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Stochastic dynamical systems developed on Riemannian manifolds

Mamajiwala, M and Roy, D (2022) Stochastic dynamical systems developed on Riemannian manifolds. In: Probabilistic Engineering Mechanics, 67 .

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Official URL: https://doi.org/10.1016/j.probengmech.2021.103179


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. © 2021 Elsevier Ltd

Item Type: Journal Article
Publication: Probabilistic Engineering Mechanics
Publisher: Elsevier Ltd
Additional Information: The copyright for this article belongs to Authors
Keywords: Additive noise; Brownian movement; Convex optimization; Dynamical systems; Geometry; Hamiltonians; Numerical methods; Stochastic systems; Vector spaces, Ito stochastic differential equations; Nonconvex optimization; Orthonormal frames; Riemannian manifold; Stochastic development; Stochastic differential equations; Stochastic dynamical system; Stochastic hamiltonian systems; Stochastics; Trapped brownian motion, Differential equations
Department/Centre: Division of Mechanical Sciences > Civil Engineering
Date Deposited: 03 Dec 2021 08:43
Last Modified: 03 Dec 2021 08:43
URI: http://eprints.iisc.ac.in/id/eprint/70601

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