  Many techniques for data science and uncertainty quantification demand
efficient tools to handle Gaussian random fields, which are defined in terms of
their mean functions and covariance operators. Recently, parameterized Gaussian
random fields have gained increased attention, due to their higher degree of
flexibility. However, especially if the random field is parameterized through
its covariance operator, classical random field discretization techniques fail
or become inefficient. In this work we introduce and analyze a new and
certified algorithm for the low-rank approximation of a parameterized family of
covariance operators which represents an extension of the adaptive cross
approximation method for symmetric positive definite matrices. The algorithm
relies on an affine linear expansion of the covariance operator with respect to
the parameters, which needs to be computed in a preprocessing step using, e.g.,
the empirical interpolation method. We discuss and test our new approach for
isotropic covariance kernels, such as Mat\'ern kernels. The numerical results
demonstrate the advantages of our approach in terms of computational time and
confirm that the proposed algorithm provides the basis of a fast sampling
procedure for parameter dependent Gaussian random fields.
