  In this paper we study rare events associated to solutions of elliptic
partial differential equations with spatially varying random coefficients. The
random coefficients follow the lognormal distribution, which is determined by a
Gaussian process. This model is employed to study the failure problem of
elastic materials in random media in which the failure is characterized by that
the strain field exceeds a high threshold. We propose an efficient importance
sampling scheme to compute small failure probabilities in the high threshold
limit. The change of measure in our scheme is parametrized by two density
functions. The efficiency of the importance sampling scheme is validated by
numerical examples.
