Laplace operators on discrete surface meshes


Séminaire Géométrie-Images: MGMI

17/06/2011 - 10:00 Mr Max Wardetzky (Göttingen) Salle 1 - Tour IRMA

As is well-known from classical Riemannian geometry, Laplace operators offer a pathway for studying smooth Riemannian manifolds. However, when turning to computations on the computer, the world around us defies to be smooth. Nonetheless, in the setting of discrete "computable" manifolds, such as polyhedral surfaces, Laplace operators are ubiquitous in applications -- spanning geometric modeling, numerical analysis, and physical simulation. Applications often require discrete operators that retain key structural properties inherent to the smooth case -- such as symmetry, locality of definition, positive semi-definitness, and a maximum principle.

Building on the smooth setting, I will present a set of natural properties for Laplace operators on discrete surface meshes. I will point out an important theoretical limitation: discrete Laplacians cannot satisfy all of these properties on general unstructured triangle meshes; retroactively, this explains the diversity of existing discrete Laplacians found in the literature. Furthermore, building on insights that date back to James Clerk Maxwell, I will provide a characterization of those triangle meshes that do allow for "perfect" Laplacians. Finally, I will present a principled construction that extends discrete Laplacians from triangle meshes to arbitrary polygonal surface meshes.