Computing the quasipotential for nongradient SDEs in 3D

Abstract

Nongradient SDEs with small white noise often arise when modeling biological and ecological time-irreversible processes. If the governing SDE were gradient, the maximum likelihood transition paths, transition rates, expected exit times, and the invariant probability distribution would be given in terms of its potential function. The quasipotential plays a similar role for nongradient SDEs. Unfortunately, the quasipotential is the solution of a functional minimization problem that can be obtained analytically only in some special cases. We propose a Dijkstra-like solver for computing the quasipotential on regular rectangular meshes in 3D. This solver results from a promotion and an upgrade of the previously introduced ordered line integral method with the midpoint quadrature rule for 2D SDEs. The key innovations that have allowed us to keep the CPU times reasonable while maintaining good accuracy are $(i)$ a new hierarchical update strategy, $(ii)$ the use of Karush-Kuhn-Tucker theory for rejecting unnecessary simplex updates, and $(iii)$ pruning the number of admissible simplexes and a fast search for them. An extensive numerical study is conducted on a series of linear and nonlinear examples where the quasipotential is analytically available or can be found at transition states by other methods. In particular, the proposed solver is applied to Tao's examples where the transition states are hyperbolic periodic orbits, and to a genetic switch model by Lv et al. (2014). The C source code implementing the proposed algorithm is available at M. Cameron's web page.

11 figures, 3 tables

References in corpus (5)

• The large deviation approach to statistical mechanics
• Ordered Line Integral Methods for Computing the Quasi-potential
• An Ordered Line Integral Method for Computing the Quasi-potential in the case of Variable Anisotropic Diffusion
• Computation of minimum action paths of the stochastic nonlinear Schroedinger equation with dissipation
• Corner cases, singularities, and dynamic factoring

Cited by in corpus (7)

• Ritz method for transition paths and quasipotentials of rare diffusive events
• Computing the quasipotential for highly dissipative and chaotic SDEs. An application to stochastic Lorenz'63
• Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning
• A Data Driven Method for Computing Quasipotentials
• Numerical computation of effective thermal equilibrium in Stochastically Switching Langevin Systems
• Quasipotentials for coupled escape problems and the gate-height bifurcation
• Sparse identification of quasipotentials via a combined data-driven method