  In this work, we describe advanced numerical tools for working with
multivariate functions and for the analysis of large data sets. These tools
will drastically reduce the required computing time and the storage cost, and,
therefore, will allow us to consider much larger data sets or finer meshes.
Covariance matrices are crucial in spatio-temporal statistical tasks, but are
often very expensive to compute and store, especially in 3D. Therefore, we
approximate covariance functions by cheap surrogates in a low-rank tensor
format. We apply the Tucker and canonical tensor decompositions to a family of
Matern- and Slater-type functions with varying parameters and demonstrate
numerically that their approximations exhibit exponentially fast convergence.
We prove the exponential convergence of the Tucker and canonical approximations
in tensor rank parameters. Several statistical operations are performed in this
low-rank tensor format, including evaluating the conditional covariance matrix,
spatially averaged estimation variance, computing a quadratic form,
determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a
large covariance matrix. Low-rank tensor approximations reduce the computing
and storage costs essentially. For example, the storage cost is reduced from an
exponential $\mathcal{O}(n^d)$ to a linear scaling $\mathcal{O}(drn)$, where
$d$ is the spatial dimension, $n$ is the number of mesh points in one
direction, and $r$ is the tensor rank. Prerequisites for applicability of the
proposed techniques are the assumptions that the data, locations, and
measurements lie on a tensor (axes-parallel) grid and that the covariance
function depends on a distance, $\Vert x-y \Vert$.
