  The quasi-potential is a key function in the Large Deviation Theory. It
characterizes the difficulty of the escape from the neighborhood of an
attractor of a stochastic non-gradient dynamical system due to the influence of
small white noise. It also gives an estimate of the invariant probability
distribution in the neighborhood of the attractor up { to} the exponential
order. We present a new family of methods for computing the quasi-potential on
a regular mesh named the Ordered Line Integral Methods (OLIMs). In comparison
with the first proposed quasi-potential finder based on the Ordered Upwind
Method (OUM) (Cameron, 2012), the new methods are 1.5 to 4 times faster, can
produce error two to three orders of magnitude smaller, and may exhibit faster
convergence. Similar to the OUM, OLIMs employ the dynamical programming
principle. Contrary to it, they (i) have an optimized strategy for the use of
computationally expensive { triangle} updates leading to a notable speed-up,
and (ii) directly solve local minimization problems using quadrature rules
instead of solving the corresponding Hamilton-Jacobi-type equation by the first
order finite difference upwind scheme. The OLIM with the right-hand quadrature
rule is equivalent to OUM. The use of higher order quadrature rules in local
minimization problems dramatically boosts up the accuracy of OLIMs. We offer a
detailed discussion on the origin of numerical errors in OLIMs and propose
rules-of-thumb for the choice of the important parameter, the update factor, in
the OUM and OLIMs. Our results are supported by extensive numerical tests on
two challenging 2D examples.
