpolation, originally introduced by Lekien and Marsden (Int J Numer Methods Eng. 2005; 63(3): 455–471), has been a cornerstone in the field of interpolation, providing continuous interpolations within three‐dimensional spaces. However, real‐world applications often demand higher levels of smoothness within ‐dimensional spaces. This paper introduces LocalPoly interpolation, a novel generalization of tricubic interpolation that extends to continuity and dimensions. A key property is the use of solely local data for interpolation, allowing for on‐demand computation of interpolation polynomials, which is particularly advantageous in scenarios where a minor subset of the space is of interest. We rigorously prove the continuity achieved by the LocalPoly interpolation method; the proof features a numerically exact method for computing polynomial coefficients. The enhanced continuity is of great relevance in optimization algorithms, where efficient convergence often relies on the availability of information. The paper explores the use of LocalPoly interpolation applied to a squared distance field in , offering insights into computational efficiency and practical implications. It also discusses future research directions to address the method's limitations in terms of dimensionality, making it a valuable addition to the toolbox of interpolation methods for various scientific and engineering applications.