Georgia Institute of Technology College of Sciences

In this talk, I will discuss some very recent results on non-expansive numerical integrators on Riemannian manifolds.

 

We shall focus on the mathematical results, but the work is motivated by neural network architectures applied to manifold-valued data, and also by some recent activities in the simulation of slender structures in mechanical engineering. In Arnold et al. (2024), we proved that when applied to non-expansive continuous models, the Geodesic Implicit Euler method is non-expansive for all stepsizes when the manifold has non-positive sectional curvature. Disappointing counter-examples showed that this cannot hold in general for positively curved spaces. In the last few weeks, we have considered the Geodesic Explicit Euler method applied to non-expansive systems on manifolds of constant sectional curvature. In this case, we have proved upper bounds for the stepsize for which the Euler scheme is non-expansive.