This talk is largely concerned with tensor trains. This recent format allows nice and simple data-sparse representation for both the high- and low-dimensional data, based on the separation of indices and tensor product representation. The algebraic techniques and comprehensive error analysis behind the tensor train computations make it possible to develop a full set of linear algebra subroutines to perform the computations with huge arrays given in the compact data-sparse format. The most impressive results of the tensor trains is the possibility to overcome the curse of dimensionality for the solution of high-dimensional problems and to build up the effective classical models of the quantum algorithms, which were considered as impossible task several years ago. However, a lot of effort is still need to be invested to establish and describe the class of problems for which the use of tensor trains is effective and to develop the rigorous convergence analysis for the algorithms based on tensor train representations.
