Second moment analysis for Robin boundary value problems on random domains

We consider the numerical solution of Robin boundary value problems on random domains. The proposed method computes the mean and the variance of the random solution with leading order in the amplitude of the random boundary perturbation relative to an unperturbed, nominal domain. The variance is computed as the trace of the solution’s two-point correlation which satisfies a deterministic boundary value problem on the tensor product of the nominal domain. We solve this moderate high-dimensional problem by either a low-rank approximation by means of the pivoted Cholesky decomposition or the combination technique. Both approaches are presented and compared by numerical experiments with respect to their efficiency.