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Nongradient SDEs with small white noise often arise when modeling biologicaland 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 itspotential function. The quasipotential plays a similar role for nongradientSDEs. Unfortunately, the quasipotential is the solution of a functionalminimization problem that can be obtained analytically only in some specialcases. We propose a Dijkstra-like solver for computing the quasipotential onregular rectangular meshes in 3D. This solver results from a promotion and anupgrade of the previously introduced ordered line integral method with themidpoint quadrature rule for 2D SDEs. The key innovations that have allowed usto keep the CPU times reasonable while maintaining good accuracy are $(i)$ anew hierarchical update strategy, $(ii)$ the use of Karush-Kuhn-Tucker theoryfor rejecting unnecessary simplex updates, and $(iii)$ pruning the number ofadmissible simplexes and a fast search for them. An extensive numerical studyis conducted on a series of linear and nonlinear examples where thequasipotential is analytically available or can be found at transition statesby other methods. In particular, the proposed solver is applied to Tao'sexamples where the transition states are hyperbolic periodic orbits, and to agenetic switch model by Lv et al. (2014). The C source code implementing theproposed algorithm is available at M. Cameron's web page.

Computing the quasipotential for nongradient SDEs in 3D