  In this paper we accomplish the development of the fast rank-adaptive solver
for tensor-structured symmetric positive definite linear systems in higher
dimensions. In [arXiv:1301.6068] this problem is approached by alternating
minimization of the energy function, which we combine with steps of the basis
expansion in accordance with the steepest descent algorithm. In this paper we
combine the same steps in such a way that the resulted algorithm works with one
or two neighboring cores at a time. The recurrent interpretation of the
algorithm allows to prove the global convergence and to estimate the
convergence rate. We also propose several strategies, both rigorous and
heuristic, to compute new subspaces for the basis enrichment in a more
efficient way. We test the algorithm on a number of high-dimensional problems,
including the non-symmetrical Fokker-Planck and chemical master equations, for
which the efficiency of the method is not fully supported by the theory. In all
examples we observe a convincing fast convergence and high efficiency of the
proposed method.
