  We introduce and compare computational techniques for sharp extreme event
probability estimates in stochastic differential equations with small additive
Gaussian noise. In particular, we focus on strategies that are scalable, i.e.
their efficiency does not degrade upon temporal and possibly spatial
refinement. For that purpose, we extend algorithms based on the Laplace method
for estimating the probability of an extreme event to infinite dimensional path
space. The method estimates the limiting exponential scaling using a single
realization of the random variable, the large deviation minimizer. Finding this
minimizer amounts to solving an optimization problem governed by a differential
equation. The probability estimate becomes sharp when it additionally includes
prefactor information, which necessitates computing the determinant of a second
derivative operator to evaluate a Gaussian integral around the minimizer. We
present an approach in infinite dimensions based on Fredholm determinants, and
develop numerical algorithms to compute these determinants efficiently for the
high-dimensional systems that arise upon discretization. We also give an
interpretation of this approach using Gaussian process covariances and
transition tubes. An example model problem, for which we provide an open-source
python implementation, is used throughout the paper to illustrate all methods
discussed. To study the performance of the methods, we consider examples of
stochastic differential and stochastic partial differential equations,
including the randomly forced incompressible three-dimensional Navier-Stokes
equations.
