  In this article, we consider elliptic diffusion problems with an anisotropic
random diffusion coefficient. We model the notable direction in terms of a
random vector field and derive regularity results for the solution's dependence
on the random parameter. It turns out that the decay of the vector field's
Karhunen-Loeve expansion entirely determines this regularity. The obtained
results allow for sophisticated quadrature methods, such as the quasi-Monte
Carlo method or the anisotropic sparse grid quadrature, in order to approximate
quantities of interest, like the solution's mean or the variance. Numerical
examples in three spatial dimensions are provided to supplement the presented
theory.
