Construction of quasi-potentials for stochastic dynamical systems: An optimization approach

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Title: Construction of quasi-potentials for stochastic dynamical systems: An optimization approach
Authors: Brackston, R
Wynn, A
Stumpf, MPH
Item Type: Journal Article
Abstract: The construction of effective and informative landscapes for stochastic dynamical systems has proven a long-standing and complex problem. In many situations, the dynamics may be described by a Langevin equation while constructing a landscape comes down to obtaining the quasipotential, a scalar function that quantifies the likelihood of reaching each point in the state space. In this work we provide a novel method for constructing such landscapes by extending a tool from control theory: the sum-of-squares method for generating Lyapunov functions. Applicable to any system described by polynomials, this method provides an analytical polynomial expression for the potential landscape, in which the coefficients of the polynomial are obtained via a convex optimization problem. The resulting landscapes are based on a decomposition of the deterministic dynamics of the original system, formed in terms of the gradient of the potential and a remaining “curl” component. By satisfying the condition that the inner product of the gradient of the potential and the remaining dynamics is everywhere negative, our derived landscapes provide both upper and lower bounds on the true quasipotential; these bounds becoming tight if the decomposition is orthogonal. The method is demonstrated to correctly compute the quasipotential for high-dimensional linear systems and also for a number of nonlinear examples.
Issue Date: 29-Aug-2018
Date of Acceptance: 9-Aug-2018
ISSN: 1539-3755
Publisher: American Physical Society
Journal / Book Title: Physical Review E
Volume: 98
Copyright Statement: © 2018 The Author(s). Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Sponsor/Funder: Biotechnology and Biological Sciences Research Council (BBSRC)
Funder's Grant Number: BB/N003608/1
Keywords: Science & Technology
Physical Sciences
Physics, Fluids & Plasmas
Physics, Mathematical
Article Number: 022136
Appears in Collections:Aeronautics
Faculty of Natural Sciences

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