CS21: Stochastic Numerics on Manifolds

Organizer: Alexander Lewis (Georg-August-Universität Göttingen)

Fundamental theorem for mean square convergence of SDEs on Riemannian manifolds

Alexander Lewis

The error rate for the Euler(-Murayama) approximation of SDEs on Riemannian manifolds in the weak sense has been established in \([1]\) and was found to be of global rate 1; reflecting the classical result known Euclidean space. However, strong convergence rates of the Euler scheme have yet to be derived. Though based on intuition, it is not unreasonable to expect that the manifold scheme has the same global rate as its Euclidean counterpart of 1/2.

By following closely to the approach laid out in the seminal works of Milstein, we show how to generate high order strong schemes on a Riemannian manifold with empty cut-locus. In particular, we show that the Euler scheme has global rate 1/2. Furthermore, we present the Milstein correction to the Euler scheme which yields a scheme of global order 1. Finally, we will formulate the manifold generalisation of the fundamental theorem of strong convergence, allowing us to obtain global convergence rates for a wide range of numerical schemes. If time permits, I will also present numerical experiments which illustrate the theoretical guarantees.

The talk will give an overview of results obtained in joint work with Karthik Bharath and Michael Tretyakov.

\([1]\) Bharath, K., Lewis, A., Sharma, A. and Tretyakov, M.V., 2023. Sampling and estimation on manifolds using the Langevin diffusion. arXiv preprint arXiv:2312.14882.

Kinetic Langevin equations on Lie groups with a geometric mechanics approach

Erwin Luesink

Kinetic Langevin equations are used for sampling from distributions and can employed in spaces with nontrivial curvature. On Lie groups, these equations can be solved with efficient numerical algorithms. In this talk, we discuss how the kinetic Langevin equations can be viewed within the framework of stochastic geometric mechanics. The benefit of doing so is that one obtains a physical interpretation as well as a multiplicative noise version of the kinetic Langevin equations.