  We study the problem of uncertainty quantification for the numerical solution
of elliptic partial differential equation boundary value problems posed on
domains with stochastically varying boundaries. We also use the uncertainty
quantification results to tackle the efficient solution of such problems. We
introduce simple transformations that map a family of domains with stochastic
boundaries to a fixed reference domain. We exploit the transformations to carry
out a prior and a posteriori error analyses and to derive an efficient Monte
Carlo sampling procedure.
