Seminarium Computational and Applied Mathematics

Abstrakt finns enbart på engelska: In recent years there has been a surge of interest in using stochastic differential equations on Riemannian manifolds for a wide range of purposes. For example in: molecular dynamics, computer vision, sampling and machine learning. As such, developing accurate algorithms has become critical to estimate solutions of manifold valued SDEs in both the weak and strong sense. Moreover, analysis of strong convergence gives us a deeper understanding of the dynamics and evolution of SDEs, which has yet to be studied in great detail on manifolds.

The error rate for the Euler(-Murayama) method 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 non-positive curvature. In particular, we show that the Euler scheme has global rate 1/2, and 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. I will also present numerical experiments which illustrate the theoretical guarantees, as well as a counterintuitive example on 2-dimensional hyperbolic space.

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.