  Low-rank tensor approximation approaches have become an important tool in the
scientific computing community. The aim is to enable the simulation and
analysis of high-dimensional problems which cannot be solved using conventional
methods anymore due to the so-called curse of dimensionality. This requires
techniques to handle linear operators defined on extremely large state spaces
and to solve the resulting systems of linear equations or eigenvalue problems.
In this paper, we present a systematic tensor-train decomposition for
nearest-neighbor interaction systems which is applicable to a host of different
problems. With the aid of this decomposition, it is possible to reduce the
memory consumption as well as the computational costs significantly.
Furthermore, it can be shown that in some cases the rank of the tensor
decomposition does not depend on the network size. The format is thus feasible
even for high-dimensional systems. We will illustrate the results with several
guiding examples such as the Ising model, a system of coupled oscillators, and
a CO oxidation model.
