  We propose a method for the accurate estimation of rare event or failure
probabilities for expensive-to-evaluate numerical models in high dimensions.
The proposed approach combines ideas from large deviation theory and adaptive
importance sampling. The importance sampler uses a cross-entropy method to find
an optimal Gaussian biasing distribution, and reuses all samples made
throughout the process for both, the target probability estimation and for
updating the biasing distributions. Large deviation theory is used to find a
good initial biasing distribution through the solution of an optimization
problem. Additionally, it is used to identify a low-dimensional subspace that
is most informative of the rare event probability. This subspace is used for
the cross-entropy method, which is known to lose efficiency in higher
dimensions. The proposed method does not require smoothing of indicator
functions nor does it involve numerical tuning parameters. We compare the
method with a state-of-the-art cross-entropy-based importance sampling scheme
using three examples: a high-dimensional failure probability estimation
benchmark, a problem governed by a diffusion equation, and a tsunami problem
governed by the time-dependent shallow water system in one spatial dimension.
