  We introduce a family of numerical algorithms for the solution of linear
system in higher dimensions with the matrix and right hand side given and the
solution sought in the tensor train format. The proposed methods are
rank--adaptive and follow the alternating directions framework, but in contrast
to ALS methods, in each iteration a tensor subspace is enlarged by a set of
vectors chosen similarly to the steepest descent algorithm. The convergence is
analyzed in the presence of approximation errors and the geometrical
convergence rate is estimated and related to the one of the steepest descent.
The complexity of the presented algorithms is linear in the mode size and
dimension and the convergence demonstrated in the numerical experiments is
comparable to the one of the DMRG--type algorithm.
