Leveraging Latent Mathematical Structure in Geometry Processing

Abstract:

Myriad problems in geometry processing are readily formulated through continuous mathematics, including variational energies, stochastic dynamics, geometric constraints, and partial differential equations. Often, the challenge in solving these formulations in practice is not the absence of a principled theoretical formulation, but that standard computational approaches are limited by design constraints or technical barriers, resulting in instability or poor scalability. Common numerical techniques resort to fragile nonlinear solvers, repeated large optimizations, or forgo analytical or physical properties of the continuous problem.

In this talk, I will present methods that tackle these limitations by recognizing and exploiting latent problem structure, including convexity, variational reformulations, and symmetry. After a prelude illustrating the value of this approach in two problems of interest in geometry---simulating elastodynamics and solving PDEs on surfaces---I will describe a new approach to resampling that leverages a time-symmetric variant of the Schrödinger bridge problem. In each of these works, we discuss how algorithmic design, when carefully guided by underlying mathematical structure, can yield numerical tools that offer practitioners greater control and flexibility.