EUROPT 2025
Abstract Submission

317. Second-order differential operators, stochastic differential equations and Brownian motions on embedded manifolds

Invited abstract in session TC-11: Advances in Manifold and Conic Optimization, stream Riemannian Manifold and Conic Optimization.

Tuesday, 14:00-16:00
Room: B100/5017

Authors (first author is the speaker)

1. DU NGUYEN
Independent
2. Stefan Sommer
Computer Science, University of Copenhagen

Abstract

The article provides a framework to simulate Riemannian Brownian and Riemannian Langevin equations on embedded manifolds in global coordinates using retraction, important in MCMC sampling on manifolds, and is relevant to simulated annealing on manifolds. We specify the conditions when a manifold M embedded in an inner product space E is an invariant manifold of a stochastic differential equation (SDE) on E, linking it with the notion of second-order differential operators on M. When M is given a Riemannian metric, we derive a simple formula for the Laplace-Beltrami operator in terms of the gradient and Hessian on E and construct the Riemannian Brownian motions on M as solutions of conservative Stratonovich and Ito SDEs on E. We derive explicitly the SDE for Brownian motions on several important manifolds in applications, including left-invariant matrix Lie groups using embedded coordinates, Stiefel, Grassmann and symmetric positive definite (SPD) manifolds. Numerically, we propose three simulation schemes to solve SDEs on manifolds. In addition to the stochastic projection method, to simulate Riemannian Brownian motions, we construct a second-order tangent retraction of the Levi-Civita connection using a given E-tubular retraction. We also propose the retractive Euler-Maruyama method to solve a SDE, taking into account the second-order term of a tangent retraction. We provide software to implement the methods in the paper, including Brownian motions of the manifolds discussed

Keywords

Status: accepted

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