Description
Point clouds provide an expressive abstraction for tasks across computational mathematics. They can be used as collocation point sets in numerical analysis, but also as a data structure for machine learning and statistical inference. I provide an introduction to how differential operators can be consistently approximated on point clouds by solving polynomial systems. This includes a meshfree geometric-computing framework that leverages polynomial regression in a unisolvent Newton-Chebyshev basis to represent complex-shaped and dynamic surfaces. Together, these render particle methods a viable choice for problems involving non-parametric or dynamic geometries, as exemplified by our work on active matter models of biological tissue morphogenesis.
