Computing minimal interpolants in $C^{1,1}(\mathbb{R}^d)$

TL;DR

This paper develops practical algorithms for constructing minimal Lipschitz gradient interpolants in $C^{1,1}(R^d)$ based on finite data, optimizing the smoothness measure with efficiency.

Contribution

It introduces efficient algorithms to compute or approximate minimal Lipschitz gradient interpolants in $C^{1,1}(R^d)$ from finite data sets.

Findings

Algorithms achieve near-minimal Lipschitz constant for gradients.

Methods are computationally efficient for practical use.

Provides a framework for optimal smooth interpolants in $C^{1,1}(R^d)$.

Abstract

We consider the following interpolation problem. Suppose one is given a finite set $E \subset \mathbb{R}^d$, a function $f: E \rightarrow \mathbb{R}$, and possibly the gradients of $f$ at the points of $E$. We want to interpolate the given information with a function $F \in C^{1,1}(\mathbb{R}^d)$ with the minimum possible value of $\mathrm{Lip} (\nabla F)$. We present practical, efficient algorithms for constructing an $F$ such that $\mathrm{Lip} (\nabla F)$ is minimal, or for less computational effort, within a small dimensionless constant of being minimal.