  Many physical and mathematical models involve random fields in their input
data. Examples are ordinary differential equations, partial differential
equations and integro--differential equations with uncertainties in the
coefficient functions described by random fields. They also play a dominant
role in problems in machine learning. In this article, we do not assume to have
knowledge of the moments or expansion terms of the random fields but we instead
have only given discretized samples for them. We thus model some measurement
process for this discrete information and then approximate the covariance
operator of the original random field. Of course, the true covariance operator
is of infinite rank and hence we can not assume to get an accurate
approximation from a finite number of spatially discretized observations. On
the other hand, smoothness of the true (unknown) covariance function results in
effective low rank approximations to the true covariance operator. We derive
explicit error estimates that involve the finite rank approximation error of
the covariance operator, the Monte-Carlo-type errors for sampling in the
stochastic domain and the numerical discretization error in the physical
domain. This permits to give sufficient conditions on the three discretization
parameters to guarantee that an error below a prescribed accuracy $\varepsilon$
is achieved.
