| Mesh generation is a tool for discretizing functions by discretizing space. Traditionally, meshes are used in scientific computing for finite element analysis. Algorithmic ideas from mesh generation can also be applied to data analysis.;Data sets often have an intrinsic geometric and topological structure. The goal of many problems in geometric inference is to expose this intrinsic structure. One important structure of a point cloud is its geometric persistent homology, a multi-scale description of the topological features of the data with respect to distances in the ambient space.;In this thesis, I bring tools from mesh generation to bear on geometric persistent homology by using a mesh to approximate distance functions induced by a point cloud. Meshes provide an efficient way to compute geometric persistent homology. I present the first time-optimal algorithm for computing quality meshes in any dimension. Then, I show how these meshes can be used to provide a substantial speedup over existing methods for computing the full geometric persistence information for range of distance functions. |
