Computing the quasipotential for highly dissipative and chaotic SDEs an application to stochastic Lorenz’63

Abstract

The study of noise-driven transitions occurring rarely on the time-scale of systems modeled by SDEs is of crucial importance for understanding such phenomena as genetic switches in living organisms and magnetization switches of the Earth. For a gradient SDE, the predictions for transition times and paths between its metastable states are done using the potential function. For a nongradient SDE, one needs to decompose its forcing into a gradient of the so-called quasipotential and a rotational component, which cannot be done analytically in general.We propose a methodology for computing the quasipotential for highly dissipative and chaotic systems built on the example of Lorenz'63 with an added stochastic term. It is based on the ordered line integral method, a Dijkstra-like quasipotential solver, and combines 3D computations in whole regions, a dimensional reduction technique, and 2D computations on radial meshes on manifolds or their unions. Our collection of source codes is available on M. Cameron's web page and on GitHub.

An efficient jet marcher for computing the quasipotential for 2D SDEs

“…All current mesh-based quasipotential solvers [5,8,9,33,6] inherit the MAP search algorithm from Sethian's and Vladimirsky's ordered upwind method (OUM) for solving static Hamilton-Jacobi equations with bounded anisotropic speed functions [29,30]. Roughly speaking, this involves an exhaustive search over rather large neighborhoods, with significant rationalizations in later solvers called ordered line integral methods (OLIMs) [8,9,33,6].…”

“…All current mesh-based quasipotential solvers [5,8,9,33,6] inherit the MAP search algorithm from Sethian's and Vladimirsky's ordered upwind method (OUM) for solving static Hamilton-Jacobi equations with bounded anisotropic speed functions [29,30]. Roughly speaking, this involves an exhaustive search over rather large neighborhoods, with significant rationalizations in later solvers called ordered line integral methods (OLIMs) [8,9,33,6]. An important aspect of the problem of computing the quasipotential is that its anisotropic speed function is unbounded, and the optimal choice of the radius for the search neighborhood remains an important issue; only rules of thumb based on detailed studies of particular systems have been proposed so far [8,9,33].…”

“…Furthermore, even if the classical solution to (11) with the boundary condition U (O) = 0 exists, it is not unique. There are at least two solutions, the quasipotential defined by (6), and U ≡ 0. In general, one can check that there are as many solutions as there are invariant subspaces for the linearized at O drift field.…”

“…There are several key algorithmic difficulties that arise in EJM but are not present in the previous first-order quasipotential solvers [5,8,9,33,6]. We describe these in detail in this section, in order to justify the decisions made in Algorithm 5 and the fail-safe method, discussed in the next section.…”

Parametric resonance for enhancing the rate of metastable transition

“…We also note overdamped Langevin (without time-dependence) is a reversible Markov process, while kinetic Langevin considered here is irreversible, and its rare event quantification, even without the time-dependent perturbation, can be much more challenging; see e.g.,[56,25,60,54,35,21,22,18,11,10,50,47,59,7,34,33]). …”