  Small random perturbations may have a dramatic impact on the long time
evolution of dynamical systems, and large deviation theory is often the right
theoretical framework to understand these effects. At the core of the theory
lies the minimization of an action functional, which in many cases of interest
has to be computed by numerical means. Here we review the theoretical and
computational aspects behind these calculations, and propose an algorithm that
simplifies the geometric minimum action method to minimize the action in the
space of arc-length parametrized curves. We then illustrate this algorithm's
capabilities by applying it to various examples from material sciences, fluid
dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models,
these examples involve stochastic (ordinary or partial) differential equations
with multiplicative or degenerate noise, Markov jump processes, and systems
with fast and slow degrees of freedom, which all violate detailed balance, so
that simpler computational methods are not applicable.
