  The tensor-train (TT) format is a data-sparse tensor representation commonly
used in high dimensional function approximations arising from computational and
data sciences. Various sequential and parallel TT decomposition algorithms have
been proposed for different tensor inputs and assumptions. In this paper, we
propose subtensor parallel adaptive TT cross, which partitions a tensor onto
distributed memory machines with multidimensional process grids, and constructs
an TT approximation iteratively with tensor elements. We derive two iterative
formulations for pivot selection and TT core construction under the distributed
memory setting, conduct communication and scaling analysis of the algorithm,
and illustrate its performance with multiple test experiments. These include up
to 6D Hilbert tensors and tensors constructed from Maxwellian distribution
functions that arise in kinetic theory. Our results demonstrate significant
accuracy with greatly reduced storage requirements via the TT cross
approximation. Furthermore, we demonstrate good to optimal strong and weak
scaling performance for the proposed parallel algorithm.
