We present our recent progress in the investigation of tensor product approximations by the Hierarchical Tucker representation as introduced by Hackbusch, where our
main focus lies in representations in terms of the special case of TT tensors (Oseledets / Tyrtishnikov).
We consider numerical methods for solving optimization problems within a prescribed format, focusing on $l_2$-approximation, linear equations and eigenvalue problems.
We present the alternating linear scheme for the TT format, which is a generalized adaptation of an alternating least square (ALS) approach for the TT format, and a modification (MALS), which corresponds to the density matrix renormalization group (DMRG) algorithm when applied to N-body fermionic systems and give a convergence analysis for the ALS scheme. Further, other iteration schemes for local optimisation are devised based on the structure of the manifold of rank-$r$ TT tensors.
We then derive the equations needed for the treatment of dynamical problems. Following an approach proposed by Lubich for the Tucker format, we derive estimates for the curvature of the TT-rank-$r$ manifold which play a crucial role for convergence estimates in dynamical low rank approximation, and which also may be extended to the HT format.