  The goal of this paper is to develop provably efficient importance sampling
Monte Carlo methods for the estimation of rare events within the class of
linear stochastic partial differential equations (SPDEs). We find that if a
spectral gap of appropriate size exists, then one can identify a lower
dimensional manifold where the rare event takes place. This allows one to build
importance sampling changes of measures that perform provably well even
pre-asymptotically (i.e. for small but non-zero size of the noise) without
degrading in performance due to infinite dimensionality or due to long
simulation time horizons. Simulation studies supplement and illustrate the
theoretical results.
