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.

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Markov Chain Monte Carlo (MCMC) is one of the most powerful methods to sample from a given probability distribution, of which the Metropolis Adjusted Langevin Algorithm (MALA) is a variant wherein the gradient of the distribution is used towards faster convergence. However, being set up in the Euclidean framework, MALA might perform poorly in higher dimensional problems or in those involving anisotropic densities as the underlying non-Euclidean aspects of the geometry of the sample space remain unaccounted for. We make use of concepts from differential geometry and stochastic calculus on Riemannian manifolds to geometrically adapt a stochastic differential equation with a non-trivial drift term. This adaptation is also referred to as a stochastic development. We apply this method specifically to the Langevin diffusion equation and arrive at a geometrically adapted Langevin dynamics. This new approach far outperforms MALA, certain manifold variants of MALA, and other approaches such as Hamiltonian Monte Carlo (HMC), its adaptive variant the no-U-turn sampler (NUTS) implemented in Stan, especially as the dimension of the problem increases where often GALA is actually the only successful method. This is evidenced through several numerical examples that include parameter estimation of a broad class of probability distributions and a logistic regression problem.

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.