  We develop a provably efficient importance sampling scheme that estimates
exit probabilities of solutions to small-noise stochastic reaction-diffusion
equations from scaled neighborhoods of a stable equilibrium. The moderate
deviation scaling allows for a local approximation of the nonlinear dynamics by
their linearized version. In addition, we identify a finite-dimensional
subspace where exits take place with high probability. Using stochastic control
and variational methods we show that our scheme performs well both in the zero
noise limit and pre-asymptotically. Simulation studies for stochastically
perturbed bistable dynamics illustrate the theoretical results.
