  Partial differential equations with random inputs have become popular models
to characterize physical systems with uncertainty coming from, e.g., imprecise
measurement and intrinsic randomness. In this paper, we perform asymptotic rare
event analysis for such elliptic PDEs with random inputs. In particular, we
consider the asymptotic regime that the noise level converges to zero
suggesting that the system uncertainty is low, but does exists. We develop
sharp approximations of the probability of a large class of rare events.
