Error Analysis of TT-Format Tensor Algorithms

The tensor train (TT) decomposition is a representation technique for arbitrary tensors, which allows efficient storage and computations. For a d-dimensional tensor with d 65 2, that decomposition consists of two ordinary matrices and d 12 2 third-order tensors. In this paper we prove that the TT decomposition of an arbitrary tensor can be computed (or approximated, for data compression purposes) by means of a backward stable algorithm based on computations with Householder matrices. Moreover, multilinear forms with tensors represented in TT format can be computed efficiently with a small backward error

From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation

We propose a new method for the approximate solution of the Lyapunov equation with rank-$1$ right-hand side, which is based on extended rational Krylov subspace approximation with adaptively computed shifts. The shift selection is obtained from the connection between the Lyapunov equation, solution of systems of linear ODEs and alternating least squares method for low-rank approximation. The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure