We develop an efficient algorithm to compute the Karhunen-Loeve expansion of stochastic fields with nonsmooth covariance kernels. Since the eigenvalues of the covariance operator do not decay exponentially, a large number of eigenpairs need to be approximated. We will employ fast methods for nonlocal operators as for example known from boundary integral equation methods. Then, each eigenpair can be computed in essentially linear complexity.

The Karhunen-Loeve expansion is applied to compute the approximate solutions of partial differential equations on stochastic domains. We considerboth, a domain transformation technique and a linearization based on the local shape derivative. Both approaches lead to a boundary value problemon a fixed nominal domain but with stochastic righthand side and/or stochastic coefficients. Stochastic Galerkin and collocation methods will be employed to solve the related boundary value problems.