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.

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

“…We start with a brief overview the OLIMs. A comprehensive description of the implementation of the OLIM in 3D is provided in [38]. It involves many technical details that are important for making the solver fast.…”

“…At each step of the main body of the OLIM, a Considered mesh point x new with the smallest tentative value becomes Accepted Front. Then the hierarchical update procedure proposed in [11] and further developed in [38] is implemented. It consists of two substeps.…”

“…This algorithm is summarized in the pseudocode below. The details of each step are elaborated in [38]. Now we outline the hierarchical update strategy.…”

“…Now we outline the hierarchical update strategy. All details of it are worked out in [38]. There are three types of updates done in the following order: one-point updates → triangle updates → simplex updates.…”

“…Second, while the OUM is practical only for 2D problems due to large CPU times in larger dimensions, the OLIMs have been successfully extended for 3D. This became possible due to the hierarchical update strategy [11,38], the use of the Karush-Kuhn-Tucker optimality conditions to eliminate unnecessary updates, and a number of implementational rationalizations.…”