BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Alexander Paul Lewis (Georg-August-University of Goettingen)
DTSTART:20250414T111500Z
DTEND:20250414T120000Z
DTSTAMP:20260417T003712Z
UID:cam/69
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/cam/69/">Fun
 damental theorem for mean square convergence of SDEs on Riemannian manifol
 ds</a>\nby Alexander Paul Lewis (Georg-August-University of Goettingen) as
  part of CAM seminar\n\nLecture held in MV:L14.\n\nAbstract\nIn recent yea
 rs there has been a surge of interest in using stochastic differential equ
 ations on Riemannian manifolds for a wide range of purposes. For example i
 n: molecular dynamics\, computer vision\, sampling and machine learning. A
 s such\, developing accurate algorithms has become critical to estimate so
 lutions of manifold valued SDEs in both the weak and strong sense. Moreove
 r\, analysis of strong convergence gives us a deeper understanding of the 
 dynamics and evolution of SDEs\, which has yet to be studied in great deta
 il on manifolds.\n\nThe error rate for the Euler(-Murayama) method on Riem
 annian manifolds in the weak sense has been established in [1] and was fou
 nd to be of global rate 1\; reflecting the classical result known Euclidea
 n space. However\, strong convergence rates of the Euler scheme have yet t
 o be derived. Though based on intuition\, it is not unreasonable to expect
  that the manifold scheme has the same global rate as its Euclidean counte
 rpart of 1/2. \n\nBy following closely to the approach laid out in the sem
 inal 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 Milste
 in correction to the Euler scheme which yields a scheme of global order 1.
  Finally\, we will formulate the manifold generalisation of the fundamenta
 l theorem of strong convergence\, allowing us to obtain global convergence
  rates for a wide range of numerical schemes. I will also present numerica
 l experiments which illustrate the theoretical guarantees\, as well as a c
 ounterintuitive example on 2-dimensional hyperbolic space. \n\nThe talk wi
 ll give an overview of results obtained in joint work with Karthik Bharath
  and Michael Tretyakov.\n\n\n[1] Bharath\, K.\, Lewis\, A.\, Sharma\, A. a
 nd Tretyakov\, M.V.\, 2023. Sampling and estimation on manifolds using the
  Langevin diffusion. arXiv preprint arXiv:2312.14882.\n
LOCATION:https://researchseminars.org/talk/cam/69/
END:VEVENT
END:VCALENDAR