Generating Discrete Morse Functions from Point Data

Abstract

If $K$ is a finite simplicial complex and $h$ is an injective map from the vertices of $K$ to $\R$, we show how to extend $h$ to a discrete Morse function in the sense of Forman in a reasonably efficient manner so that the resulting discrete Morse function mirrors the large-scale behavior of $h$. A concrete algorithm is given for the case where $K$ is a subcomplex of $\R^3$.