Classical multivariate Hermite coordinate interpolation on n-dimensional grids

TL;DR

This paper develops a novel, algebraically simpler Hermite interpolation method for n-dimensional grids, proving its uniqueness, continuity, and demonstrating high accuracy in numerical examples compared to existing spline methods.

Contribution

It introduces a new compact closed form for multivariate Hermite interpolation, proving its uniqueness and spline-like continuity on n-dimensional grids.

Findings

Unique interpolation polynomial proven for n-dimensional grids

Compact closed form simplifies computation across dimensions

Numerical examples show high accuracy and favorable comparison to spline methods

Abstract

In this work, we study the Hermite interpolation on $n$-dimensional non-equally spaced, rectilinear grids over a field $\Bbbk $ of characteristic zero, given the values of the function at each point of the grid and the partial derivatives up to a maximum degree. First, we prove the uniqueness of the interpolating polynomial, and we further obtain a compact closed form that uses a single summation, irrespective of the dimensionality, which is algebraically simpler than the only alternative closed form for the $n$-dimensional classical Hermite interpolation [1]. We provide the remainder of the interpolation in integral form; we derive the ideal of the interpolation and express the interpolation remainder using only polynomial divisions, in the case of interpolating a polynomial function. Moreover, we prove the continuity of Hermite polynomials defined on adjacent $n$-dimensional grids, thus establishing spline behavior. Finally, we perform illustrative numerical examples to showcase the applicability and high accuracy of the proposed interpolant, in the simple case of few points, as well as hundreds of points on 3D-grids using a spline-like interpolation, which compares favorably to state-of-the-art spline interpolation methods.