Mesh-free methods for boundary value problems (BVPs) can be convenient on manifolds where generating a mesh may be difficult or when the manifold is not known explicitly but is determined by data.  Moreover, BVPs are important in machine learning since they provide a rigorous method of regularization for many regression problems.  In this talk we introduce the tools required to solve basic BVPs using only points sampled on an unknown manifold embedded in Euclidean space.  The key advance, discovered by Ryan Vaughn, is that the Diffusion Maps algorithm from machine learning is a consistent estimator of the Dirichlet energy (or weak-sense Laplacian) for manifolds with boundary.  Explaining this surprising and challenging result will form the core of the presentation and provides many insights valuable to kernel-based machine learning methods.  This result is also the key to developing the other components necessary for solving BVPs, including an estimator for the distance-to-boundary function and a boundary integral estimator.  Finally, these tools are combined to solve the weak formulation of standard BVPs on manifolds.

Zoom meeting: Link