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.
Categories: math.NA cs.DS cs.NA math.CA