This paper provides a detailed study of ordered line integral methods (OLIMs), “a new family of methods for computing the quasi-potential on a regular mesh.” In the systems behavior field, “the quasi-potential is a key function in ... large deviation theory.” The influence of small white noise in stochastic nongradient dynamical systems is also taken into account. It gives some possibilities to “escape from the neighborhood of an attractor.” As the authors show, the new methods are up to four times quicker and capable of producing much smaller computation errors with much faster convergence.

After the introduction in section 1, section 2 reviews OLIMs. Section 3 presents numerical test results, with two examples, based on performance comparisons of the OLIMs (four versions) and the original ordered upwind method (OUM)-based quasi-potential solver [1]. Section 4 discusses the different factors that influence errors, and section 5 provides a summary.

For those who are focused on 2D cases with isotropic diffusion and looking for numerical solutions that balance accuracy with central processing unit (CPU) time, this is a very valuable and comprehensive paper.