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We introduce a family of numerical algorithms for the solution of linearsystem in higher dimensions with the matrix and right hand side given and thesolution sought in the tensor train format. The proposed methods arerank--adaptive and follow the alternating directions framework, but in contrastto ALS methods, in each iteration a tensor subspace is enlarged by a set ofvectors chosen similarly to the steepest descent algorithm. The convergence isanalyzed in the presence of approximation errors and the geometricalconvergence rate is estimated and related to the one of the steepest descent.The complexity of the presented algorithms is linear in the mode size anddimension and the convergence demonstrated in the numerical experiments iscomparable to the one of the DMRG--type algorithm.

Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems