Multivariate Newton Interpolation

TL;DR

This paper extends Newton interpolation to multiple dimensions, introduces an efficient algorithm for polynomial interpolation, and demonstrates that unisolvent Newton-Chebyshev nodes effectively approximate Sobolev functions without Runge's phenomenon.

Contribution

It generalizes Newton interpolation to multivariate spaces, develops the PIP-SOLVER algorithm, and introduces unisolvent Newton-Chebyshev nodes for stable, high-dimensional polynomial approximation.

Findings

The PIP-SOLVER computes solutions in quadratic time relative to node count.

Unisolvent Newton-Chebyshev nodes avoid Runge's phenomenon in high dimensions.

Arbitrary Sobolev functions can be uniformly approximated using the proposed method.

Abstract

For $m,n \in \mathbb{N}$, $m\geq 1$ and a given function $f : \mathbb{R}^m\longrightarrow \mathbb{R}$, the polynomial interpolation problem (PIP) is to determine a unisolvent node set $P_{m,n} \subseteq \mathbb{R}^m$ of $N(m,n):=|P_{m,n}|=\binom{m+n}{n}$ points and the uniquely defined polynomial $Q_{m,n,f}\in \Pi_{m,n}$ in $m$ variables of degree $\mathrm{deg}(Q_{m,n,f})\leq n \in \mathbb{N}$ that fits $f$ on $P_{m,n}$, i.e., $Q_{m,n,f}(p) = f(p)$, $\forall\, p \in P_{m,n}$. For $m=1$ the solution to the PIP is well known. In higher dimensions, however, no closed framework was available. We here present a generalization of the classic Newton interpolation from one-dimensional to arbitrary-dimensional spaces. Further we formulate an algorithm, termed PIP-SOLVER, based on a multivariate divided difference scheme that computes the solution $Q_{m,n,f}$ in $\mathcal{O}\big(N(m,n)^2\big)$ time using $\mathcal{O}\big(mN(m,n)\big)$ memory. Further, we introduce unisolvent Newton-Chebyshev nodes and show that these nodes avoid Runge's phenomenon in the sense that arbitrary periodic Sobolev functions $f \in H^k(\Omega,\mathbb{R}) \subsetneq C^0(\Omega,\mathbb{R})$, $\Omega =[-1,1]^m$ of regularity $k >m/2$ can be uniformly approximated, i.e., $ \lim_{n\rightarrow \infty}||\,f -Q_{m,n,f} \,||_{C^0(\Omega)}= 0$. Numerical experiments demonstrate the computational performance and approximation accuracy of the PIP-SOLVER in practice. We expect the presented results to be relevant for many applications, including numerical solvers, quadrature, non-linear optimization, polynomial regression, adaptive sampling, Bayesian inference, and spectral analysis.