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

TL;DR

The paper introduces the efficient jet marching (EJM) algorithm, a second-order accurate method for computing the quasipotential and its gradient in 2D SDEs, enabling precise estimates of invariant measures and escape times.

Contribution

The EJM algorithm improves upon previous methods by achieving second-order accuracy and reducing computation time through targeted search neighborhoods.

Findings

EJM achieves second-order accuracy for quasipotential calculations.

EJM reduces computation time with targeted search neighborhoods.

EJM enables precise estimation of escape times and invariant measures.

Abstract

We present a new algorithm, the efficient jet marching method (EJM), for computing the quasipotential and its gradient for two-dimensional SDEs. The quasipotential is a potential-like function for nongradient SDEs that gives asymptotic estimates for the invariant probability measure, expected escape times from basins of attractors, and maximum likelihood escape paths. The quasipotential is a solution to an optimal control problem with an anisotropic cost function which can be solved for numerically via Dijkstra-like label-setting methods. Previous Dijkstra-like quasipotential solvers have displayed in general 1st order accuracy in the mesh spacing. However, by utilizing higher order interpolations of the quasipotential as well as more accurate approximations of the minimum action paths (MAPs), EJM achieves second-order accuracy for the quasipotential and nearly second-order for its gradient. Moreover, by using targeted search neighborhoods for the fastest characteristics following the ideas of Mirebeau, EJM also enjoys a reduction in computation time. This highly accurate solver enables us to compute the prefactor for the WKB approximation for the invariant probability measure and the Bouchet-Reygner sharp estimate for the expected escape time for the Maier-Stein SDE. Our codes are available on GitHub.