  We introduce a Tucker tensor cross approximation method that constructs a
low-rank representation of a $d$-dimensional tensor by sparsely sampling its
fibers. These fibers are selected using the discrete empirical interpolation
method (DEIM). Our proposed algorithm is referred to as DEIM fiber sampling
(DEIM-FS). For a rank-$r$ approximation of an $\mathcal{O}(N^d)$ tensor,
DEIM-FS requires access to only $dNr^{d-1}$ tensor entries, a requirement that
scales linearly with the tensor size along each mode. We demonstrate that
DEIM-FS achieves an approximation accuracy close to the Tucker-tensor
approximation obtained via higher-order singular value decomposition at a
significantly reduced cost. We also present DEIM-FS (iterative) that does not
require access to singular vectors of the target tensor unfolding and can be
viewed as a black-box Tucker tensor algorithm. We employ DEIM-FS to reduce the
computational cost associated with solving nonlinear tensor differential
equations (TDEs) using dynamical low-rank approximation (DLRA). The
computational cost of solving DLRA equations can become prohibitive when the
exact rank of the right-hand side tensor is large. This issue arises in many
TDEs, especially in cases involving non-polynomial nonlinearities, where the
right-hand side tensor has full rank. This necessitates the storage and
computation of tensors of size $\mathcal{O}(N^d)$. We show that DEIM-FS results
in significant computational savings for DLRA by constructing a low-rank Tucker
approximation of the right-hand side tensor on the fly. Another advantage of
using DEIM-FS is to significantly simplify the implementation of DLRA
equations, irrespective of the type of TDEs. We demonstrate the efficiency of
the algorithm through several examples including solving high-dimensional
partial differential equations.
