  Gaussian random fields over infinite-dimensional Hilbert spaces require the
definition of appropriate covariance operators. The use of elliptic PDE
operators to construct covariance operators allows to build on fast PDE solvers
for manipulations with the resulting covariance and precision operators.
However, PDE operators require a choice of boundary conditions, and this choice
can have a strong and usually undesired influence on the Gaussian random field.
We propose two techniques that allow to ameliorate these boundary effects for
large-scale problems. The first approach combines the elliptic PDE operator
with a Robin boundary condition, where a varying Robin coefficient is computed
from an optimization problem. The second approach normalizes the pointwise
variance by rescaling the covariance operator. These approaches can be used
individually or can be combined. We study properties of these approaches, and
discuss their computational complexity. The performance of our approaches is
studied for random fields defined over simple and complex two- and
three-dimensional domains.
