Small random perturbations may have a dramatic impact on the long timeevolution of dynamical systems, and large deviation theory is often the righttheoretical framework to understand these effects. At the core of the theorylies the minimization of an action functional, which in many cases of interesthas to be computed by numerical means. Here we review the theoretical andcomputational aspects behind these calculations, and propose an algorithm thatsimplifies the geometric minimum action method to minimize the action in thespace of arc-length parametrized curves. We then illustrate this algorithm'scapabilities by applying it to various examples from material sciences, fluiddynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models,these examples involve stochastic (ordinary or partial) differential equationswith multiplicative or degenerate noise, Markov jump processes, and systemswith fast and slow degrees of freedom, which all violate detailed balance, sothat simpler computational methods are not applicable.
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