Stochastic dynamical systems developed on Riemannian manifolds

Abstract

We propose a method for developing the flows of stochastic dynamical systems, posed as Ito's stochastic differential equations, on a Riemannian manifold identified through a suitably constructed metric. The framework used for the stochastic development, viz. an orthonormal frame bundle that relates a vector on the tangent space of the manifold to its counterpart in the Euclidean space of the same dimension, is the same as that used for developing a standard Brownian motion on the manifold. Mainly drawing upon some aspects of the energetics so as to constrain the flow according to any known or prescribed conditions, we show how to expediently arrive at a suitable metric, thus briefly demonstrating the application of the method to a broad range of problems of general scientific interest. These include simulations of Brownian dynamics trapped in a potential well, a numerical integration scheme that reproduces the linear increase in the mean energy of conservative dynamical systems under additive noise and non-convex optimization. The simplicity of the method and the sharp contrast in its performance vis-á-vis the correspondent Euclidean schemes in our numerical work provide a compelling evidence to its potential, especially in the context of numerical schemes for systems with the ready availability of an energy functional, e.g. those in nonlinear elasticity.

Geometrically adapted Langevin dynamics for Markov chain Monte Carlo simulations

“…Next, consider the simplified MMALA. In this approach, the connection term (last term in (36)) is dropped, apparently for the sake of simplification. Moreover, it is claimed that the invariant distribution remains unchanged despite dropping this term on account of the acceptance probability, which is clearly not true for two simple reasons.…”

“…(a) q is an isomorphism between R d and T x M -the tangent space at x on M (b) The horizontal motion of q t on M : The tangent vectors (q 0 E 1 , q 0 E 2 ) at T x0 M are parallelly transported according to the curvature of M The stochastic counterpart of ( 13) is arrived at by interpreting it in the Stratonovich sense and then determining its Ito representation as in [36]. In the present work, we adopt a slightly different route, in that we start with the equation for Brownian motion on a Riemannian manifold and find how an additional drift applied to the Euclidean SDE manifests itself on the Riemannian manifold.…”

“…Here, every point x in M is furnished with an orthonormal frame Q that serves as an isomorphism between the Euclidean space R d and T x M , the d-dimensional tangent space to M at the point x. The procedure that we adopt largely follows the article by [27] and may be considered both an alternative and extension of the procedure explored in [36] to reach the same result. To start with consider the following SDE in…”

“…This last version has exactly been arrived at in a follow-up article by [55] by a slightly different approach. It is done with the objective of correcting the proposal as per (36) so that it converges to the target distribution. In order to achieve this, a couple of adjustments are made to (36) so that it becomes equal to the multiplicatively driven Langevin system [32].…”

“…Stochastic development is the framework that is used for the derivation of the equation for Brownian motion on a manifold, leading to the celebrated Laplace-Beltrami operator. We extend this approach for a general SDE, which is also applied to other interesting problems in [36]. A brief review of the relevant literature is provided in section 2 followed by detailed derivation.…”