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

Abstract

We consider the following interpolation problem. Suppose one is given a finite set $E \subset \mathbb R^d$, a function $f \colon E \to \mathbb R$, 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 Lip$(\nabla F)$. We present practical, efficient algorithms for constructing an $F$ such that Lip$(\nabla F)$ is minimal, or for less computational effort, within a small dimensionless constant of being minimal.